The theory of dynamical systems plays an important role in the mathematical description of timedependent processes in various fields, such as physics and technology, biology or economics. It includes the study of systems of ordinary differential equations, partial differential equations, delaydifferential equations and iterated mappings.
The research in this field at WIAS is mainly focussed to develop the mathematical background for problems from the dynamics of semiconductor lasers and nonlinear optics. This leads to the following topics:
 theory of singularly perturbed systems and asymptotic methods
 bifurcation theory and numerical bifurcation analysis by pathfollowing methods
 dynamics of delaydifferential equations
 collective dynamics in large coupled systems
 selforganization and control of spatiotemporal patterns
An important topic in the field of delaydifferential equations related to models of lasers with optical feedback is the limit of large delay time. Significant mathematical problems are asymptotic descriptions of the spectra for equilibria and periodic solutions.
A further major focus are highdimensional effects and complex dynamics in large coupled systems. Here in particular collective dynamics and pattern formation in systems of coupled oscillators are of interest.
Publications
Monographs

M. Tlidi, R. Lefever, A.G. Vladimirov, On Vegetation Clustering, Localized Bare Soil Spots and Fairy Circles, in: Dissipative Solitons: From Optics to Biology and Medicine, N. Akhmediev, A. Ankiewicz, eds., 751 of Lecture Notes in Physics, Springer, Berlin, Heidelberg, 2008, pp. 381402, (Chapter Published).
Articles in Refereed Journals

I. Franović, S. Yanchuk, S. Eydam, I. Bačić, M. Wolfrum, Dynamics of a stochastic excitable system with slowly adapting feedback, Chaos. An Interdisciplinary Journal of Nonlinear Science, 30 (2020), pp. 083109/1083109/11, DOI 10.1063/1.5145176 .
Abstract
We study an excitable active rotator with slowly adapting nonlinear feedback and noise. Depending on the adaptation and the noise level, this system may display noiseinduced spiking, noiseperturbed oscillations, or stochastic busting. We show how the system exhibits transitions between these dynamical regimes, as well as how one can enhance or suppress the coherence resonance, or effectively control the features of the stochastic bursting. The setup can be considered as a paradigmatic model for a neuron with a slow recovery variable or, more generally, as an excitable system under the influence of a nonlinear control mechanism. We employ a multiple timescale approach that combines the classical adiabatic elimination with averaging of rapid oscillations and stochastic averaging of noiseinduced fluctuations by a corresponding stationary FokkerPlanck equation. This allows us to perform a numerical bifurcation analysis of a reduced slow system and to determine the parameter regions associated with different types of dynamics. In particular, we demonstrate the existence of a region of bistability, where the noiseinduced switching between a stationary and an oscillatory regime gives rise to stochastic bursting. 
O. Melchert, C. Brée, A. Tajalli, A. Pape, R. Arkhipov, S. Willms, I. Babushkin, D. Skryabin, G. Steinmeyer, U. Morgner, A. Demircan, Alloptical supercontinuum switching, Nature Photonics, 3 (2020), pp. 146/1146/8, DOI 10.1038/s42005020004141 .
Abstract
Efficient alloptical switching is a challenging task as photons are bosons and cannot immediately interact with each other. Consequently, one has to resort to nonlinear optical interactions, with the Kerr gate being the classical example. However, the latter requires strong pulses to switch weaker ones. Numerous approaches have been investigated to overcome the resulting lack of fanout capability of alloptical switches, most of which relied on types of resonant enhancement of lightmatter interaction. Here we experimentally demonstrate a novel approach that utilizes switching between different portions of soliton fission induced supercontinua, exploiting an optical event horizon. This concept enables a high switching efficiency and contrast in a dissipation free setting. Our approach enables fanout, does not require critical biasing, and is at least partially cascadable. Controlling complex soliton dynamics paves the way towards building alloptical logic gates with advanced functionalities. 
M. Tlidi, E. BerríosCaro , D. PintoRamo, A.G. Vladimirov, M.G. Clerc, Interaction between vegetation patches and gaps: A selforganized response to water scarcity, Physica D. Nonlinear Phenomena, 414 (2020), pp. 132708/1132708/12, DOI 10.1016/j.physd.2020.132708 .
Abstract
The dynamics of ecological systems are often described by integrodifferential equations that incorporate nonlocal interactions associated with facilitative, competitive interactions between plants, and seed dispersion. In the weakgradient limit, these models can be reduced to a simple partialdifferential equation in the form of a nonvariational Swift?Hohenberg equation. In this contribution, we perform this reduction for any type of kernels provided that their Taylor series converge. Some parameters such as linear and nonlinear diffusion coefficients are affected by the spatial form of the kernel. In particular, Gaussian and exponential kernels are used to evaluate all coefficients of the reduced model. This weak gradient approximation is greatly useful for the investigation of periodic and localized vegetation patches, and gaps. Based on this simple model, we investigate the interaction between twowell separated patches and gaps. In the case of patches, the interaction is always repulsive. As a consequence, bounded states of patches are excluded. However, when two gaps are close to one another, they start to interact through their oscillatory tails. The interaction alternates between attractive and repulsive depending on the distance separating them. This allows for the stabilization of bounded gaps and clusters of them. The analytical formula of the interaction potential is derived for both patches and gaps interactions and checked by numerical investigation of the model equation. 
A. Boni, H.J. Wünsche, H. Wenzel, P. Crump, Impact of the capture time on the series resistance of quantumwell diode lasers, Semiconductor Science and Technology, 35 (2020), pp. 085032/1085032/9, DOI 10.1088/13616641/ab9723 .
Abstract
Electrons and holes injected into a semiconductor heterostructure containing quantum wellsare captured with a finite time. We show theoretically that this very fact can cause a considerableexcess contribution to the series resistivity and this is one of the main limiting factors to higherefficiency for GaAs based highpower lasers. The theory combines a standard microscopicbasedmodel for the captureescape processes in the quantum well with a driftdiffusion description ofcurrent flow outside the quantum well. Simulations of five GaAsbased devices differing in theirAlcontent reveal the rootcause of the unexpected and until now unexplained increase of theseries resistance with decreasing heat sink temperature measured recently. The finite capturetime results in resistances in excess of the bulk layer resistances (decreasing with increasingtemperature) from 1 mΩ up to 30 mΩ in good agreement with experiment. 
M. Kantner, Th. Koprucki, Beyond just ``flattening the curve'': Optimal control of epidemics with purely nonpharmaceutical interventions, Journal of Mathematics in Industry, 10 (2020), pp. 23/123/23, DOI 10.1186/s13362020000913 .
Abstract
When effective medical treatment and vaccination are not available, nonpharmaceutical interventions such as social distancing, home quarantine and farreaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptibleexposedinfectiousrecovered) model and continuoustime optimal control theory, we compute the optimal nonpharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of diseaserelated deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socioeconomic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a singleintervention scenario that goes beyond heuristically motivated interventions and simple ?flattening of the curve?. Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socioeconomic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID19 pandemic in Germany. 
S. Eydam, I. Franović, M. Wolfrum, Leapfrog patterns in systems of two coupled FitzHughNagumo units, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 99 (2019), pp. 042207/1042207/9, DOI 10.1103/PhysRevE.99.042207 .
Abstract
We study a system of two identical FitzHughNagumo units with a mutual linear coupling in the fast variables. While an attractive coupling always leads to synchronous behavior, a repulsive coupling can give rise to dynamical regimes with alternating spiking order, called leapfrogging. We analyze various types of periodic and chaotic leapfrogging regimes, using numerical pathfollowing methods to investigate their emergence and stability, as well as to obtain the complex bifurcation scenario which organizes their appearance in parameter space. In particular, we show that the stability region of the simplest periodic leapfrog pattern has the shape of a locking cone pointing to the canard transition of the uncoupled system. We also discuss the role of the timescale separation in the coupled FitzHughNagumo system and the relation of the leapfrog solutions to the theory of mixedmode oscillations in multiple timescale systems. 
S. Eydam, M. Wolfrum, The link between coherence echoes and mode locking, Chaos. An Interdisciplinary Journal of Nonlinear Science, 29 (2019), pp. 103114/1103114/10, DOI 10.1063/1.5114699 .
Abstract
We investigate the appearance of sharp pulses in the mean field of Kuramototype globally coupled phase oscillator systems. In systems with exactly equidistant natural frequencies self organized periodic pulsations of the mean field, called mode locking, have been described re cently as a new collective dynamics below the synchronization threshold. We show here that mode locking can appear also for frequency combs with modes of finite width, where the natu ral frequencies are randomly chosen from equidistant frequency intervals. In contrast to that, so called coherence echoes, which manifest themselves also as pulses in the mean field, have been found in systems with completely disordered natural frequencies as the result of two consecutive stimulations applied to the system. We show that such echo pulses can be explained by a stimula tion induced mode locking of a subpopulation representing a frequency comb. Moreover, we find that the presence of a second harmonic in the interaction function, which can lead to the global stability of the modelocking regime for equidistant natural frequencies, can enhance the echo phenomenon significantly. The nonmonotonous behavior of echo amplitudes can be explained as a result of the linear dispersion within the selforganized modelocked frequency comb. Fi nally we investigate the effect of small periodic stimulations on oscillator systems with disordered natural frequencies and show how the global coupling can support the stimulated pulsation by increasing the width of locking plateaus. 
A. Stephan, H. Stephan, Memory equations as reduced Markov processes, Discrete and Continuous Dynamical Systems, 39 (2019), pp. 21332155, DOI 10.3934/dcds.2019089 .
Abstract
A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we give an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as the change of the type of some quasiparticles along oneway loops. Typically, the arising Markov process does not have the detailed balance property. The method leads to a more realisitc modeling of memory equations. Moreover, it carries over the large number of investigation tools for Markov processes to memory equations, like the calculation of the equilibrium state, the asymptotic behavior and so on. The method can be used for an approximative solution of some degenerate memory equations like delay differential equations. 
P. Kravetc, D. Rachinskii, A.G. Vladimirov, Periodic pulsating dynamics of slowfast delayed systems with a periodic close to the delay, European Journal of Applied Mathematics, 30 (2019), pp. 3962, DOI 10.1017/S0956792517000377 .
Abstract
We consider slow?fast delayed systems and discuss pulsating periodic solutions, which are characterised by specific properties that (a) the period of the periodic solution is close to the delay, and (b) these solutions are formed close to a bifurcation threshold. Such solutions were previously found in models of modelocked lasers. Through a case study of population models, this work demonstrates the existence of similar solutions for a rather wide class of delayed systems. The periodic dynamics originates from the Hopf bifurcation on the positive equilibrium. We show that the continuous transformation of the periodic orbit to the pulsating regime is simultaneous with multiple secondary almost resonant Hopf bifurcations, which the equilibrium undergoes over a short interval of parameter values. We derive asymptotic approximations for the pulsating periodic solution and consider scaling of the solution and its period with the small parameter that measures the ratio of the time scales. The role of competition for the realisation of the bifurcation scenario is highlighted. 
K.R. Schneider, The point charge oscillator: Qualitative and analytical investigations, Mathematical Modelling and Analysis. Matematinis Modeliavimis ir Analize. The Baltic Journal on Mathematical Applications, Numerical Analysis and Differential Equations, 24 (2019), pp. 372384, DOI 10.3846/mma.2019.023 .
Abstract
We determine the global phase portrait of a mathematical model describing the point charge oscillator. It shows that the family of closed orbits describing the point charge oscillations has two envelopes: an equilibrium point and a homoclinic orbit to an equilibrium point at infinity. We derive an expression for the growth rate of the primitive perod Τ_{α} of the oscillation with the amplitude α as α tends to infinity. Finally, we determine an exact relation between period and amplitude by means of the Jacobi elliptic function cn. 
S. Yanchuk, S. Ruschel, J. Sieber, M. Wolfrum, Temporal dissipative solitons in timedelay feedback systems, Physical Review Letters, 123 (2019), pp. 053901/1053901/6, DOI 10.1103/PhysRevLett.123.053901 .
Abstract
Localized states are a universal phenomenon observed in spatially distributed dissipative nonlinear systems. Known as dissipative solitons, autosolitons, spot or pulse solitons, these states play an important role in data transmission using optical pulses, neural signal propagation, and other processes. While this phenomenon was thoroughly studied in spatially extended systems, temporally localized states are gaining attention only recently, driven primarily by applications from fiber or semiconductor lasers. Here we present a theory for temporal dissipative solitons (TDS) in systems with timedelayed feedback. In particular, we derive a system with an advanced argument, which determines the profile of the TDS. We also provide a complete classification of the spectrum of TDS into interface and pseudocontinuous spectrum. We illustrate our theory with two examples: a generic delayed phase oscillator, which is a reduced model for an injected laser with feedback, and the FitzHughNagumo neuron with delayed feedback. Finally, we discuss possible destabilization mechanisms of TDS and show an example where the TDS delocalizes and its pseudocontinuous spectrum develops a modulational instability. 
B. Jahnel, Ch. Külske, Attractor properties for irreversible and reversible interacting particle systems, Communications in Mathematical Physics, 366 (2019), pp. 139172, DOI 10.1007/s00220019033524 .
Abstract
We consider translationinvariant interacting particle systems on the lattice with finite local state space admitting at least one Gibbs measure as a timestationary measure. The dynamics can be irreversible but should satisfy some mild nondegeneracy conditions. We prove that weak limit points of any trajectory of translationinvariant measures, satisfying a nonnullness condition, are Gibbs states for the same specification as the timestationary measure. This is done under the additional assumption that zero entropy loss of the limiting measure w.r.t. the timestationary measure implies that they are Gibbs measures for the same specification.We also give an alternate version of the last condition such that the nonnullness requirement can be dropped. For dynamics admitting a reversible Gibbs measure the alternative condition can be verified, which yields the attractor property for such dynamics. This generalizes convergence results using relative entropy techniques to a large class of dynamics including irreversible and nonergodic ones. We use this to show synchronization for the rotation dynamics exhibited in citeJaKu12 possibly at low temperature, and possibly nonreversible. We assume the additional regularity properties on the dynamics: 1 There is at least one stationary measure which is a Gibbs measure. 2 Zero loss of relative entropy density under dynamics implies the Gibbs property. 
B. Jahnel, Ch. Külske, Gibbsian representation for point processes via hyperedge potentials, Journal of Theoretical Probability, pp. published online on 03.11.2019, urlhttps://doi.org/10.1007/s10959019009607, DOI 10.1007/s10959019009607 .
Abstract
We consider marked point processes on the ddimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing uniformly absolutely convergent Hamiltonians in terms of hyperedge potentials in the sense of Georgii [2]. These potentials are natural generalizations of physical multibody potentials which are useful in models of stochastic geometry. 
A. Pimenov, J. Javaloyes, S.V. Gurevich, A.G. Vladimirov, Light bullets in a timedelay model of a wideaperture modelocked semiconductor laser, Philosophical Transactions of the Royal Society A : Mathematical, Physical & Engineering Sciences, 376 (2018), pp. 20170372/120170372/14, DOI 10.1098/rsta.2017.0372 .
Abstract
Recently, a mechanism of formation of light bullets (LBs) in wideaperture passively modelocked lasers was proposed. The conditions for existence and stability of these bullets, found in the long cavity limit, were studied theoretically under the mean field (MF) approximation using a Haustype model equation. In this paper we relax the MF approximation and study LB formation in a model of a wideaperture three section laser with a long diffractive section and short absorber and gain sections. To this end we derive a nonlocal delaydifferential equation (NDDE) model and demonstrate by means of numerical simulations that this model supports stable LBs. We observe that the predictions about the regions of existence and stability of the LBs made previously using MF laser models agree well with the results obtained using the NDDE model. Moreover, we demonstrate that the general conclusions based upon the Haus model that regard the robustness of the LBs remain true in the NDDE model valid beyond the MF approximation, when the gain, losses and diffraction per cavity roundtrip are not small perturbations anymore. 
D.R.M. Renger, P. Koltai , From large deviations to transport semidistances: Coherence analysis for finite Lagrangian data, Journal of NonNewtonian Fluid Mechanics, 28 (2018), pp. 19151957, DOI 10.1007/s0033201894710 .
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic timereversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradientflow, steepestascent, or maximalentropyproduction equation. Onsager's original theorem is limited to closetoequilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic timereversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows. 
I. Bačić, S. Yanchuk, M. Wolfrum, I. Franović, Noiseinduced switching in two adaptively coupled excitable systems, European Physical Journal Special Topics, 227 (2018), pp. 10771090, DOI 10.1140/epjst/e20188000846 .
Abstract
We demonstrate that the interplay of noise and plasticity gives rise to slow stochastic fluctuations in a system of two adaptively coupled active rotators with excitable local dynamics. Depending on the adaptation rate, two qualitatively different types of switching behavior are observed. For slower adaptation, one finds alternation between two modes of noiseinduced oscillations, whereby the modes are distinguished by the different order of spiking between the units. In case of faster adaptation, the system switches between the metastable states derived from coexisting attractors of the corresponding deterministic system, whereby the phases exhibit a burstinglike behavior. The qualitative features of the switching dynamics are analyzed within the framework of fastslow analysis. 
I. Omelchenko, O.E. Omel'chenko, A. Zakharova, E. Schöll, Optimal design of the tweezer control for chimera states, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 97 (2018), pp. 012216/1012216/9, DOI 10.1103/PhysRevE.97.012216 .
Abstract
Chimera states are complex spatiotemporal patterns, which consist of coexisting domains of spatially coherent and incoherent dynamics in systems of coupled oscillators. In small networks, chimera states usually exhibit short lifetimes and erratic drifting of the spatial position of the incoherent domain. A tweezer feedback control scheme can stabilize and fix the position of chimera states. We analyse the action of the tweezer control in small nonlocally coupled networks of Van der Pol and FitzHughNagumo oscillators, and determine the ranges of optimal control parameters. We demonstrate that the tweezer control scheme allows for stabilization of chimera states with different shapes, and can be used as an instrument for controlling the coherent domains size, as well as the maximum average frequency difference of the oscillators. 
O. Burylko, A. Mielke, M. Wolfrum, S. Yanchuk, Coexistence of Hamiltonianlike and dissipative dynamics in chains of coupled phase oscillators with skewsymmetric coupling, SIAM Journal on Applied Dynamical Systems, 17 (2018), pp. 20762105, DOI 10.1137/17M1155685 .
Abstract
We consider rings of coupled phase oscillators with anisotropic coupling. When the coupling is skewsymmetric, i. e. when the anisotropy is balanced in a specific way, the system shows robustly a coexistence of Hamiltonianlike and dissipative regions in the phase space. We relate this phenomenon to the timereversibility property of the system. The geometry of lowdimensional systems up to five oscillators is described in detail. In particular, we show that the boundary between the dissipative and Hamiltonianlike regions consists of families of heteroclinic connections. For larger chains with skewsymmetric coupling, some sufficient conditions for the coexistence are provided, and in the limit of N → ∞ oscillators, we formally derive an amplitude equation for solutions in the neighborhood of the synchronous solution. It has the form of a nonlinear Schrödinger equation and describes the Hamiltonianlike region existing around the synchronous state similarly to the case of finite rings. 
I. Franović, O.E. Omel'chenko, M. Wolfrum, Phasesensitive excitability of a limit cycle, Chaos. An Interdisciplinary Journal of Nonlinear Science, 28 (2018), pp. 071105/1071105/6, DOI 10.1063/1.5045179 .
Abstract
The classical notion of excitability refers to an equilibrium state that shows under the influence of perturbations a nonlinear thresholdlike behavior. Here, we extend this concept by demonstrating how periodic orbits can exhibit a specific form of excitable behavior where the nonlinear thresholdlike response appears only after perturbations applied within a certain part of the periodic orbit, i.e the excitability happens to be phase sensitive. As a paradigmatic example of this concept we employ the classical FitzHughNagumo system. The relaxation oscillations, appearing in the oscillatory regime of this system, turn out to exhibit a phase sensitive nonlinear thresholdlike response to perturbations, which can be explained by the nonlinear behavior in the vicinity of the canard trajectory. Triggering the phase sensitive excitability of the relaxation oscillations by noise we find a characteristic nonmonotone dependence of the mean spiking rate of the relaxation oscillation on the noise level. We explain this nonmonotone dependence as a result of an interplay of two competing effects of the increasing noise: the growing efficiency of the excitation and the degradation of the nonlinear response. 
P. Gurevich, S. Reichelt, Pulses in FitzHughNagumo systems with rapidly oscillating coefficients, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 16 (2018), pp. 833856.
Abstract
This paper is devoted to pulse solutions in FitzHughNagumo systems that are coupled parabolic equations with rapidly periodically oscillating coefficients. In the limit of vanishing periods, there arises a twoscale FitzHughNagumo system, which qualitatively and quantitatively captures the dynamics of the original system. We prove existence and stability of pulses in the limit system and show their proximity on any finite time interval to pulselike solutions of the original system. 
O.O. Omel'chenko, M. Wolfrum, E. Knobloch, Stability of spiral chimera states on a torus, SIAM Journal on Applied Dynamical Systems, 17 (2018), pp. 97127, DOI 10.1137/17M1141151 .
Abstract
We study destabilization mechanisms of spiral coherenceincoherence patterns known as spiral chimera states that form on a twodimensional lattice of nonlocally coupled phase oscillators. For this purpose we employ the linearization of the OttAntonsen equation that is valid in the continuum limit and perform a detailed twoparameter stability analysis of a $D_4$symmetric chimera state, i.e., a fourcore spiral state. We identify fold, Hopf and paritybreaking bifurcations as the main mechanisms whereby spiral chimeras can lose stability. Beyond these bifurcations we find new spatiotemporal patterns, in particular, quasiperiodic chimeras, $D_2$symmetric spiral chimeras as well as drifting states. 
U. Bandelow, A. Ankiewicz, S. Amiranashvili, N. Akhmediev, SasaSatsuma hierarchy of integrable evolution equations, Chaos. An Interdisciplinary Journal of Nonlinear Science, 28 (2018), pp. 053108/1053108/11, DOI 10.1063/1.5030604 .
Abstract
We present the infinite hierarchy of SasaSatsuma evolution equations. The corresponding Lax pairs are given, thus proving its integrability. The lowest order member of this hierarchy is the nonlinear Schrödinger equation, while the next one is the SasaSatsuma equation that includes thirdorder terms. Up to sixth order terms of the hierarchy are given in explicit form, while the provided recurrence relation allows one to explicitly write all higherorder terms. The whole hierarchy can be combined into a single general equation. Each term in this equation contains a real independent coefficient that provides the possibility of adapting the equation to practical needs. A few examples of exact solutions of this general equation with an infinite number of terms are also given explicitly. 
O.E. Omel'chenko, The mathematics behind chimera states, Nonlinearity, 31 (2018), pp. R121R164, DOI 10.1088/13616544/aaaa07 .
Abstract
Chimera states are selforganized spatiotemporal patterns of coexisting coherence and incoherence. We give an overview of the main mathematical methods used in studies of chimera states, focusing on chimera states in spatially extended coupled oscillator systems. We discuss the continuum limit approach to these states, OttAntonsen manifold reduction, finite size chimera states, control of chimera states and the influence of system design on the type of chimera state that is observed. 
A.G. Vladimirov, S.V. Gurevich, M. Tlidi, Effect of Cherenkov radiation on localizedstate interaction, Physical Review A, 97 (2018), pp. 013816/1013816/6, DOI 10.1103/PhysRevA.97.013816 .
Abstract
We study theoretically the interaction of temporal localized states in all fiber cavities and microresonatorbased optical frequency comb generators. We show that Cherenkov radiation emitted in the presence of third order dispersion breaks the symmetry of the localized structrures interaction and greatly enlarges their interaction range thus facilitating the experimental observation of the dissipative soliton bound states. Analytical derivation of the reduced equations governing slow time evolution of the positions of two interacting localized states in a generalized LugiatoLefever model with the third order dispersion term is performed. Numerical solutions of the model equation are in close agreement with analytical predictions. 
S. Eydam, M. Wolfrum, Mode locking in systems of globally coupled phase oscillators, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 96 (2017), pp. 052205/1052205/8, DOI 10.1103/PhysRevE.96.052205 .
Abstract
We investigate the dynamics of a Kuramototype system of globally coupled phase oscillators with equidistant natural frequencies and a coupling strength below the synchronization threshold. It turns out that in such cases one can observe a stable regime of sharp pulses in the mean field amplitude with a pulsation frequency given by spacing of the natural frequencies. This resembles a process known as modelocking in laser and relies on the emergence of a phase relation induced by the nonlinear coupling. We discuss the role of the first and second harmonic in the phaseinteraction function for the stability of the pulsations and present various bifurcating dynamical regimes such as periodically and chaotically modulated modelocking, transitions to phase turbulence and intermittency. Moreover, we study the role of the system size and show that in certain cases one can observe typeII supertransients, where the system reaches the globally stable modelocking solution only after an exponentially long transient of phase turbulence. 
A. Pimenov, S. Slepneva, G. Huyet, A.G. Vladimirov, Dispersive timedelay dynamical systems, Physical Review Letters, 118 (2017), pp. 193901/1193901/6.
Abstract
We present a theoretical approach to model the dynamics of a dispersive nonlinear system using a set of delay differential equations with distributed delay term. We illustrate the use of this approach by considering a frequency swept laser comprimising a semiconductor optical amplifier (SOA), a tunable bandpass filter and a long dispersive fiber delay line. We demonstrate that this system exhibits a rich spectrum of dynamical behaviors which are in agreement with the experimental observations. In particular, the multimode modulational instability observed experimentally in the laser in the anomalous dispersion regime and leading to a turbulent laser output was found analytically in the limit of large delay time. 
S. Olmi, D. AnguloGarcia, A. Imparato, A. Torcini, Exact firing time statistics of neurons driven by discrete inhibitory noise, Scientific Reports, 7 (2017), pp. 1577/11577/15, DOI 10.1038/s41598017016588 .
Abstract
Neurons in the intact brain receive a continuous and irregular synaptic bombardment from excitatory and inhibitory presynaptic neurons, which determines the firing activity of the stimulated neuron. In order to investigate the influence of inhibitory stimulation on the firing time statistics, we consider Leaky IntegrateandFire neurons subject to inhibitory instantaneous postsynaptic potentials. In particular, we report exact results for the firing rate, the coefficient of variation and the spike train spectrum for various synaptic weight distributions. Our results are not limited to stimulations of infinitesimal amplitude, but they apply as well to finite amplitude postsynaptic potentials, thus being able to capture the effect of rare and large spikes. The developed methods are able to reproduce also the average firing properties of heterogeneous neuronal populations. 
TH. Erneux, J. Javaloyes, M. Wolfrum, S. Yanchuk, Introduction to focus issue: Timedelay dynamics, Chaos. An Interdisciplinary Journal of Nonlinear Science, 27 (2017), pp. 114201/1114201/5, DOI 10.1063/1.5011354 .
Abstract
The field of dynamical systems with time delay is an active research area that connects practically all scientific disciplines including mathematics, physics, engineering, biology, neuroscience, physiology, economics, and many others. This Focus Issue brings together contributions from both experimental and theoretical groups and emphasizes a large variety of applications. In particular, lasers and optoelectronic oscillators subject to timedelayed feedbacks have been explored by several authors for their specific dynamical output, but also because they are ideal testbeds for experimental studies of delay induced phenomena. Topics include the control of cavity solitons, as light spots in spatially extended systems, new devices for chaos communication or random number generation, higher order locking phenomena between delay and laser oscillation period, and systematic bifurcation studies of modelocked laser systems. Moreover, two original theoretical approaches are explored for the socalled Low Frequency Fluctuations, a particular chaotical regime in laser output which has attracted a lot of interest for more than 30?years. Current hot problems such as the synchronization properties of networks of delaycoupled units, novel stabilization techniques, and the large delay limit of a delay differential equation are also addressed in this special issue. In addition, analytical and numerical tools for bifurcation problems with or without noise and two reviews on concrete questions are proposed. The first review deals with the rich dynamics of simple delay climate models for El Nino Southern Oscillations, and the second review concentrates on neuromorphic photonic circuits where optical elements are used to emulate spiking neurons. Finally, two interesting biological problems are considered in this Focus Issue, namely, multistrain epidemic models and the interaction of glucose and insulin for more effective treatment. 
V. Klinshov, D. Shchapin, S. Yanchuk, M. Wolfrum, O. D'huys, V. Nekorkin, Embedding the dynamics of a single delay system into a feedforward ring, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 96 (2017), pp. 042217/1042217/9.
Abstract
We investigate the relation between the dynamics of a single oscillator with delayed selffeedback and a feedforward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the selffeedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where stability of periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example we demonstrate how the complex bifurcation scenario of simultaneously emerging multijittering solutions can be transferred from a single oscillator with delayed pulse feedback to multijittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHughNagumo type. 
D. Puzyrev, A.G. Vladimirov, A. Pimenov, S.V. Gurevich, S. Yanchuk, Bound pulse trains in arrays of coupled spatially extended dynamical systems, Physical Review Letters, 119 (2017), pp. 163901/1163901/6, DOI 10.1103/PhysRevLett.119.163901 .
Abstract
We study the dynamics of an array of nearestneighbor coupled spatially distributed systems each generating a periodic sequence of short pulses. We demonstrate that, unlike a solitary system generating a train of equidistant pulses, an array of such systems can produce a sequence of clusters of closely packed pulses, with the distance between individual pulses depending on the coupling phase. This regime associated with the formation of locally coupled pulse trains bounded due to a balance of attraction and repulsion between them is different from the pulse bound states reported earlier in different laser, plasma, chemical, and biological systems. We propose a simplified analytical description of the observed phenomenon, which is in good agreement with the results of direct numerical simulations of a model system describing an array of coupled modelocked lasers. 
A.P. Willis, Y. Duguet, O. Omel'chenko, M. Wolfrum, Surfing the edge: Finding nonlinear solutions using feedback control, Journal of Fluid Mechanics, 831 (2017), pp. 579591.
Abstract
Many transitional wallbounded shear flows are characterised by the coexistence in statespace of laminar and turbulent regimes. Probing the edge boundarz between the two attractors has led in the last decade to the numerical discovery of new (unstable) solutions to the incompressible NavierStokes equations. However, the iterative bisection method used to achieve this can become prohibitively costly for large systems. Here we suggest a simple feedback control strategy to stabilise edge states, hence accelerating their numerical identification by several orders of magnitude. The method is illustrated for several configurations of cylindrical pipe flow. Traveling waves solutions are identified as edge states, and can be isolated rapidly in only one short numerical run. A new branch of solutions is also identified. When the edge state is a periodic orbit or chaotic state, the feedback control does not converge precisely to solutions of the uncontrolled system, but nevertheless brings the dynamics very close to the original edge manifold in a single run. We discuss the opportunities offered by the speed and simplicity of this new method to probe the structure of both state space and parameter space. 
M. Hintermüller, C.N. Rautenberg, M. Mohammadi, M. Kanitsar, Optimal sensor placement: A robust approach, SIAM Journal on Control and Optimization, 55 (2017), pp. 36093639.
Abstract
We address the problem of optimally placing sensor networks for convectiondiffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. The paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the wellposedness of the optimization problem and finalizes with a range of numerical tests. 
M. Hintermüller, C.N. Rautenberg, On the uniqueness and numerical approximation of solutions to certain parabolic quasivariational inequalities, Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 74 (2017), pp. 135.
Abstract
A class of abstract nonlinear evolution quasivariational inequality (QVI) problems in function space is considered. The abstract framework developed in this paper includes constraint sets of obstacle and gradient type. The paper address the existence, uniqueness and approximation of solutions when the constraint set mapping is of a special form. Uniqueness is addressed through contractive behavior of a nonlinear mapping whose fixed points are solutions to the QVI. An axiomatic semidiscrete approximation scheme is developed, which is proven to be convergent and which is numerically implemented. The paper ends by a report on numerical tests for several nonlinear constraints of gradienttype. 
O. Omel'chenko, L. Recke, V. Butuzov, N. Nefedov, Timeperiodic boundary layer solutions to singularly perturbed parabolic problems, Journal of Differential Equations, 262 (2017), pp. 48234862.
Abstract
In this paper, we present a result of implicit function theorem type, which was designed for application to singularly perturbed problems. This result is based on fixed point iterations for contractive mappings, in particular, no monotonicity or sign preservation properties are needed. Then we apply our abstract result to timeperiodic boundary layer solutions (which are allowed to be nonmonotone with respect to the space variable) in semilinear parabolic problems with two independent singular perturbation parameters. We prove existence and local uniqueness of those solutions, and estimate their distance to certain approximate solutions. 
A. Pimenov, D. Rachinskii, Robust homoclinic orbits in planar systems with Preisach hysteresis operator, Journal of Physics: Conference Series, 727 (2016), pp. 012012/1012012/15, DOI 10.1088/17426596/727/1/012012 .
Abstract
We construct examples of robust homoclinic orbits for systems of ordinary differential equations coupled with the Preisach hysteresis operator. Existence of such orbits is demonstrated for the first time. We discuss a generic mechanism that creates robust homoclinic orbits and a method for finding them. An example of a homoclinic orbit in a population dynamics model with hysteretic response of the prey to variations of the predator is studied numerically 
I. Omelchenko, O. Omel'chenko, A. Zakharova, M. Wolfrum, E. Schöll, Tweezers for chimeras in small networks, Physical Review Letters, 116 (2016), pp. 114101/1114101/5.
Abstract
We propose a control scheme which can stabilize and fix the position of chimera states in small networks. Chimeras consist of coexisting domains of spatially coherent and incoherent dynamics in systems of nonlocally coupled identical oscillators. Chimera states are generically difficult to observe in small networks due to their short lifetime and erratic drifting of the spatial position of the incoherent domain. The control scheme, like a tweezer, might be useful in experiments, where usually only small networks can be realized. 
D. Davino, P. Krejčí, A. Pimenov, D. Rachinskii, C. Visone, Analysis of an operatordifferential model for magnetostrictive energy harvesting, Communications in Nonlinear Science and Numerical Simulation, 39 (2016), pp. 504519.
Abstract
We present a model of, and analysis of an optimization problem for, a magnetostrictive harvesting device which converts mechanical energy of the repetitive process such as vibrations of the smart material to electrical energy that is then supplied to an electric load. The model combines a lumped differential equation for a simple electronic circuit with an operator model for the complex constitutive law of the magnetostrictive material. The operator based on the formalism of the phenomenological Preisach model describes nonlinear saturation effects and hysteresis losses typical of magnetostrictive materials in a thermodynamically consistent fashion. We prove wellposedness of the full operatordifferential system and establish global asymptotic stability of the periodic regime under periodic mechanical forcing that represents mechanical vibrations due to varying environmental conditions. Then we show the existence of an optimal solution for the problem of maximization of the output power with respect to a set of controllable parameters (for the periodically forced system). Analytical results are illustrated with numerical examples of an optimal solution. 
K. Panajotov, D. Puzyrev, A.G. Vladimirov, S.V. Gurevich, M. Tlidi, Impact of timedelayed feedback on spatiotemporal dynamics in the LugiatoLefever model, Physical Review A, 93 (2016), pp. 043835/1043835/7.
Abstract
We analyze the impact of delayed optical feedback (OF) on the spatiotemporal dynamics of the LugiatoLefever model. First, we carry out linear stability analysis and reveal the role of the OF strength and phase on the shape of the bistable curve as well as on Turing, AndronovHopf, and travelingwave instability regions. Further, we demonstrate how the OF impacts the spatial dynamics by shifting the regions with different spatial eigenvalue spectra. In addition, we reveal a clustering behavior of cavity solitons as a function of the OF strength at fixed OF phase. Depending on the feedback parameters, OF can also induce a drift bifurcation of a stationary cavity soliton, as well as an AndronovHopf bifurcation of a drifting soliton. We present an analytical expression for the threshold of the drift bifurcation and show that above a certain value of the OF strength the system enters a region of spatiotemporal chaos. 
D. Puzyrev, A.G. Vladimirov, S.V. Gurevich, S. Yanchuk, Modulational instability and zigzagging of dissipative solitons induced by delayed feedback, Physical Review A, 93 (2016), pp. 041801/1041801/5.
Abstract
We report a destabilization mechanism of localized solutions in spatially extended systems which is induced by delayed feedback. Considering a model of a wideaperture laser with a saturable absorber and delayed optical feedback, we demonstrate the appearance of multiple coexistent laser cavity solitons. We show that at large delays apart from the drift and phase instabilities the soliton can exhibit a delayinduced modulational instability associated with the translational neutral mode. The combination of drift and modulational instabilities produces a zigzagging motion of the solitons, which are either periodic, with the period close to the delay time, or chaotic, with lowfrequency fluctuations in the direction of the soliton motion. The same type of modulational instability is demonstrated for localized solutions of the cubicquintic complex GinzburgLandau equation. 
K.R. Schneider, A. Grin, Study of the bifurcation of a multiple limit cycle of the second kind by means of a DulacCherkas function: A case study, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. World Scientific, Singapore., 26 (2016), 1650229, DOI 10.1142/S0218127416502291 .
Abstract
We consider a generalized pendulum equation depending on the scalar parameter $mu$ having for $mu=0$ a limit cycle $Gamma$ of the second kind and of multiplicity three. We study the bifurcation behavior of $Gamma$ for $1 le mu le (sqrt5+3)/2$ by means of a DulacCherkas function. 
O. Omel'chenko, M. Wolfrum, Is there an impact of small phase lags in the Kuramoto model?, Chaos. An Interdisciplinary Journal of Nonlinear Science, 26 (2016), pp. 094806/1094806/6.
Abstract
We discuss the influence of small phase lags on the synchronization transitions in the Kuramoto model for a large inhomogeneous population of globally coupled phase oscillators. Without a phase lag, all unimodal distributions of the natural frequencies give rise to a classical synchronization scenario, where above the onset of synchrony at the Kuramoto threshold there is an increasing synchrony for increasing coupling strength. We show that already for arbitrarily small phase lags there are certain unimodal distributions of natural frequencies such that for increasing coupling strength synchrony may decrease and even complete incoherence may regain stability. Moreover, our example allows a qualitative understanding of the mechanism for such nonuniversal synchronization transitions. 
M. Wolfrum, S. Gurevich, O. Omel'chenko, Turbulence in the OttAntonsen equation for arrays of coupled phase oscillators, Nonlinearity, 29 (2016), pp. 257270.
Abstract
In this paper we study the transition to synchrony in an onedimensional array of oscillators with nonlocal coupling. For its description in the continuum limit of a large number of phase oscillators, we use a corresponding OttAntonsen equation, which is an integrodifferential equation for the evolution of the macroscopic profiles of the local mean field. Recently, it has been reported that in the spatially extended case at the synchronization threshold there appear partially coherent plane waves with different wave numbers, which are organized in the wellknown Eckhaus scenario. In this paper, we show that for KuramotoSakaguchi phase oscillators the phase lag parameter in the interaction function can induce a BenjaminFeir type instability of the partially coherent plane waves. The emerging collective macroscopic chaos appears as an intermediate stage between complete incoherence and stable partially coherent plane waves. We give an analytic treatment of the BenjaminFeir instability and its onset in a codimensiontwo bifurcation in the OttAntonsen equation as well as a numerical study of the transition from phase turbulence to amplitude turbulence inside the BenjaminFeir unstable region. 
M. Kantner, E. Schöll, S. Yanchuk, Delayinduced patterns in a twodimensional lattice of coupled oscillators, Scientific Reports, 5 (2015), pp. 8522/18522/9.
Abstract
We show how a variety of stable spatiotemporal periodic patterns can be created in 2Dlattices of coupled oscillators with nonhomogeneous coupling delays. The results are illustrated using the FitzHughNagumo coupled neurons as well as coupled limit cycle (StuartLandau) oscillators. A "hybrid dispersion relation" is introduced, which describes the stability of the patterns in spatially extended systems with large timedelay. 
L. Lücken, J.P. Pade, K. Knauer, Classification of coupled dynamical systems with multiple delays: Finding the minimal number of delays, SIAM Journal on Applied Dynamical Systems, 14 (2015), pp. 286304.
Abstract
In this article we study networks of coupled dynamical systems with timedelayed connections. If two such networks hold different delays on the connections it is in general possible that they exhibit different dynamical behavior as well. We prove that for particular sets of delays this is not the case. To this aim we introduce a componentwise timeshift transformation (CTT) which allows to classify systems which possess equivalent dynamics, though possibly different sets of connection delays. In particular, we show for a large class of semiflows (including the case of delay differential equations) that the stability of attractors is invariant under this transformation. Moreover we show that each equivalence class which is mediated by the CTT possesses a representative system in which the number of different delays is not larger than the cycle space dimension of the underlying graph. We conclude that the 'true' dimension of the corresponding parameter space of delays is in general smaller than it appears at first glance. 
E. Meca Álvarez, I. Mercader, L. RamirezPiscina, Transitions between symmetric and nonsymmetric regimes in binarymixture convection, Physica D. Nonlinear Phenomena, 303 (2015), pp. 3949.

A. Pimenov, T.C. Kelly, A. Korobeinikov, J.A. O'Callaghan, D. Rachinskii, Adaptive behaviour and multiple equilibrium states in a predatorprey model, Theoretical Population Biology, 101 (2015), pp. 2430.
Abstract
There is evidence that multiple stable equilibrium states are possible in reallife ecological systems. In order to verify a hypothesis that such a multitude of equilibrium states can be caused by adapting of animal behaviour to changes of environmental conditions, we consider a simple predatorprey model where prey changes a mode of behaviour in response to the pressure of predation. This model exhibits two stable coexisting equilibrium states with basins of attraction separated by a separatrix of a saddle point. 
S. Yanchuk, L. Lücken, M. Wolfrum, A. Mielke, Spectrum and amplitude equations for scalar delaydifferential equations with large delay, Discrete and Continuous Dynamical Systems, 35 (2015), pp. 537553.
Abstract
The subject of the paper are scalar delaydifferential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classification is similar to the one for parabolic partial differential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delaydifferential equations close to the destabilization threshold. 
S. Yanchuk, P. Perlikowski, M. Wolfrum, A. Stefański, T. Kapitaniak, Amplitude equations for collective spatiotemporal dynamics in arrays of coupled systems, Chaos. An Interdisciplinary Journal of Nonlinear Science, 25 (2015), pp. 033113/1033113/8.
Abstract
We study the coupling induced destabilization in an array of identical oscillators coupled in a ring structure where the number of oscillators in the ring is large. The coupling structure includes different types of interactions with several next neighbors. We derive an amplitude equation of GinzburgLandau type, which describes the destabilization of a uniform stationary state and closeby solutions in the limit of a large number of nodes. Studying numerically an example of unidirectionally coupled Duffing oscillators, we observe a coupling induced transition to collective spatiotemporal chaos, which can be understood using the derived amplitude equations. 
T. Jonsson, S. Berg, M. Emmerson, A. Pimenov, The context dependency of species keystone status during food web disassembly, Food Webs, 5 (2015), pp. 110.

V. Klinshov, L. Lücken, D. Shchapin, V. Nekorkin, S. Yanchuk, Multistable jittering in oscillators with pulsatile delayed feedback, Physical Review Letters, 114 (2015), pp. 178103/1178103/5.
Abstract
Oscillatory systems with timedelayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in recent years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. At the bifurcation point numerous regimes with nonequal interspike intervals emerge. We show that the number of the emerging, socalled “jittering” regimes grows emphexponentially with the delay value. Although this appears as highly degenerate from a dynamical systems viewpoint, the “multijitter” bifurcation occurs robustly in a large class of systems. We observe it not only in a paradigmatic phasereduced model, but also in a simulated HodgkinHuxley neuron model and in an experiment with an electronic circuit. 
V. Klinshov, L. Lücken, D. Shchapin, V. Nekorkin, S. Yanchuk, Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 92 (2015), pp. 042914/1042914/15.
Abstract
Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillator's phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous “jittering” regimes with nonequal interspike intervals (ISIs). The number of the emergent solutions increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how each periodic solution consisting of different ISIs implies the appearance of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback. 
V.Z. Tronciu, M. Radziunas, Ch. Kürbis, H. Wenzel, A. Wicht, Numerical and experimental investigations of microintegrated external cavity diode lasers, Optical and Quantum Electronics, 47 (2015), pp. 14591464.

O. Omel'chenko, L. Recke, Existence, local uniqueness and asymptotic approximation of spike solutions to singularly perturbed elliptic problems, Hiroshima Mathematical Journal, 45 (2015), pp. 3589.
Abstract
This paper concerns general singularly perturbed second order semilinear elliptic equations on bounded domains $Omega subset R^n$ with nonlinear natural boundary conditions. The equations are not necessarily of variational type. We describe an algorithm to construct sequences of approximate spike solutions, we prove existence and local uniqueness of exact spike solutions close to the approximate ones (using an Implicit Function Theorem type result), and we estimate the distance between the approximate and the exact solutions. Here ”spike solution” means that there exists a point in $Omega$ such that the solution has a spikelike shape in a vicinity of such point and that the solution is approximately zero away from this point. The spike shape is not radially symmetric in general and may change sign. 
M. Radziunas, V.Z. Tronciu, E. Luvsandamdin, Ch. Kürbis, A. Wicht, H. Wenzel, Study of microintegrated externalcavity diode lasers: Simulations, analysis and experiments, IEEE J. Quantum Electron., 51 (2015), pp. 2000408/12000408/8.
Abstract
This paper reports the results of numerical and experimental investigations of the dynamics of an external cavity diode laser device composed of a semiconductor laser and a distant Bragg grating, which provides an optical feedback. Due to the influence of the feedback, this system can operate at different dynamic regimes. The traveling wave model is used for simulations and analysis of the nonlinear dynamics in the considered laser device. Based on this model, a detailed analysis of the optical modes is performed, and the stability of the stationary states is discussed. It is shown, that the results obtained from the simulation and analysis of the device are in good agreement with experimental findings. 
M. Wolfrum, O. Omel'chenko, J. Sieber, Regular and irregular patterns of selflocalized excitation in arrays of coupled phase oscillators, Chaos. An Interdisciplinary Journal of Nonlinear Science, 25 (2015), pp. 053113/1053113/7.
Abstract
We study a system of phase oscillators with nonlocal coupling in a ring that supports selforganized patterns of coherence and incoherence, called chimera states. Introducing a global feedback loop, connecting the phase lag to the order parameter, we can observe chimera states also for systems with a small number of oscillators. Numerical simulations show a huge variety of regular and irregular patterns composed of localized phase slipping events of single oscillators. Using methods of classical finite dimensional chaos and bifurcation theory, we can identify the emergence of chaotic chimera states as a result of transitions to chaos via period doubling cascades, torus breakup, and intermittency. We can explain the observed phenomena by a mechanism of selfmodulated excitability in a discrete excitable medium. 
R.M. Arkhipov, I. Babushkin, M.K. Lebedev, Y.A. Tolmachev, M.V. Arkhipov, Transient Cherenkov radiation from an inhomogeneous string excited by an ultrashort laser pulse at superluminal velocity, Physical Review A, 89 (2014), pp. 043811/1043811/10.
Abstract
An optical response of onedimensional string made of dipoles with a periodically varying density excited by a spot of light moving along the string at the superluminal (subluminal) velocity is studied. We consider in details the spectral and temporal dynamics of the Cherenkov radiation, which occurs in such system in the transient regime. We point out the resonance character of radiation and the appearance of a new Dopplerlike frequency in the spectrum of the transient Cherenkov radiation. Possible applications of the effect as well as different string topologies are discussed 
A. Pimenov, T. Habruseva, D. Rachinskii, S.P. Hegarty, H. Guillaume, A.G. Vladimirov, Effect of dynamical instability on timing jitter in passively modelocked quantumdot lasers, Optics Letters, 39 (2014), pp. 68156818.
Abstract
We study the effect of noise on the dynamics of passively modelocked semiconductor lasers both experimentally and theoretically. A method combining analytical and numeri cal approaches for estimation of pulse timing jitter is proposed. We investigate how the presence of dynamical features such as wavelength bistability affects timing jitter. 
A. Pimenov, D. Rachinskii, Homoclinic orbits in a twopatch predatorprey model with Preisach hysteresis operator, Mathematica Bohemica, 139 (2014), pp. 285298.
Abstract
Systems of operatordifferential equations which hysteresis operators can have unstable equilibrium points with an open basin of attraction. In this paper, a numerical example of a robust homoclinic loop is presented for the first time in a population dynamics model with hysteretic response of prey to variations of predator. A mechanism creating this homoclinic trajectory is discussed. 
A. Pimenov, D. Rachinskii, Robust homoclinic orbits in planar systems with Preisach hysteresis operator, Journal of Physics: Conference Series, 727 (2016), pp. 012012/1012012/15, DOI 10.1088/17426596/727/1/012012 .
Abstract
We construct examples of robust homoclinic orbits for systems of ordinary differential equations coupled with the Preisach hysteresis operator. Existence of such orbits is demonstrated for the first time. We discuss a generic mechanism that creates robust homoclinic orbits and a method for finding them. An example of a homoclinic orbit in a population dynamics model with hysteretic response of the prey to variations of the predator is studied numerically 
M. Radszuweit, H. Engel, M. Bär, An active poroelastic model for mechanochemical patterns in protoplasmic droplets of Physarum polycephalum, PLOS ONE, 9 (2014), pp. e99220/1e99220/15.
Abstract
Motivated by recent experimental studies, we derive and analyze a twodimensional model for the contraction patterns observed in protoplasmic droplets of Physarum polycephalum. The model couples a description of an active poroelastic twophase medium with equations describing the spatiotemporal dynamics of the intracellular free calcium concentration. The poroelastic medium is assumed to consist of an active viscoelastic solid representing the cytoskeleton and a viscous fluid describing the cytosol. The equations for the poroelastic medium are obtained from continuum force balance and include the relevant mechanical fields and an incompressibility condition for the twophase medium. The reactiondiffusion equations for the calcium dynamics in the protoplasm of Physarum are extended by advective transport due to the flow of the cytosol generated by mechanical stress. Moreover, we assume that the active tension in the solid cytoskeleton is regulated by the calcium concentration in the fluid phase at the same location, which introduces a mechanochemical coupling.
A linear stability analysis of the homogeneous state without deformation and cytosolic flows exhibits an oscillatory Turing instability for a large enough mechanochemical coupling strength. Numerical simulations of the model equations reproduce a large variety of wave patterns, including traveling and standing waves, turbulent patterns, rotating spirals and antiphase oscillations in line with experimental observations of contraction patterns in the protoplasmic droplets. 
S. Slepneva, B. O'Shaughnessy, S.P. Hegarty, A.G. Vladimirov, H.C. Lyu, K. Karnowski, M. Wojtkowski, G. Huyet, Dynamics of a short cavity swept source OCT laser, Optics Express, 22 (2014), pp. 1817718185.

J. Sieber, O. Omel'chenko, M. Wolfrum, Controlling unstable chaos: Stabilizing chimera states by feedback, Physical Review Letters, 112 (2014), pp. 054102/1054102/5.
Abstract
We present a control scheme that is able to find and stabilize a chaotic saddle in a system with a large number of interacting particles. This allows us to track a high dimensional chaotic attractor through a bifurcation where it loses its attractivity. Similar to a classical delayed feedback control, the scheme is noninvasive, however, only in an appropriately relaxed sense considering the chaotic regime as a statistical equilibrium displaying random fluctuations as a finite size effects. We demonstrate the control scheme for so called chimera states, which are coherenceincoherence patterns in coupled oscillator systems. The control makes chimera states observable close to coherence, for small numbers of oscillators, and for random initial conditions. 
O. Omel'chenko, M. Wolfrum, C. Laing, Partially coherent twisted states in arrays of coupled phase oscillators, Chaos. An Interdisciplinary Journal of Nonlinear Science, 24 (2014), pp. 023102/1023102/9.
Abstract
We consider a onedimensional array of phase oscillators with nonlocal coupling and a Lorentzian distribution of natural frequencies. The primary objects of interest are partially coherent states that are uniformly "twisted" in space. To analyze these we take the continuum limit, perform an Ott/Antonsen reduction, integrate over the natural frequencies and study the resulting spatiotemporal system on an unbounded domain. We show that these twisted states and their stability can be calculated explicitly. We find that stable twisted states with different wave numbers appear for increasing coupling strength in the wellknown Eckhaus scenario. Simulations of finite arrays of oscillators show good agreement with results of the analysis of the infinite system. 
R.M. Arkhipov, A. Pimenov, M. Radziunas, A.G. Vladimirov, D. Arsenjević, D. Rachinskii, H. Schmeckebier, D. Bimberg, Hybrid modelocking in edgeemitting semiconductor lasers: Simulations, analysis and experiments, IEEE J. Select. Topics Quantum Electron., 19 (2013), pp. 1100208/11100208/6.
Abstract
Hybrid modelocking in a two section edgeemitting semiconductor laser is studied numerically and analytically using a set of three delay differential equations. In this set the external RF signal applied to the saturable absorber section is modeled by modulation of the carrier relaxation rate in this section. Estimation of the locking range where the pulse repetition frequency is synchronized with the frequency of the external modulation is performed numerically and the effect of the modulation shape and amplitude on this range is investigated. Asymptotic analysis of the dependence of the locking range width on the laser parameters is carried out in the limit of small signal modulation. Our numerical simulations indicate that hybrid modelocking can be also achieved in the cases when the frequency of the external modulation is approximately twice larger and twice smaller than the pulse repetition frequency of the free running passively modelocked laser fP . Finally, we provide an experimental demonstration of hybrid modelocking in a 20 GHz quantumdot laser with the modulation frequency of the reverse bias applied to the absorber section close to fP =2. 
M. Kantner, S. Yanchuk, Bifurcation analysis of delayinduced patterns in a ring of HodgkinHuxley neurons, Philosophical Transactions of the Royal Society A : Mathematical, Physical & Engineering Sciences, 371 (2013), pp. 20120470/120120470/13.

M.V. Arkhipov, R.M. Arkhipov, S.A. Pulkin, Effects of inversionless oscillation in twolevel media from the point of view of specificities of the spatiotemporal propagation dynamics of radiation, Optics and Spectroscopy, 114 (2013), pp. 831837.
Abstract
We report the results of computer simulation of the emission of radiation by an extended twolevel medium in a ring cavity. The cases of using strong external monochromatic, quasimonochromatic, and biharmonic radiation for pumping the twolevel medium are analyzed. It is shown that the emission of radiation with spectral content different from that of the pump radiation, which is interpreted as the inversionless oscillation, is the result of the spatiotemporal dynamics of light propagation in an extended twolevel medium imbedded in a cavity. The appearance of this radiation is not related to known resonances of amplification of a weak probe field in a thin layer of the twolevel system (the effect of inversionless oscillation) induced by strong resonance monochromatic or biharmonic field, as was thought before. 
R. Čiegis, A. Mirinavičius, M. Radziunas, Comparison of split step solvers for multidimensional Schrödinger problems, Computational Methods in Applied Mathematics, 13 (2013), pp. 237250.
Abstract
Hybrid modelocking in a two section edgeemitting semiconductor laser is studied numerically and analytically using a set of three delay differential equations. In this set the external RF signal applied to the saturable absorber section is modeled by modulation of the carrier relaxation rate in this section. Estimation of the locking range where the pulse repetition frequency is synchronized with the frequency of the external modulation is performed numerically and the effect of the modulation shape and amplitude on this range is investigated. Asymptotic analysis of the dependence of the locking range width on the laser parameters is carried out in the limit of small signal modulation. Our numerical simulations indicate that hybrid modelocking can be also achieved in the cases when the frequency of the external modulation is approximately twice larger and twice smaller than the pulse repetition frequency of the free running passively modelocked laser fP . Finally, we provide an experimental demonstration of hybrid modelocking in a 20 GHz quantumdot laser with the modulation frequency of the reverse bias applied to the absorber section close to fP =2. 
J. Sieber, M. Wolfrum, M. Lichtner, S. Yanchuk, On the stability of periodic orbits in delay equations with large delay, Discrete and Continuous Dynamical Systems, 33 (2013), pp. 31093134.
Abstract
We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay. 
O. Omel'chenko, M. Wolfrum, Bifurcations in the SakaguchiKuramoto model, Physica D. Nonlinear Phenomena, 263 (2013), pp. 7485.
Abstract
We analyze the SakaguchiKuramoto model of coupled phase oscillators in a continuum limit given by a frequency dependent version of the OttAntonsen system. Based on a selfconsistency equation, we provide a detailed analysis of partially synchronized states, their bifurcation from the completely incoherent state and their stability properties. We use this method to analyze the bifurcations for various types of frequency distributions and explain the appearance of nonuniversal synchronization transitions. 
O. Omel'chenko, Coherenceincoherence patterns in a ring of nonlocally coupled phase oscillators, Nonlinearity, 26 (2013), pp. 24692498.
Abstract
We consider a paradigmatic spatially extended model of nonlocally coupled phase oscillators which are uniformly distributed within a onedimensional interval and interact depending on the distance between their sites modulo periodic boundary conditions. This model can display peculiar spatiotemporal patterns consisting of alternating patches with synchronized (coherent) or irregular (incoherent) oscillator dynamics, hence the name coherenceincoherence pattern, or chimera state. For such patterns we formulate a general bifurcation analysis scheme based on a hierarchy of continuum limit equations. This gives us possibility to classify known coherenceincoherence patterns and to suggest directions for searching new ones. 
B. Fiedler, C. Rocha, M. Wolfrum, A permutation characterization of Sturm global attractors of Hamiltonian type, Journal of Differential Equations, 252 (2012), pp. 588623.
Abstract
We consider Neumann boundary value problems of the form $u_t = u_xx + f $ on the interval $0 leq x leq pi$ for dissipative nonlinearities $f = f (u)$. A permutation characterization for the global attractors of the semiflows generated by these equations is well known, even in the general case $f = f (x, u, u_x )$. We present a permutation characterization for the global attractors in the restrictive class of nonlinearities $f = f (u)$. In this class the stationary solutions of the parabolic equation satisfy the second order ODE $v^primeprime + f (v) = 0$ and we obtain the permutation characterization from a characterization of the set of $2pi$periodic orbits of this planar Hamiltonian system. Our results are based on a diligent discussion of this mere pendulum equation. 
B. Fiedler, C. Rocha, M. Wolfrum, Sturm global attractors for S$^1$equivariant parabolic equations, Networks Heterogeneous Media, 7 (2012), pp. 617659.
Abstract
We consider Neumann boundary value problems of the form $u_t = u_xx + f $ on the interval $0 leq x leq pi$ for dissipative nonlinearities $f = f (u)$. A permutation characterization for the global attractors of the semiflows generated by these equations is well known, even in the general case $f = f (x, u, u_x )$. We present a permutation characterization for the global attractors in the restrictive class of nonlinearities $f = f (u)$. In this class the stationary solutions of the parabolic equation satisfy the second order ODE $v^primeprime + f (v) = 0$ and we obtain the permutation characterization from a characterization of the set of $2pi$periodic orbits of this planar Hamiltonian system. Our results are based on a diligent discussion of this mere pendulum equation. 
CH. Otto, K. Lüdge, A.G. Vladimirov, M. Wolfrum, E. Schöll, Delay induced dynamics and jitter reduction of passively modelocked semiconductor lasers subject to optical feedback, New Journal of Physics, 14 (2012), pp. 113033/1113033/29.

T. Girnyk, M. Hasler, Y. Maistrenko, Multistability of twisted states in nonlocally coupled Kuramototype models, Chaos, Solitons and Fractals, 22 (2012), pp. 013114/1013114/10.
Abstract
A ring of N identical phase oscillators with interactions between Lnearest neighbors is considered, where L ranges from 1 (local coupling) to N/2 (global coupling). The coupling function is a simple sinusoid, as in the Kuramoto model, but with a minus sign which has a profound influence on its behavior. Without limitation of the generality the frequency of the freerunning oscillators can be set to zero. The resulting system is of gradient type and therefore all its solutions converge to an equilibrium point. All socalled qtwisted states, where the phase difference between neighboring oscillators on the ring is 2 pi q/N are equilibrium points, where q is an integer. Their stability in the limit N > inf. is discussed along the line of1. In addition we prove that when a twisted state is asymptotically stable for the infinite system, it is also asymptotically stable for sufficiently large N. Note that for smaller N, the same qtwisted states may become unstable and other qtwisted states may become stable. Finally, the existence of additional equilibrium states, called here multitwisted states, is shown by numerical simulation. The phase difference between neighboring oscillators is approximately 2 pi q/N in one sector of the ring, 2 pi q/N in another sector, and it has intermediate values between the two sectors. Our numerical investigation suggests that the number of different stable multitwisted states grows exponentially as N > inf. It is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discretetime translational dynamical system where the spacevariable (position on the ring) plays the role of time. The qtwisted states are then fixed points and the multitwisted states are periodic solutions of period N that are close to a heteroclinic cycle. Due to the apparently exponentially fast growing number of such stable periodic solutions, the system shows spatial chaos as N > 1. 
O. Omel'chenko, M. Wolfrum, Nonuniversal transitions to synchrony in the SakaguchiKuramoto model, Physical Review Letters, 109 (2012), pp. 164101/1164101/4.
Abstract
We investigate the transition to synchrony in a system of phase oscillators that are globally coupled with a phase lag (SakaguchiKuramoto model). We show that for certain unimodal frequency distributions there appear unusual types of synchronization transitions, where synchrony can decay with increasing coupling, incoherence can regain stability for increasing coupling, or multistability between partially synchronized states and/or the incoherent state can appear. Our method is a bifurcation analysis based on a frequency dependent version of the OttAntonsen method and allows for a universal description of possible synchronization transition scenarios for any given distribution of natural frequencies. 
O. Omel'chenko, M. Wolfrum, S. Yanchuk, Y. Maistrenko, O. Sudakov, Stationary patterns of coherence and incoherence in twodimensional arrays of nonlocally coupled phase oscillators, Phys. Rev. E (3), 85 (2012), pp. 036210/1036210/5.
Abstract
Recently it has been shown that large arrays of identical oscillators with nonlocal coupling can have a remarkable type of solutions that display a stationary macroscopic pattern of coexisting regions with coherent and incoherent motion, often caled chimera states. We present here a detailed numerical study of the appearance of such solutions in twodimensional arrays of coupled phase oscillators. We discover a variety of stationary patterns, including circular spots, stripe patterns, and patterns of multiple spirals. Here, the stationarity means that for increasing system size the locally averaged phase distributions tend to the stationary profile given by the corresponding thermodynamic limit equation. 
M. Wolfrum, The Turing bifurcation in network systems: Collective patterns and single differentiated nodes, Physica D. Nonlinear Phenomena, 241 (2012), pp. 13511357.
Abstract
We study the emergence of patterns in a diffusively coupled network that undergoes a Turing instability. Our main focus is the emergence of stable solutions with a single differentiated node in systems with large and possibly irregular network topology. Based on a meanfield approach, we study the bifurcations of such solutions for varying system parameters and varying degree of the differentiated node. Such solutions appear typically before the onset of Turing instability and provide the basis for the complex scenario of multistability and hysteresis that can be observed in such systems. Moreover, we discuss the appearance of stable collective patterns and present a codimensiontwo bifurcation that organizes the interplay between collective patterns and patterns with single differentiated nodes. 
A.G. Vladimirov, R. Lefever, M. Tlidi, Relative stability of multipeak localized patterns of cavity solitons, Physical Review A, 84 (2011), pp. 043848/1043848/4.
Abstract
We study the relative stability of different onedimensional (1D) and twodimensional (2D) clusters of closely packed localized peaks of the SwiftHohenberg equation. In the 1D case, we demonstrate numerically the existence of a spatial Maxwell transition point where all clusters involving up to 15 peaks are equally stable. Above (below) this point, clusters become more (less) stable when their number of peaks increases. In the 2D case, since clusters involving more than two peaks may exhibit distinct spatial arrangements, this point splits into a set of Maxwell transition pointsWe study the relative stability of different onedimensional (1D) and twodimensional (2D) clusters of closely packed localized peaks of the SwiftHohenberg equation. In the 1D case, we demonstrate numerically the existence of a spatial Maxwell transition point where all clusters involving up to 15 peaks are equally stable. Above (below) this point, clusters become more (less) stable when their number of peaks increases. In the 2D case, since clusters involving more than two peaks may exhibit distinct spatial arrangements, this point splits into a set of Maxwell transition points 
I. Babushkin, U. Bandelow, A. Vladimirov, Rotational symmetry breaking in smallarea circular vertical cavity surface emitting lasers, Optics Communications, 284 (2011), pp. 12991302.
Abstract
We investigate theoretically the dynamics of three loworder transverse modes in a smallarea vertical cavity surface emitting laser. We demonstrate the breaking of axial symmetry of the transverse field distribution in such a device. In particular, we show that if the linewidth enhancement factor is sufficiently large dynamical regimes with broken axial symmetry can exist up to very high diffusion coefficients 10 um^2/ns. 
M. Lichtner, M. Wolfrum, S. Yanchuk, The spectrum of delay differential equations with large delay, SIAM Journal on Mathematical Analysis, 43 (2011), pp. 788802.

M. Wolfrum, O. Omel'chenko, Chimera states are chaotic transients, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 84 (2011), pp. 015201(R)/1015201(R)/4.
Abstract
Spatiotemporal chaos and turbulence are universal concepts for the explanation of irregular behavior in various physical systems. Recently, a remarkable new phenomenon, called "chimera states", has been described, where in a spatially homogeneous system regions of irregular incoherent motion coexist with regular synchronized motion, forming a self organized pattern in a population of nonlocally coupled oscillators. Whereas most of the previous studies of chimera states focused their attention to the case of large numbers of oscillators employing the thermodynamic limit of infinitely many oscillators, we investigate here the properties of chimera states in populations of finite size using concepts from deterministic chaos. Our calculations of the Lyapunov spectrum show that the incoherent motion, which is described in the thermodynamic limit as a stationary behavior, in finite size systems appears as weak spatially extensive chaos. Moreover, for sufficiently small populations the chimera states reveal their transient nature: after a certain timespan we observe a sudden collapse of the chimera pattern and a transition to the completely coherent state. Our results indicate that chimera states can be considered as chaotic transients, showing the same properties as typeII supertransients in coupled map lattices. 
M. Wolfrum, O. Omel'chenko, S. Yanchuk, Y. Maistrenko, Spectral properties of chimera states, Chaos. An Interdisciplinary Journal of Nonlinear Science, 21 (2011), pp. 0013112/1013112/8.
Abstract
Chimera states are particular trajectories in systems of phase oscillators with nonlocal coupling that display a pattern of coherent and incoherent motion. We present here a detailed analysis of the spectral properties for such trajectories. First, we study numerically their Lyapunov spectrum and its behavior for an increasing number of oscillators. The spectra demonstrate the hyperchaotic nature of the chimera states and show a correspondence of the Lyapunov dimension with the number of incoherent oscillators. Then, we pass to the thermodynamic limit equation and present an analytic approach to the spectrum of a corresponding linearized evolution operator. We show that in this setting, the chimera state is neutrally stable and that the continuous spectrum coincides with the limit of the hyperchaotic Lyapunov spectrum obtained for the finite size systems. 
A.G. Vladimirov, U. Bandelow, G. Fiol, D. Arsenijević, M. Kleinert, D. Bimberg, A. Pimenov, D. Rachinskii, Dynamical regimes in a monolithic passively modelocked quantum dot laser, Journal of the Optical Society of America. B, 27 (2010), pp. 21022109.

P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, P. Mosiolek, T. Kapitaniak, Routes to complex dynamics in a ring of unidirectionally coupled systems, Chaos. An Interdisciplinary Journal of Nonlinear Science, 20 (2010), pp. 013111/1013111/9.

M. Tlidi, A.G. Vladimirov, D. Turaev, G. Kozyreff, D. Pieroux, T. Erneux, Spontaneous motion of localized structures and localized patterns induced by delayed feedback, The European Physical Journal D. Atomic, Molecular, Optical and Plasma Physics, 59 (2010), pp. 5965.

O.E. Omel'chenko, Y.L. Maistrenko, P.A. Tass, Chimera states induced by spatially modulated delayed feedback, Phys. Rev. E (3), 82 (2010), pp. 066201/1066201/13.

O.E. Omel'chenko, M. Wolfrum, Y.L. Maistrenko, Chimera states as chaotic spatiotemporal patterns, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 81 (2010), pp. 065201(R)/1065201R)/4.

M. Wolfrum, S. Yanchuk, A multiple time scale approach to the stability of external cavity modes in the LangKobayashi system using the limit of large delay, SIAM Journal on Applied Dynamical Systems, 9 (2010), pp. 519535.

M. Wolfrum, S. Yanchuk, P. Hövel, E. Schöll, Complex dynamics in delaydifferential equations with large delay, The European Physical Journal, Special Topics, 191 (2010), pp. 91103.

M. Tlidi, A.G. Vladimirov, D. Pieroux, D. Turaev, Spontaneous motion of cavity solitons induced by a delayed feedback, Physical Review Letters, 103 (2009), pp. 103904/1103904/4.

V.Z. Tronciu, Excitability and coherence resonance of DFB laser with passive dispersive reflector, Moldavian Journal of the Physical Sciences, 7 (2008), pp. 218223.

S. Yanchuk, M. Wolfrum, Destabilization patterns in large regular networks, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 77 (2008), pp. 026212/1026212/7.
Abstract
We describe a generic mechanism for the destabilization in large regular networks of identical coupled oscillators. Based on a reduction method for the spectral problem, we first present a criterion for this type of destabilization. Then, we investigate the related bifurcation scenario, showing the existence of a large number of coexisting periodic solutions with different frequencies, spatial patterns, and stability properties. Even for unidirectional coupling this can be understood in analogy to the wellknown Eckhaus scenario for diffusive systems. 
V.F. Butuzov, N.N. Nefedov, L. Recke, K.R. Schneider, Existence and stability of solutions with periodically moving weak internal layers, Journal of Mathematical Analysis and Applications, 348 (2008), pp. 508515.
Abstract
We consider the periodic parabolic differential equation $ep^2 Big( fracpartial^2 upartial x^2 fracpartial upartial t Big)=f(u,x,t,ep)$ under the assumption that $ve$ is a small positive parameter and that the degenerate equation $f(u,x,t,0) =0$ has two intersecting solutions. We derive conditions such that there exists an asymptotically stable solution $u_p(x,t,ep)$ which is $T$periodic in $t$, satisfies noflux boundary conditions and tends to the stable composed root of the degenerate equation as $eprightarrow 0$. 
D. Turaev, M. Radziunas, A.G. Vladimirov, Chaotic soliton walk in periodically modulated media, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 77 (2008), pp. 06520/106520/4.

M. Lichtner, Spectral mapping theorem for linear hyperbolic systems, Proceedings of the American Mathematical Society, 136 (2008), pp. 20912101.

D. Turaev, A.G. Vladimirov, S. Zelik, Chaotic bound state of localized structures in the complex GinzburgLandau equation, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 75 (2007), pp. 045601/1045601/4.

M. Lichtner, M. Radziunas, L. Recke, Wellposedness, smooth dependence and center manifold reduction for a semilinear hyperbolic system from laser dynamics, Mathematical Methods in the Applied Sciences, 30 (2007), pp. 931960.

A.G. Vladimirov, D.V. Skryabin, G. Kozyreff, P. Mandel, M. Tlidi, Bragg localized structures in a passive cavity with transverse modulation of the refractive index and the pump, Optics Express, 14 (2006), pp. 16.

S. Yanchuk, A. Stefanski, T. Kapitaniak, J. Wojewoda, Dynamics of an array of mutually coupled semiconductor lasers, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 73 (2006), pp. 016209/1016209/7.

S. Yanchuk, M. Wolfrum, P. Hövel, E. Schöll, Control of unstable steady states by strongly delayed feedback, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 74 (2006), pp. 026201/1026201/7.

M. Nizette, D. Rachinskii, A. Vladimirov, M. Wolfrum, Pulse interaction via gain and loss dynamics in passive modelocking, Physica D. Nonlinear Phenomena, 218 (2006), pp. 95104.

A. Politi, F. Ginelli, S. Yanchuk, Y. Maistrenko, From synchronization to Lyapunov exponents and back, Physica D. Nonlinear Phenomena, 224 (2006), pp. 90101.

D.I. Rachinskii, A. Vladimirov, U. Bandelow, B. Hüttl, R. Kaiser, Qswitching instability in a modelocked semiconductor laser, Journal of the Optical Society of America. B, 23 (2006), pp. 663670.

A. Yulin, D. Skryabin, A.G. Vladimirov, Modulation instability of discrete solitons in coupled waveguides with group velocity dispersion, Optics Express, 14 (2006), pp. 1234712352.

M. Wolfrum, S. Yanchuk, Eckhaus instability in systems with large delay, Physical Review Letters, 96 (2006), pp. 220201/1220201/4.

TH. Koprucki, M. Baro, U. Bandelow, Th. Tien, F. Weik, J.W. Tomm, M. Grau, M.Ch. Amann, Electronic structure and optoelectronic properties of strained InAsSb/GaSb multiple quantum wells, Applied Physics Letters, 87 (2005), pp. 181911/1181911/3.

A. Vladimirov, D. Turaev, Model for passive mode locking in semiconductor lasers, Physical Review A, 72 (2005), pp. 033808/1033808/13.

S. Yanchuk, Discretization of frequencies in delay coupled oscillators, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 72 (2005), pp. 036205/1036205/5.

S. Yanchuk, Properties of stationary states of delay equations with large delay and applications to laser dynamics, Mathematical Methods in the Applied Sciences, 28 (2005), pp. 363377.

K. Gatermann, M. Wolfrum, Bernstein's second theorem and Viro's method for sparse polynomial systems in chemistry, Adv. Appl. Math., 34 (2005), pp. 252294.

D.I. Rachinskii, K.R. Schneider, Dynamic Hopf bifurcations generated by nonlinear terms, Journal of Differential Equations, 210 (2005), pp. 6586.

M. Wolfrum, J. Härterich, Describing a class of global attractors via symbol sequences, Discrete and Continuous Dynamical Systems, 12 (2005), pp. 531554.

S. Yanchuk, A. Stefanski, J. Wojewoda, T. Kapitaniak, Simple estimation of synchronization threshold in ensembles of diffusively coupled chaotic systems, Phys. Rev. E (3), 70 (2004), pp. 026217/1026217/11.

B. Fiedler, C. Rocha, M. Wolfrum, Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle, Journal of Differential Equations, 201 (2004), pp. 99138.

K.R. Schneider, S. Yanchuk, L. Recke, Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit, Phys. Rev. E (3), 69 (2004), pp. 056221/1056221/12.

S.V. Fedorov, N.N. Rosanov, A.N. Shatsev, N.A. Veretenov, A.G. Vladimirov, Topologically multicharged and multihumped rotating solitons in wideaperture lasers with saturable absorber, IEEE J. Quantum Electron., 39 (2003), pp. 216226.

D. Rachinskii, K.R. Schneider, Delayed loss of stability in systems with degenerate linear parts, Zeitschrift fur Analysis und ihre Anwendungen. Journal for Analysis and its Applications, 22 (2003), pp. 433453.

M. Tlidi, A.G. Vladimirov, P. Mandel, Interaction and stability of periodic and localized structures in optical bistable systems, IEEE J. Quantum Electron., 39 (2003), pp. 197205.

K.R. Schneider, E. Shchetinina, Oneparametric families of canard cycles: Two explicitly solvable examples, Mathematical Methods in the Applied Sciences, 2 (2003), pp. 7475.

K.R. Schneider, S. Yanchuk, Complete synchronization of symmetrically coupled autonomous systems, Applicable Analysis. An International Journal, 82 (2003), pp. 11271143.

K.R. Schneider, V.A. Sobolev, E. Shchepakina, New type of travelling wave solutions, Mathematical Methods in the Applied Sciences, 26 (2003), pp. 13491361.

A.G. Vladimirov, G. Kozyreff, P. Mandel, Synchronization of weakly stable oscillators and semiconductor laser arrays, Europhysics Letters, 61 (2003), pp. 613619.

A.G. Vladimirov, G. Kozyreff, P. Mandel, Synchronization of weakly stable oscillators and semiconductor laser arrays, Europhysics Letters, 61 (2003), pp. 613619.

S. Yanchuk, T. Kapitaniak, Manifestation of riddling in the presence of small parameter mismatch between coupled systems, Phys. Rev. E (3), 68, 017202 (2003), 4.

S. Yanchuk, Y. Maistrenko, E. Mosekilde, Synchronization of timecontinuous chaotic oscillators, Chaos. An Interdisciplinary Journal of Nonlinear Science, 13 (2003), pp. 388400.

S. Yanchuk, G. Kristensen, I. Shushko, Dynamical approach to complex regional economic growth based on Keynesian model for China, Chaos, Solitons and Fractals, 18 (2003), pp. 937952.

M. Wolfrum, A sequence of order relations: Encoding heteroclinic connections in scalar parabolic PDE, Journal of Differential Equations, 183 (2002), pp. 5678.

M. Wolfrum, Geometry of heteroclinic cascades in scalar parabolic differential equations, Journal of Dynamics and Differential Equations, 14 (2002), pp. 207241.
Contributions to Collected Editions

M. Kolarczik, F. Böhm, U. Woggon, N. Owschimikow, A. Pimenov, M. Wolfrum, A.G. Vladimirov, S. Meinecke, B. Lingnau, L. Jaurigue, K. Lüdge, Coherent and incoherent dynamics in quantum dots and nanophotonic devices, in: Semiconductor Nanophotonics, M. Kneissl, A. Knorr, S. Reitzenstein, A. Hoffmann, eds., 194 of Springer Series in SolidState Sciences, Springer, Cham, 2020, pp. 91133, DOI 10.1007/9783030356569_4 .
Abstract
The interest in coherent and incoherent dynamics in novel semiconductor gain media and nanophotonic devices is driven by the wish to understand the optical gain spectrally, dynamically, and energetically for applications in optical amplifiers, lasers or specially designed multisection devices. This chapter is devoted to the investigation of carrier dynamics inside nanostructured gain media as well as to the dynamics of the resulting light output. It is structured into two parts. The first part deals with the characterization of ultrafast and complex carrier dynamics via the optical response of the gain medium with pumpprobe methods, twocolor fourwave mixing setups and quantumstate tomography. We discuss the optical nonlinearities resulting from lightmatter coupling and charge carrier interactions using microscopically motivated rateequation models. In the second part, nanostructured modelocked lasers are investigated, with a focus on analytic insights about the regularity of the pulsed light emission. A method for efficiently predicting the timing fluctuations is presented and used to optimize the device properties. Finally, one specific design of a modelocked laser with tapered gain section is discussed which draws the attention to alternative ways of producing very stable and high intensity laser pulses. 
A.V. Kovalev, E.A. Viktorov, N. Rebrova, A.G. Vladimirov, G. Huyet, Theoretical study of modelocked lasers with nonlinear loop mirrors, in: Proc. SPIE 10682, Semiconductor Lasers and Laser Dynamics VIII, K. Panayotov, M. Sciamanna, R. Michalzik, eds., SPIE Digital Library, 2018, pp. 1068226/11068226/6.

M. Wolfrum, Enumeration of positive meanders, in: Patterns of Dynamics. PaDy 2016, P. Gurevich, J. Hell, B. Sandstede, A. Scheel, eds., 205 of Springer Proceedings in Mathematics & Statistics, Springer, Cham, 2018, pp. 203212, DOI 10.1007/9783319641737_13 .
Abstract
Meanders are geometrical objects, defined by a nonselfintersecting curve, intersecting several times through an infinite straight line. The subclass of positive meanders has been defined and used extensively for the study of the attractors of scalar parabolic PDEs. In this paper, we use bracket sequences and winding numbers to investigate the class of positive meanders. We prove a theorem about possible combinations of bracket sequences to obtain a meander with prescribed winding numbers and present an algorithm to compute the number of positive meanders with a given number of intersection points. 
W.W. Ahmed, S. Kumar, R. Herrero, M. Botey, M. Radziunas, K. Staliunas, Suppression of modulation instability in pump modulated flatmirror VECSELs, in: Nonlinear Optics and its Applications IV, B.J. Eggleton, N.G.R. Broderick, A.L. Gaeta, eds., 9894 of Proceedings of SPIE, SPIE Digital Library, 2016, pp. 989406/1989406/7.
Abstract
We show that modulation instability (MI) can be suppressed in vertical external cavity surface emitting lasers (VECSELs) by introducing a periodic spatiotemporal modulation of the pump profile which in turn allows a simple flatmirror configuration. The stability analysis of such pump modulated flatmirror VECSELs is performed by a modified Floquet method and results are confirmed by full numerical integration of the model equations. It is found that the amplitude of the modulation as well as its spatial and temporal frequencies are crucial parameters for high spatial beam quality emission. We identify regions of complete and partial stabilization in parameter space for VECSELs with different external cavity lengths. The proposed method is shown to efficiently stabilize VECSELs with cavity lengths ranging from millimetres up to centimetres. However, the applicability of this method becomes limited for micrometerlong cavities due to strong intrinsic relaxation oscillations. 
D. Turaev, A.G. Vladimirov, S. Zelik, Interaction of spatial and temporal cavity solitons in modelocked lasers and passive cavities, in: Laser Optics (LO), 2016 International Conference, IEEE, New York, 2016, pp. 37628.
Abstract
We study interaction of wellseparated localized structures of light in the presence of periodic perturbations. Oscillating localized structures were found to emit weakly decaying dispersive waves leading to a strong enhancement of the interaction and formation of new types of bound states. We discuss the applicability of our analytical results to the interpretation of experimental and numerical data reported earlier. 
A.G. Vladimirov, G. Huyet, A. Pimenov, Delay differential models in multimode laser dynamics: Taking chromatic dispersion into account, in: Semiconductor Lasers and Laser Dynamics VII, 9892 of Proceedings of SPIE, SPIE, Bellingham, Washington, 2016, pp. 98920I/198920I/7.
Abstract
A set of differential equations with distributed delay is derived for modeling of multimode ring lasers with intracavity chromatic dispersion. Analytical stability analysis of continuous wave regimes is performed and it is demonstrated that sufficiently strong anomalous dispersion can destabilize these regimes. © (2016) COPYRIGHT Society of PhotoOptical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only. 
U. Bandelow, S. Amiranashvili, N. Akhmediev, Limitation for ultrashort solitons in nonlinear optical fibers by cusp formation, in: CLEO®/Europe  EQEC 2015: Conference Digest, OSA Technical Digest (Online) (Optical Society of America, 2015), paper EI2.3 THU, 2015, pp. 11.

A. Glitzky, A. Mielke, L. Recke, M. Wolfrum, S. Yanchuk, D2  Mathematics for optoelectronic devices, in: MATHEON  Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 243256.

S. Amiranashvili, A. Demircan, C. Brée, G. Steinmeyer, F. Mitschke, Manipulating light by light in optical fibers, in: 3rd Bonn Humboldt Award Winners' Forum ``Frontiers in Quantum Optics: Taming the World of Atoms and Photons  100 Years after Niels Bohr'', Bonn, October 912, 2013, Networking Guide, pp. 5859.

P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, T. Kapitaniak, Dynamics of a large ring of unidirectionally coupled duffing oscillators, in: IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design, Proceedings of the IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design, held Aberdeen, UK, 2730 July 2010, M. Wiercigroch, G. Rega, eds., 32 of IUTAM Bookseries, Springer, Dordrecht et al., 2013, pp. 6372.

C. Brée, S. Amiranashvili, U. Bandelow, Spatiotemporal pulse propagation in nonlinear dispersive optical media, in: Proceedings of the 12th International Conference on Numerical Simulation of Optoelectronic Devices, NUSOD'12, J. Piprek, W. Lu, eds., IEEE Conference Publications Management Group, New Jersey, USA, 2012, pp. 131132.

D. Turaev, A.G. Vladimirov, S. Zelik, Strong enhancement of interaction of optical pulses induced by oscillatory instability, in: CLEO/Europe and EQEC 2009 Conference Digest (Optical Society of America, 2009), poster EH.P.13 WED, 2009, pp. 11.

L. Recke, M. Wolfrum, S. Yanchuk, Dynamics of coupled semiconductor lasers, in: Analysis and Control of Complex Nonlinear Processes in Physics, Chemistry and Biology, Chapter 6, L. SchimanskyGeier, B. Fiedler, J. Kurths, E. Schöll, eds., World Scientific, New Jersey [et al.], 2007, pp. 185212.

A.G. Vladimirov, D.V. Skryabin, M. Tlidi, Localized structures of light in nonlinear devices with intracavity photonic bandgap material, in: 2007 European Conference on Lasers and ElectroOptics and the European Quantum Electronics Conference (CLEO®/EuropeIQEC) Conference Digest (oral presentation IG4MON), IEEE, 2007, pp. 11.

J. Ehrt, J. Härterich, Convergence to stationary states in spatially inhomogeneous balance laws, in: Hyperbolic Problems. Theory, Numerics and Applications I, F. Asakura, S. Kawashima, A. Matsumura, S. Nishibata, K. Nishihara, eds., Yokohama Publishers, Yokohama, 2006, pp. 367374.

M. Wolfrum, The concept of adjacency for stationary and nonstationary solutions of scalar semilinear parabolic PDE, in: EQUADIFF 2003. Proceedings of the International Conference on Differential Equations, Hasselt, Belgium, 2226 July 2003, F. Dumortier, H. Broer, J. Mawhin, A. Vanderbauwhede, S. Verduyn Lunel, eds., World Scientific, Singapore, 2005, pp. 678684.

S. Yanchuk, K.R. Schneider, Complete synchronization of symmetrically coupled autonomous systems, in: EQUADIFF 2003. Proceedings of the International Conference on Differential Equations, Hasselt, Belgium, 2226 July 2003, F. Dumortier, H. Broer, J. Mawhin, A. Vanderbauwhede, S. Verduyn Lunel, eds., World Scientific, Singapore, 2005, pp. 494496.

S. Yanchuk, M. Wolfrum, Instabilities of equilibria of delaydifferential equations with large delay, in: Proceedings of the ENOC2005 Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven, The Netherlands, August 712, 2005, D.H. VAN Campen, M.D. Lazurko, W.P.J.M. caps">caps">van der Oever, eds., 2005, pp. 10601065.

M. Wolfrum, S. Yanchuk, Synchronous and asynchronous instabilities of two lasers with a long delayed coupling, in: Proceedings of the ENOC2005 Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven, The Netherlands, August 712, 2005, D.H. VAN Campen, M.D. Lazurko, W.P.J.M. caps">caps">van der Oever, eds., 2005, pp. 20692073.

S. Yanchuk, K.R. Schneider, L. Recke, Dynamics of two F2F coupled lasers: Instantaneous coupling limit, in: Proceeding of SPIE: Semiconductor Lasers and Laser Dynamics Conference ``Photonics Europe'', 5452, SPIE, Washington, USA, 2004, pp. 5162.

Y.L. Maistrenko, O. Popovych, S. Yanchuk, Synchronization and clustering in ensembles of coupled chaotic oscillators, in: Synchronization: Theory and Application. Proceedings of the NATO Advanced Study Institute, A. Pikovsky, Y.L. Maistrenko, eds., 109 of NATO Sci. Ser. II Math. Phys. Chem., Kluwer Acad. Publishers, Dordrecht, 2003, pp. 101138.
Preprints, Reports, Technical Reports

S. Yanchuk, M. Wolfrum, T. Pereira, D. Turaev, Absolute stability and absolute hyperbolicity in systems with discrete timedelays, Preprint no. 2824, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2824 .
Abstract, PDF (425 kByte)
An equilibrium of a delay differential equation (DDE) is absolutely stable, if it is locally asymptotically stable for all delays. We present criteria for absolute stability of DDEs with discrete timedelays. In the case of a single delay, the absolute stability is shown to be equivalent to asymptotic stability for sufficiently large delays. Similarly, for multiple delays, the absolute stability is equivalent to asymptotic stability for hierarchically large delays. Additionally, we give necessary and sufficient conditions for a linear DDE to be hyperbolic for all delays. The latter conditions are crucial for determining whether a system can have stabilizing or destabilizing bifurcations by varying time delays. 
S. Amiranashvili, M. Radziunas, U. Bandelow, K. Busch, R. Čiegis, Additive splitting methods for parallel solution of evolution problems, Preprint no. 2767, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2767 .
Abstract, PDF (312 kByte)
We demonstrate how a multiplicative splitting method of order P can be used to construct an additive splitting method of order P + 3. The weight coefficients of the additive method depend only on P, which must be an odd number. Specifically we discuss a fourthorder additive method, which is yielded by the LieTrotter splitting. We provide error estimates, stability analysis, and numerical examples with the special discussion of the parallelization properties and applications to nonlinear optics. 
M. Kantner, Th. Koprucki, Beyond just ``flattening the curve'': Optimal control of epidemics with purely nonpharmaceutical interventions, Preprint no. 2748, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2748 .
Abstract, PDF (3116 kByte)
When effective medical treatment and vaccination are not available, nonpharmaceutical interventions such as social distancing, home quarantine and farreaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptibleexposedinfectiousrecovered) model and continuoustime optimal control theory, we compute the optimal nonpharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of diseaserelated deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socioeconomic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a singleintervention scenario that goes beyond heuristically motivated interventions and simple "flattening of the curve". Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socioeconomic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID19 pandemic in Germany. 
M. Ghani Varzaneh, S. Riedel, A dynamical theory for singular stochastic delay differential equations II: Nonlinear equations and invariant manifolds, Preprint no. 2701, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2701 .
Abstract, PDF (383 kByte)
Building on results obtained in [GVRS], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory and a semiinvertible Multiplicative Ergodic Theorem for cocycles acting on measurable fields of Banach spaces obtained in [GVR]. 
K.R. Schneider, A. Grin, Global bifurcation analysis of limit cycles for a generalized van der Pol system, Preprint no. 2639, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2639 .
Abstract, PDF (189 kByte)
We present a new approach for the global bifurcation analysis of limit cycles for a generalized van der Pol system. It is based on the existence of a DulacCherkas function and on applying two topologically equivalent systems: one of them is a rotated vector field, the other one is a singularly perturbed system. 
K.R. Schneider, A. Grin, Lower and upper bounds for the number of limit cycles on a cylinder, Preprint no. 2638, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2638 .
Abstract, PDF (295 kByte)
We consider autonomous systems with cylindrical phase space. Lower and upper bounds for the number of limit cycles surrounding the cylinder can be obtained by means of an appropriate DulacCherkas function. We present different possibilities to improve these bounds including the case that the exact number of limit cycles can be determined. These approaches are based on the use of several DulacCherkas functions or on applying some factorized Dulac function.
Talks, Poster

M. Kantner, Mathematical modeling and optimal control of the COVID19 pandemic (online talk), Mathematisches Kolloquium, Bergische Universität Wuppertal, April 27, 2021.

W. Matthias, Temporal dissipative solitons in systems of delaydifferential equations, SIAM Conference on Dynamical Systems (Online Event), Minisymposium 184 ``Traveling Pulses in Delay and Lattice Differential Equations'', May 27, 2021, Portland, Oregon, USA, May 27, 2021.

M. Wolfrum, Dynamics of a stochastic excitable system with slowly adapting feedback, Workshop on Control of SelfOrganizing Nonlinear Systems 2020, September 2  3, 2020, TU Berlin, September 3, 2020.

M. Wolfrum, Temporal dissipative solitons in systems with time delay, Séminaire Orléans, Institut Denis Poisson, Orléans, France, January 23, 2020.

S. Eydam, A. Gerdes, Extensive chaos, cluster and chimera states in globallycoupled StuartLandau systems, SFB 910: Workshop on ''Control of SelfOrganizing Nonlinear Systems'', Lutherstadt Wittenberg, August 20  22, 2019.

S. Eydam, Modelocked solutions in systems of globallycoupled phase oscillators, XXXIX. Dynamics Days Europe, September 2  6, 2019, University of Rostock, September 6, 2019.

S. Eydam, Modelocked solutions in systems of globallycoupled phase oscillators, WIAS Workshop ,,Optical Solitons and Frequency Comb Generation'', September 18  20, 2019, WIAS Berlin, September 18, 2019.

M. Wolfrum, Phasesensitive excitability of a limit cycle, SFB 910: Workshop on ''Control of SelfOrganizing Nonlinear Systems'', August 20  22, 2019, SFB 910, Lutherstadt Wittenberg, August 21, 2019.

M. Wolfrum, Phasesensitive excitability of a limit cycle, XXXIX Dynamics Days Europe, September 2  6, 2019, University of Rostock, September 3, 2019.

M. Wolfrum, Temporal dissipative solitons in a DDE model of a ring laser with optical injection, Equadiff 2019, July 8  12, 2019, Leiden University, Netherlands, July 9, 2019.

M. Wolfrum, Temporal dissipative solitons in systems with time delay, 11th Colloquium on the Qualitative Theory of Differential Equations, University of Szeged, Bolyai Institute, Hungary, June 20, 2019.

M. Wolfrum, The relation of Chimeras, bump states, and Turing patterns in arrays of coupled oscillators, School and Workshop on Patterns of Synchrony: Chimera States and Beyond, May 6  17, 2019, International Centre for Theoretical Physics, Trieste, Italy, May 16, 2019.

S. Reichelt, Pulses in FitzHughNagumo systems with rapidly oscillating coefficients, SIAM Annual Meeting, Minisymposium 101 ``Multiscale Analysis and Simulation on Heterogeneous Media'', July 9  13, 2018, Society for Industrial and Applied Mathematics, Oregon Convention Center (OCC), Portland, USA, July 12, 2018.

S. Eydam, Bifurcations of modelocked solutions, Workshop ,,Dynamics in Coupled Oscillator Systems", WIAS, Berlin, November 19, 2018.

S. Eydam, Mode locking in systems of globally coupled phase oscillators, Dynamics Days Europe, Loughborough, UK, September 3  7, 2018.

S. Eydam, Modelocking in systems of globally coupled phase oscillators, International Conference on Control of SelfOrganizing Nonlinear Systems, Warnemünde, September 9  13, 2018.

A. Pimenov, Analysis of temporal localized structures in a time delay model of a ring laser, 675. WEHeraeus Seminar: Delayed Complex Systems 2018, Bad Honnef, July 2  5, 2018.

A. Pimenov, Effect of chromatic dispersion in a delayed model of a modelocked laser, Workshop ,,Nonlinear Dynamics in Semiconductor Lasers (NDSL2018)'', June 18  20, 2018, WIAS, Berlin, June 20, 2018.

U. Bandelow, Ultrashort solitons and their control in the regime of event horizons in nonlinear dispersive optical media, George Stegeman Symposium, University of Central Florida, Orlando, USA, March 13, 2018.

A.G. Vladimirov, Delay models in nonlinear laser dynamics, Dynamics Days Europe 2018, September 3  7, 2018, Loughborough University, UK, September 6, 2018.

M. Wolfrum, Dynamics of coupled oscillator systems and their continuum limits, Kolloquium der Arbeitsgruppe Modellierung, Numerik, Differentialgleichungen, Technische Universität Berlin, Institut für Mathematik, June 12, 2018.

M. Wolfrum, Phase solitons in DDEs with large delay, 14th IFAC Workshop on Time Delay Systems, June 28  30, 2018, Budapest University of Technology and Economics, Hungary, June 29, 2018.

M. Wolfrum, Phasesensitive excitability of a limit cycle, International Conference on Control of SelfOrganizing Nonlinear Systems, Warnemünde, September 9  13, 2018.

S. Reichelt, Pulses in FitzHughNagumo systems with periodic coefficients, Seminar ``Dynamical Systems and Applications'', Technische Universität Berlin, Institut für Mathematik, May 3, 2017.

S. Reichelt, Traveling waves in FitzHughNagumo systems with rapidly oscillating coefficients, Workshop ``Control of Selforganizing Nonlinear Systems'', August 29  31, 2017, Collaborative Research Center 910: Control of selforganizing nonlinear systems: Theoretical methods and concepts of application, Lutherstadt Wittenberg, August 30, 2017.

S. Eydam, Modelocking in systems of globally coupled phase oscillators, Workshop on Control of SelfOrganizing Nonlinear Systems, August 28  31, 2017, TU Berlin/SFB 910, Lutherstadt Wittenberg, August 31, 2017.

S. Eydam, Phase oscillator mode locking, SCL Seminar, Belgrade Institute of Physics, Serbia, September 28, 2017.

S. Eydam, Phase oscillator modelocking, Forschungsseminar ``Applied Dynamical Systems'', TU Berlin, June 14, 2017.

M. Kantner, Modeling of quantum dot based singlephoton LEDs on a device level, MATHEON Workshop 10th Annual Meeting ``Photonic Devices'', February 9  10, 2017, KonradZuseZentrum für Informationstechnik Berlin, February 10, 2017.

O. Omel'chenko, Bifurcations mediating appearance of chimera states, XXXVII Dynamics Days Europe, Minisymposium 3 ``Complex Networks: Delays And Collective Dynamics'', June 5  9, 2017, University of Szeged, Faculty of Science and Informatics, Hungary, June 8, 2017.

O. Omel'chenko, Bifurcations mediating the appearance of chimera states, SIAM Conference on Applications of Dynamical Systems (DS 17), Minisymposium ``Large Scale Dynamics In Coupled Systems On Networks'', May 21  25, 2017, Society for Industrial and Applied Mathematics (SIAM), Snowbird, USA, May 24, 2017.

O. Omel'chenko, Controlling unstable complex dynamics: From coupled oscillators to fluid mechanics, XV Latin American Workshop on NonLinear Phenomena, November 6  10, 2017, Facultad de Ciencias y Astronomía, Universidad de La Serena, Chile, November 7, 2017.

O. Omel'chenko, Introduction to chimera states, Seminar of the Scientific Computing Laboratory, University of Belgrade, Institue of Physics, Serbia, May 4, 2017.

O. Omel'chenko, Noninvasive model reconstruction from a partially synchronized state, XXXVII Dynamics Days Europe, Minisymposium 14b ``Synchronization Patterns In Networks: Theory and Applications'', June 5  9, 2017, University of Szeged, Faculty of Science and Informatics, Hungary, June 8, 2017.

O. Omel'chenko, Stabilizing control scheme: From chimera states to edge states, Internal seminar of the Prof. E. Knobloch group, Department of Physics, University of California, Berkeley, USA, May 17, 2017.

M. Wolfrum, Chimera states in systems of coupled phase oscillators, Emerging Topics in Network Dynamical Systems, June 6  9, 2017, Lorentz Center, Leiden, Netherlands, June 6, 2017.

M. Wolfrum, Dynamics of coupled oscillator systems and their continuum limits, CIMWIAS Workshop ``Topics in Applied Analysis and Optimisation'', December 6  8, 2017, International Center for Mathematics, University of Lisbon, Portugal, December 6, 2017.

S. Reichelt, Competing patterns in antisymmetrically coupled SwiftHohenberg equations, International Conference on Control of Complex Systems and Networks, SFB 910 ``Control of SelfOrganizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Heringsdorf/Usedom, September 4  8, 2016.

S. Eydam, Modelocking in systems of coupled phase oscillators, Seminar Applied Dynamical Systems, Technische Universität Berlin, Berlin, July 13, 2016.

S. Eydam, Modelocking in systems of phase oscillators with higher harmonic coupling, International Conference on Control of Complex Systems and Networks, SFB 910 ``Control of SelfOrganizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Heringsdorf/Usedom, September 4  8, 2016.

S. Eydam, Modelocking in systems of phase oscillators with higher harmonic interaction, Workshop on Synchronization and Oscillators with Generalized Coupling, Exeter, UK, April 20  22, 2016.

M. Kantner, Modeling and simulation of carrier dynamics in quantum dot based singlephoton sources, Nonlinear Dynamics in Semiconductor Lasers, WIAS, Berlin, June 15, 2016.

U. Bandelow, Nonlinear dynamical effects in photonics: Modeling, simulation and analysis, Coloquio del Instituto de Física, Pontificia Universidad Católica de Chile, Santiago, December 14, 2016.

U. Bandelow, Ultrashort solitons that do not want to be too short in duration, XIX Conference on Nonequilibrium Statistical Mechanics and Nonlinear Physics (MEDYFINOL 2016), Universidad de los Andes, Universidad de Mar del Plata, and Instituto Tecnológico de Buenos Aires, Valdivia, Chile, December 7, 2016.

O. Omel'chenko, Asymptotics of traveling coherenceincoherence patterns, Contemporary Problems of Mathematical Physics and Computational Mathematics, Lomonosov Moscow State University, Russian Federation, November 2, 2016.

O. Omel'chenko, Chimera states in nonlocally coupled oscillators: Their variety and control, 4th International Conference on Complex Dynamical Systems and Applications, National Institute of Technology, Durgapur, India, February 16, 2016.

O. Omel'chenko, Creative control for chimera states, Workshop on Synchronization and Oscillators with Generalized Coupling, University of Exeter, UK, April 21, 2016.

O. Omel'chenko, Meanfield equation for coherenceincoherence patterns, 7th European Congress of Mathematics (7ECM), Minisymposium 37 ``Propagation Phenomena in Discrete Media'', July 18  22, 2016, Technische Universität Berlin, July 22, 2016.

O. Omel'chenko, Nonuniversal transitions to synchrony in globally coupled phase oscillator, International Workshop on Nonlinear Complex Dynamical Systems, Indian Statistical Institute, Kolkata, February 19, 2016.

O. Omel'chenko, On the limitations of the Kuramoto model, Dynamics Days Latin America and the Caribbean, Benemérita Universidad Autónoma de Puebla, Mexico, October 28, 2016.

O. Omel'chenko, Patterns of coherence and incoherence, Patterns of Dynamics Conference in Honor of Bernold Fiedler, July 25  29, 2016, Free University of Berlin, Berlin, July 29, 2016.

O. Omel'chenko, Regular and irregular patterns of selflocalized excitation in arrays of coupled phase oscillators, International Conference on Control of Complex Systems and Networks, SFB 910 ``Control of SelfOrganizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Heringsdorf/Usedom, September 4  8, 2016.

O. Omel'chenko, Spike solutions to singularly perturbed elliptic problems, The 13th Annual Workshop on Numerical Methods for Problems with Layer Phenomena, Lomonosov Moscow State University, Russian Federation, April 7, 2016.

A. Vladimirov, Delay differential equation models of frequency swept laser light sources, International Conference on Structural Nonlinear Dynamics and Diagnosis (CSNDD'2016), University of Hassan II Casablanca, Marrakech, Morocco, May 24, 2016.

A. Vladimirov, Interaction of spatial and temporal cavity solitons in modelocked lasers and passive cavities, 17th International Conference ``Laser Optics 2016'', June 27  July 1, 2016, Saint Petersburg, Russian Federation, June 29, 2016.

A.G. Vladimirov, Distributed delay differential equation models in laser dynamics, Volga Neuroscience Meeting 2016, July 24  30, 2016, from Saint Petersburg to Nizhny Novgorod, Russian Federation, July 28, 2016.

A.G. Vladimirov, Interaction of temporal cavity solitons in driven fiber resonators and modelocked lasers, International Tandem Workshop on Pattern Dynamics in Nonlinear Optical Cavities, August 15  19, 2016, MaxPlanckInstitut für Physik komplexer Systeme, Dresden, August 15, 2016.

A.G. Vladimirov, Nonlinear dynamics of a frequency swept laser, Quantum Optics Seminar, SaintPetersburg State University, SaintPetersburg, Russian Federation, January 12, 2016.

M. Wolfrum, Emergence of collective behavior in coupled oscillator systems, Workshop ''Dynamics in Networks with Special Properties'', January 25  29, 2016, Mathematical Biosciences Institute (MBI), Columbus, USA, January 27, 2016.

M. Wolfrum, Emergence of collective behavior in coupled oscillator systems, Wednesdays@NICO, Northwestern University, Northwestern Institute on Complex Systems, Evanston, USA, January 20, 2016.

M. Wolfrum, Synchronization transitions in systems of coupled phase oscillators, Workshop on Synchronization and Oscillators with Generalized Coupling, April 20  22, 2016, University of Exeter, UK, April 21, 2016.

M. Wolfrum, Synchronization transitions in systems of coupled phase oscillators, Arbeitsgruppenseminar ``Chemische Physik fern vom Gleichgewicht'', Technische Universität München, Fachbereich Physik, March 18, 2016.

M. Wolfrum, Synchronization transitions in systems of coupled phase oscillators, Oberseminar Angewandte Mathematik, Westfälische WilhelmsUniversität Münster, Fachbereich Mathematik und Informatik, June 22, 2016.

L. Lücken, Interplay of noise and synaptic plasticity in coupled neural oscillators, Workshop ``Dynamics and Stability of Interacting Nonlinear Oscillators and their Applications'', WIAS Berlin, Berlin, February 13, 2015.

S. Yanchuk, Delayinduced patterns in a twodimensional lattice of coupled oscillators, 10th Colloquium on the Qualitative Theory of Differential Equations, July 1  4, 2015, University of Szeged, Bolyai Institute, Szeged, Hungary, July 4, 2015.

S. Yanchuk, How time delays influence dynamics, School of the International Research Training Group 1740 ``Dynamical Phenomena in Complex Networks'', July 20  21, 2015, HumboldtUniversität zu Berlin, Institut für Physik, Berlin, July 20, 2015.

U. Bandelow, Limitation for ultrashort solitons in nonlinear optical fibers by cusp formation, CLEO/EuropeEQEC 2015 Conference, June 21  25, 2015, München, June 25, 2015.

O. Omel'chenko, Chimera states in systems with control, EQUADIFF 2015, Minisymposium 3 ``Coupled Oscillator Systems and their MeanField Dynamics'', July 6  10, 2015, Lyon, France, July 9, 2015.

O. Omel'chenko, Creative control of chimera states, XXXV Dynamics Days Europe 2015, Minisymposium ``Controlling Complex Networks: Interplay of Structure, Noise, and Delay'', September 6  11, 2015, University of Exeter, Centre for Systems, Dynamics and Control, UK, September 9, 2015.

O. Omel'chenko, Paradoxes of the Kuramoto model, Seminar of the Department of Mathematics, Lomonosov Moscow State University, Russian Federation, November 25, 2015.

A.G. Vladimirov, Application of delay differential equations to the analysis of nonlinear dynamics in modelocked lasers, Colloquium Nonlinear Sciences, Universität Münster, Center for Nonlinear Sciences, May 19, 2015.

M. Wolfrum, Chimera states with global feedback, Workshop on Control of SelfOrganizing Nonlinear Systems, Wittenberg, September 14  16, 2015.

M. Wolfrum, Nonuniversal transitions to synchrony in the SakaguchiKuramoto model, International Workshop on Dynamics of Coupled Oscillators: 40 Years of the Kuramoto Model, July 27  31, 2015, MaxPlanckInstitut fúr Physik Komplexer Systeme, Dresden, July 30, 2015.

S. Amiranashvili, Elementary processes behind turbulent states in optical fibers, Weak Chaos and Weak Turbulence, February 3  7, 2014, MaxPlanckInstitut für Physik komplexer Systeme, Dresden, February 5, 2014.

S. Amiranashvili, Extreme waves in optical fibers, Wave Interaction (WIN2014), April 23  26, 2014, Johannes Kepler University, Linz, Austria, April 24, 2014.

S. Amiranashvili, Solitons who do not want to be too short, Workshop on Abnormal Wave Events (WAWE2014), June 5  6, 2014, Nice, France, June 5, 2014.

U. Bandelow, Basic equations of classical soliton theory: Solutions and applications, BMSWIAS Summer School ``Applied Analysis for Materials'', August 25  September 5, 2014, Berlin Mathematical School, Technische Universität Berlin.

O. Omel'chenko, Bifurcation analysis of chimera states, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 13: Nonlocally Coupled Dynamical Systems: Analysis and Applications, July 7  11, 2014, Madrid, Spain, July 7, 2014.

O. Omel'chenko, Eckhaus scenario for partially coherent twisted states in arrays of coupled phase oscillators, XXXIV Dynamics Days Europe, September 7  12, 2014, Bayreuth, September 11, 2014.

O. Omel'chenko, Spike solutions to singularly perturbed elliptic problems, Workshop ``Modern Problems of Mathematical Physics'', November 28  29, 2014, Lomonosov Moscow State University, Russian Federation, November 28, 2014.

A.G. Vladimirov, Delay differential equations in laser dynamics, International ConferenceSchool Hamiltonian Dynamics, Nonautonomous Systems, and Patterns in PDE's, December 10  15, 2014, Nishni Novgorod, Russian Federation, December 14, 2014.

M. Wolfrum, Stabilizing chimera states by feedback control, Colloquium ``Applications of Dynamical Networks'' of the Collaborative Research Center 910, Technische Universität Berlin, June 20, 2014.

S. Amiranashvili, Solitons that are too short in duration, International Workshop: Extreme Nonlinear Optics & Solitons, October 28  30, 13, WIAS Berlin, October 28, 2013.

O. Omel'chenko, Thermodynamic limit approach for bifurcation analysis of chimera states, Forschungsseminar ``Dynamische Systeme'', Freie Universität Berlin, January 10, 2013.

O. Omel'chenko, M. Wolfrum, Generalizing the OttAntonsen method for coupled phase oscillators, Dynamics Days BerlinBrandenburg, October 1  2, 2013, Technische Universität Berlin, October 2, 2013.

O. Omel'chenko, Nonuniversal transitions to synchrony in the SakaguchiKuramoto model, XXXIII Dynamics Days Europe, Minisymposium MS6 ``Collective Behavior in Networks of Oscillators'', June 3  7, 2013, Madrid, Spain, June 4, 2013.

O. Omel'chenko, Nonuniversal transitions to synchrony in the SakaguchiKuramoto model, SIAM Conference on Applications of Dynamical Systems (DS13), May 19  23, 2013, Snowbird, USA, May 22, 2013.

O. Omel'chenko, Synchronization phenomena in large size systems of coupled oscillators, Bogolyubov Readings DIF2013 ``Differential Equations, Theory of Functions and Their Applications'', June 23  30, 2013, Sevastopol, Ukraine, June 24, 2013.

M. Wolfrum, The Turing bifurcation on networks: Collective patterns and single differentiated nodes, International Conference on Dynamics of Differential Equations, March 16  20, 2013, Georgia Institute of Technology, Atlanta, USA, March 18, 2013.

M. Wolfrum, Chimera states: Patterns of coherence and incoherence in coupled oscillator systems, Dynamical Systems and Mathematical Biology Seminar, Georgia State University, Atlanta, USA, March 12, 2013.

M. Wolfrum, The Turing bifurcation on networks: Collective patterns and differentiated nodes, Kolloquium SFB 910, Technische Universität Berlin, Berlin, January 11, 2013.

M. Wolfrum, The Turing bifurcation on networks: Collective patterns and single differentiated nodes, Applied Dynamics Seminar, University of Maryland, Washington, USA, March 7, 2013.

S. Amiranashvili, Tiny waves we should never ignore, OSA  The Optical Society, Topical Meeting ``Nonlinear Photonics'', June 17  21, 2012, Colorado Springs, USA, June 18, 2012.

R. Arkhipov, M. Radziunas, A. Vladimirov, Theoretical analysis of hybrid modelocked quantum dot semiconductor lasers, International Conference ``Laser Optics 2012'', St. Petersburg, Russian Federation, June 25  29, 2012.

R. Arkhipov, M.V. Arkhipov, S.A. Pulkin, Numerical simulations of lasing without population inversion in twolevel optically dense medium, International Conference ``Laser Optics 2012'', St. Petersburg, Russian Federation, June 25  29, 2012.

R. Arkhipov, Hybrid modelocking in semiconductor quantum dot lasers: Simulation, analysis and comparison with experiments, ITN PROPHET MidTerm Review Meeting, October 9  11, 2012, Paris, France, October 11, 2012.

R. Arkhipov, Numerical analysis of hybrid modelocking in semiconductor quantum dot lasers, XIV AllRussian Scientific SchoolSeminar ``Wave Phenomena in Inhomogeneous Media'' (Waves2012), Zvenigorod, Russian Federation, May 21  26, 2012.

R. Arkhipov, Spectral and temporal characteristics of resonant medium radiation excited at the superluminal velocity, International Symposium Advances in Nonlinear Photonics, September 23  27, 2012, Belarusian State University, Minsk, Belarus, September 26, 2012.

R. Arkhipov, The new principle of the alloptical streak camera based on ultrafast laser beam deflection by lightinduced coherent photonic crystal, International Symposium Advances in Nonlinear Photonics, September 23  27, 2012, Belarusian State University, Minsk, Belarus, September 25, 2012.

R. Arkhipov, Theoretical investigation of hybrid modelocking in twosection semiconductor quantum dot lasers, International Symposium Advances in Nonlinear Photonics, September 23  27, 2012, Belarusian State University, Minsk, Belarus, September 24, 2012.

I. Babushkin, Emission and control of coherent broadband THz radiation using plasmagenerating femtosecond light pulses, IPHTKolloquium, Institut für Photonische Technologien (IPHT), Jena, November 20, 2012.

O. Omel'chenko, Bifurcation analysis of chimera states, International Workshop: Coupled Networks, Patterns and Complexity, WIAS Berlin, November 21, 2012.

O. Omel'chenko, Chimera states: Spatiotemporal patterns of synchrony and disorder, Universität Hamburg, Department of Mathematics, November 12, 2012.

O. Omel'chenko, Coherenceincoherence patterns in systems of nonlocally coupled phase oscillators, Statistical Physics and Nonlinear Dynamics & Stochastic Processes, HumboldtUniversität zu Berlin, Institut für Physik, Berlin, June 18, 2012.

O. Omel'chenko, Nonuniversal transitions to synchrony in the SakaguchiKuramoto model, Seminar Applied Analysis, HumboldtUniversität zu Berlin, October 29, 2012.

O. Omel'chenko, What are chimera states, Westfälische WilhelmsUniversität Münster, Center for Nonlinear Science, November 6, 2012.

O. Omel'chenko, Coherenceincoherence patterns in systems of nonlocally coupled phase oscillators, XXXII Dynamics Days Europe, September 2  7, 2012, University of Gothenburg, Sweden, September 4, 2012.

O. Omel'chenko, Synchronization transition in the SakaguchiKuramoto model, 7th Crimean School and Workshop ``Emergent Dynamics of Oscillatory Networks'', May 20  27, 2012, Mellas, Crimea, Ukraine, May 22, 2012.

M. Wolfrum, Chimera states: Patterns of coherence and incoherence in coupled oscillator systems, Workshop ``Dynamics of Patterns'', December 16  21, 2012, Mathematisches Forschungsinstitut Oberwolfach, December 21, 2012.

M. Wolfrum, The Turing instability in irregular network systems, Jahrestagung der Deutschen MathematikerVereinigung (DMV) 2012, Minisymposium ``Dynamical Systems'', September 18  20, 2012, Universität des Saarlandes, Fakultät für Mathematik und Informatik, Saarbrücken, September 20, 2012.

T. Girnyk, Two groups of globally coupled Kuramoto oscillators, Uni Potsdam, April 11, 2011.

O. Omelchenko, What does thermodynamic limit tell us about Chimera states?, SIAM Conference on Applications of Dynamical Systems (DS11), May 22  26, 2011, Society for Industrial and Applied Mathematics, Snowbird, Utah, USA, May 26, 2011.

V. Tronciu, Semiconductor lasers  Key elements for chaos based communication systems, Università di Pavia, Ph.D. School of Electrical and Electronic Engineering and Computer Science, Italy, September 23, 2011.

M. Wolfrum, Mechanisms of semistrong pulse interaction in the Schnakenberg model, Seminar z kvalitativnej teorie diferencialnych rovnic, Comenius University, Bratislava, Slovakia, November 10, 2011.

M. Wolfrum, Stability properties of equilibria and periodic solutions in systems with large delay, Equadiff 2011, August 1  5, 2011, University of Loughborough, UK, August 2, 2011.

M. Wolfrum, Stability properties of equilibria and periodic solutions in systems with large delay, The Sixth International Conference on Differential and Functional Differential Equations (DFDE 2011), August 17  21, 2011, Steklov Mathematical Institute, Moscow, Russian Federation, August 19, 2011.

J. Ehrt, Cascades of heteroclinic connections in viscous balance laws, 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, May 25  28, 2010, Technische Universität Dresden, May 27, 2010.

A.G. Vladimirov, Interaction of dissipative solitons and pulses in laser systems, Université Libre de Bruxelles, Optique Nonlinéaire Théorique, Belgium, April 21, 2010.

A.G. Vladimirov, Localized structures of light and their interaction, Imperial College London, Department of Applied Mathematics, UK, April 27, 2010.

A.G. Vladimirov, Nonlinear dynamics in lasers, Technische Universität Berlin, Institut für Festkörperphysik, March 24, 2010.

T. Girnyk, Multistability of twisted states in nonlocally coupled Kuramototype models, Universität Potsdam, Institut für Physik und Astronomie, October 25, 2010.

T. Girnyk, Multistability of twisted states in nonlocally coupled Kuramototype models, École Polytechnique Fédérale de Lausanne, Laboratory of Nonlinear Systems (EPFLLANOS), Switzerland, November 17, 2010.

T. Girnyk, Stability of twisted states in repulsive Kuramoto models, Research Group Seminar, Freie Universität Berlin, research group ``Nonlinear Dynamics'', December 2, 2010.

M. Lichtner, Stability of delay differential equations with large delay, Dynamical System Seminar, Portsmouth University, Department of Mathematics, UK, March 17, 2010.

O.E. Omel'chenko, Coupling and motion of chimera states, Humboldt Kolleg Ukraine ``Mathematics and Life Sciences: Possibilities, Interlacements and Limits'', August 5  8, 2010, Kiev, Ukraine, August 7, 2010.

O.E. Omel'chenko, Dynamical properties of chimera states, Dynamics Days Europe, September 6  10, 2010, University of Bristol, Department of Engineering Mathematics, UK, September 6, 2010.

O.E. Omel'chenko, Moving chimera states, International Workshop ``Nonlinear Dynamics on Networks'', July 5  9, 2010, National Academy of Sciences of Ukraine, Kiev, July 6, 2010.

O.E. Omel'chenko, On the dynamical nature of chimera states, The 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, May 25  28, 2010, Technische Universität Dresden, May 25, 2010.

M. Wolfrum, Chimera states are chaotic transients, International Workshop ``Nonlinear Dynamics on Networks'', July 5  9, 2010, National Academy of Sciences of Ukraine, Kiev, July 6, 2010.

M. Wolfrum, Mechanisms of semistrong pulse interaction in the Schnakenberg model, Emerging Topics in Dynamical Systems and Partial Differential Equations (DSPDEs'10), May 31  June 4, 2010, International Center for Numerical Methods in Engineering, Barcelona, Spain, June 1, 2010.

M. Wolfrum, Mechanisms of semistrong pulse interaction in the Schnakenberg model, Localized Structures in Dissipative Nonlinear Systems, October 18  20, 2010, WIAS, October 19, 2010.

M. Wolfrum, Routes to complex dynamics in a ring of unidirectionally coupled systems, Dynamics Days Europe 2010, September 6  10, 2010, University of Bristol, UK, September 7, 2010.

M. Wolfrum, Scaling properties of the spectrum for DDEs with large delay, Applied Maths Seminar, University of Exeter, Institute of Applied Mathematics, UK, November 22, 2010.

M. Wolfrum, Scaling properties of the spectrum for ODEs with large delay, 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, May 25  28, 2010, Technische Universität Dresden, May 25, 2010.

A.G. Vladimirov, Enhancement of interaction of dissipative solitons above selfpulsing instability threshold, CPNLW09 Soliton 2009 ``Solitons in Their Roaring Forties: Coherence and Persistence in Nonlinear Waves'', January 6  9, 2009, Nice University, Nice, France, January 8, 2009.

A.G. Vladimirov, Spontaneous motion of dissipative solitons under the effect of delay, Australasian Conference on Optics, Lasers and Spectroscopy and Australian Conference on Optical Fibre Technology in association with the International Workshop on Dissipative Solitons (ACOLS ACOFT DS 2009), November 29  December 3, 2009, University of Adelaide, Australia, December 1, 2009.

A.G. Vladimirov, Strong enhancement of interaction of optical pulses induced by oscillatory instability, European Conference on Lasers and ElectroOptics and the XIth European Quantum Electronics Conference 2009 (CLEOtextsuperscript®/Europe  EQEC 2009, Munich, June 14  19, 2009.

U. Bandelow, Semiconductor laser instabilities and dynamics (short course), 9th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD) 2009, September 14  18, 2009, Gwangju Institute of Science and Technology (GIST), Korea (Republic of), September 16, 2009.

M. Wolfrum, Asymptotic properties of the Floquet spectrum for delay differential equations with large delay, Seminario ISC, Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Florence, Italy, April 30, 2009.

M. Wolfrum, Delay differential equations with large delay, Symposium ``Evolution Equations, Related Topics and Applications'', September 9  11, 2009, Helmholtz Zentrum München, September 9, 2009.

M. Wolfrum, The Eckhaus scenario in delay differential equation with large delay, International Workshop ``Trends in Bifurcation Analysis: Methods and Applications (TBA 2009)'', June 3  5, 2009, Milan, Italy, June 5, 2009.

J. Ehrt, Normally hyperbolic manifolds for viscous balance laws, Work Seminar ``Modelling, Analysis and Simulation'', Centrum voor Wiskunde & Informatica, Amsterdam, Netherlands, August 22, 2008.

J. Ehrt, Semistrong interaction of pulses, Work Seminar ``Modelling, Analysis and Simulation'', Centrum voor Wiskunde & Informatica, Amsterdam, Netherlands, October 23, 2008.

S. Yanchuk, Bifurcations in lattices of unidirectionally coupled oscillators, Jour fixe, Graduiertenkolleg ``Analysis, Numerics, and Optimization of Multiphase Problems'', HumboldtUniversität zu Berlin, April 17, 2008.

S. Yanchuk, Destabilization in chains of coupled oscillators, Seminar of Work Group ``Neuromodulation'', Forschungszentrum Jülich, Institut für Neurowissenschaften und Biophysik, Teilinstitut Medizin, April 29, 2008.

U. Bandelow, Modeling and analysis of masteroscillator poweramplifier seminconductor lasers, University of Washington, Seattle, USA, October 16, 2008.

U. Bandelow, Short pulses in nonlinear optical fibers: Models and applications, Colloquium ``Nonlinear Dynamics in Complex Optical Systems'', HumboldtUniversität zu Berlin, Institut für Physik, June 19, 2008.

M. Wolfrum, Delaydifferential equations with large delay, Seminar of the Working Group ``Dynamische Systeme'', Universität Hamburg, Department Mathematik, January 16, 2008.

M. Wolfrum, The Eckhaus scenario in delaydifferential equations with large delay, Workshop ``Dynamics of Patterns'', December 14  20, 2008, Mathematisches Forschungsinstitut Oberwolfach, December 19, 2008.

S. Yanchuk, Eckhaus instability in systems with large delay, International Conference on Differential Equations (EQUADIFF 07), August 5  11, 2007, Vienna University of Technology, Austria, August 7, 2007.

S. Yanchuk, How size of a large system effects its dynamics?, European Conference on Complex Systems, October 1  6, 2007, Dresden, October 4, 2007.

U. Bandelow, Efficient modeling and analysis of dynamical effects in semiconductor laser devices, University of Nottingham, George Green Institute, UK, July 6, 2007.

U. Bandelow, Feedback enhanced modulation bandwidth, Dynamics Days Europe, University of Loughborough, UK, July 12, 2007.

U. Bandelow, Nichtlineare Effekte in Halbleiterlasern und optischen Fasern, Habilitandenkolloquium, HumboldtUniversität zu Berlin, Institut für Physik, April 17, 2007.

U. Bandelow, Semiconductor laser instabilities and dynamics (Short Course SC 0702), 7th International Conference ``Numerical Simulation of Optoelectronic Devices'' (NUSOD'07), University of Delaware, Newark, USA, September 25, 2007.

J. Ehrt, Slowmotion of multipulse solutions in reactiondiffusion systems by semistrong interaction, International Conference on Differential Equations (EQUADIFF 07), August 5  11, 2007, Vienna University of Technology, Austria, August 7, 2007.

M. Lichtner, Invariant manifold theorem for semilinear hyperbolic systems, EQUADIFF 07, August 5  11, 2007, Technische Universität Wien, Austria, August 7, 2007.

M. Wolfrum, Delay differential equations with large delay, Dynamical Systems Seminar, University of Minnesota, School of Mathematics, Minneapolis, USA, March 5, 2007.

S. Yanchuk, Amplitude equations for delay differential equations with large delay, Research Seminar Applied Analysis, Humboldt University of Berlin, Institute of Mathematics, April 27, 2006.

S. Yanchuk, Bifurcation theory for singularly perturbed systems with delay, National Academy of Sciences of Ukraine, Institute of Mathematics, Kiev, October 16, 2006.

S. Yanchuk, Bifurcations in systems with large delay, National Academy of Sciences of Ukraine, Institute of Mathematics, Kiev, May 29, 2006.

S. Yanchuk, Bifurcations in systems with long delay, Seminar of the Magnetoencephalography (MEG) Group, Research Center Jülich, Institute of Medicine, April 19, 2006.

S. Yanchuk, Hopf bifurcation for systems with large delay, Workshop ``Complex Dynamics and Delay Effects in Coupled Systems'', September 11  13, 2006, HumboldtUniversität zu Berlin, September 11, 2006.

S. Yanchuk, Typical instabilities in systems with large delay, 6th Crimean School and Workshops ``Nonlinear Dynamics, Chaos and Applications'', May 15  26, 2006, Yalta, Crimea, Ukraine, May 24, 2006.

U. Bandelow, Modeling and simulation of optoelectronic devices, Kickoff Workshop ``Materials in New Light'', HumboldtUniversität zu Berlin, Institut für Physik, Berlin, January 6, 2006.

U. Bandelow, Modellierung und Simulation optoelektronischer Bauelemente, Berliner Industriegespräche, Deutsche Physikalische Gesellschaft, MagnusHaus, Berlin, September 6, 2006.

U. Bandelow, Simulation and analysis of spatiotemporal effects in complex laser structures, Kickoff Workshop ``Materials in New Light'', HumboldtUniversität zu Berlin, Institut für Physik, Berlin, January 6, 2006.

M. Lichtner, A spectral gap mapping theorem and smooth invariant center manifolds for semilinear hyperbolic systems, 6th AIMS International Conference on Dynamical Systems, Differential Equations & Applications, June 25  28, 2006, Université de Poitiers, France, June 28, 2006.

A. Vladimirov, Dynamics of light pulses in modelocked lasers, 6th Crimean School and Workshops ``Nonlinear Dynamics, Chaos and Applications'', May 15  26, 2006, Yalta, Crimea, Ukraine, May 20, 2006.

A. Vladimirov, Laser dissipative solitons and their interaction, Minisymposium on Dissipative Solitons, WIAS, Berlin, April 20, 2006.

A. Vladimirov, Localized structures of light in laser systems and their weak interactions, Technische Universität Berlin, June 14, 2006.

A. Vladimirov, Nonlinear dynamics and bifurcations in multimode and spatially distributed laser systems, June 20  23, 2006, St. Petersburg State University, Russian Federation, June 20, 2006.

A. Vladimirov, Nonlinear dynamics in multimode and spatially extended laser systems, Moscow State University, Physics Faculty, Russian Federation, November 10, 2006.

A. Vladimirov, Transverse Bragg dissipative solitons in a Kerr cavity with refractive index modulation, Laser Optics Conference, June 26  30, 2006, St. Petersburg, Russian Federation, June 28, 2006.

M. Wolfrum, Describing a class of global attractors via symbol sequences, 6th AIMS International Conference on Dynamical Systems, Differential Equations & Applications, June 25  28, 2006, Université de Poitiers, France, June 28, 2006.

M. Wolfrum, Dynamics of chemical systems with mass action kinetics, Colloquium in Memory of Karin Gatermann, Universität Hamburg, Fachbereich Mathematik, January 7, 2006.

M. Wolfrum, Instabilities of laser systems with delay, 6th Crimean School and Workshops ``Nonlinear Dynamics, Chaos and Applications'', May 15  26, 2006, Yalta, Crimea, Ukraine, May 19, 2006.

M. Wolfrum, Systems of delay differential equations with large delay, Seminario do Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, Departamento de Matemática, Lisbon, Portugal, March 28, 2006.

A. Vladimirov, Interaction of dissipative solitons in laser systems, Ben Gurion University of the Negev, Department of Mathematics, Beer Sheva, Israel, November 17, 2005.

A. Vladimirov, Theoretical analysis of dynamical instabilities in a modelocked semiconductor laser, Workshop ``Nonlinear Dynamics in Photonics'', May 2  4, 2005, WIAS, Berlin, May 3, 2005.

S. Yanchuk, Appearance of patterns in delay coupled laser arrays, Universität Potsdam, January 31, 2005.

S. Yanchuk, Bifurcations in systems with large delay, SFB 555 Symposium, FritzHaberInstitut der MaxPlanckGesellschaft, Berlin, May 27, 2005.

S. Yanchuk, Instabilities of equililbria of delaydifferential equations with large delay, ENOC 2005 (EUROMECH Nonlinear Oscillations Conference), August 7  12, 2005, Eindhoven, Netherlands, August 9, 2005.

S. Yanchuk, Normal forms for systems with large delay, National Academy of Sciences of Ukraine, Institute of Mathematics, Kiev, October 10, 2005.

S. Yanchuk, Properties of the LangKobayashi model with large delay, Workshop ``Nonlinear Dynamics in Photonics'', May 2  4, 2005, WIAS, Berlin, May 2, 2005.

M. Nizette, A. Vladimirov, M. Wolfrum, D. Rachinskii, Delay differential equations for passive modelocking, International Quantum Electronics Conference, München, June 12  17, 2005.

D. Turaev, S. Zelik, A. Vladimirov, Chaotic bound state of localized structures in the complex GinzburgLandau equation, Conference Digest ``Nonlinear Guided Waves and their Applications'', Dresden, September 6  9, 2005.

U. Bandelow, Analyse dynamischer Effekte in Optoelektronik und Photonik, Institutsseminar, FerdinandBraunInstitut für Höchstfrequenztechnik, Berlin, December 9, 2005.

V. Tronciu, Resonant coupling of a semiconductor laser to a FabryPerot resonator, Minisymposium ``Laser + Resonator'', WIAS, Berlin, February 17, 2005.

M. Wolfrum, Systems of delay differential equations with large delay, OttovonGuerickeUniversität Magdeburg, Institut für Analysis und Numerik, June 14, 2005.

S. Yanchuk, Dynamics of two F2F coupled lasers: Instantaneous coupling limit, SPIE Photonics Europe 2004 Conference ``Semiconductor Lasers and Laser Dynamics'', April 27  30, 2004, Strasbourg, France, April 28, 2004.

S. Yanchuk, Intermittent synchronization in a system of coupled lasers, WIAS Workshop ``Synchronization and Highdimensional Chaos in Coupled Systems'', November 15  16, 2004, Berlin, November 15, 2004.

S. Yanchuk, Pattern formation in systems with large delay, Seminar ``Synchronization and Chaos'', National Academy of Sciences of Ukraine, Institute of Mathematics, Kiev, December 28, 2004.

S. Yanchuk, Singularly perturbed delaydifferential equations. What do they have in common with ODEs and maps?, Seminar ``Nonlinear Oscillations'', National Academy of Sciences of Ukraine, Institute of Mathematics, Kiev, July 12, 2004.

U. Bandelow, 40 GHz modelocked semiconductor lasers: Theory, simulation and experiments, Annual Meeting 2004 of the Optical Society of America (OSA) ``Frontiers in Optics'', October 10  14, 2004, Rochester, USA, October 11, 2004.

K.R. Schneider, Invariant manifolds for random dynamical systems with two time scales, Moscow State University, Faculty of Physics, Russian Federation, September 16, 2004.

K.R. Schneider, Invariante Mannigfaltigkeiten für zufällige dynamische Systeme mit schnellen und langsamen Variablen, Workshop GAMMFachausschuss "`Dynamik und Regelungstheorie"' und VDI/VDEGMAAusschuss 1.40 "`Theoretische Verfahren der Regelungstechnik"', Universität Kassel, Regelungstechnik und Systemdynamik, March 8, 2004.

K.R. Schneider, Systeme mit schnellen und langsamen Variablen unter zufälligen Einwirkungen, Colloquium ``Singularly Disturbed Systems and Complex Dynamics'', June 16, 2004, Moscow State University, Faculty of Physics, Russian Federation, June 16, 2004.

U. Bandelow, Report on WIAS activities concerning COST Action 288, Kickoff Meeting for the Cost Action 288, COST TIST Secretariat, Brussels, Belgium, April 7, 2003.

U. Bandelow, Simulation of modelocked lasers based on a distributed timedomain model, WIAS Workshop ``Dynamics of Semiconductor Lasers'', September 15  17, 2003, Berlin, September 17, 2003.

K.R. Schneider, Canard solutions of finite and infinitedimensional dynamical systems, Moscow State University, Faculty of Physics, Russian Federation, October 1, 2003.

K.R. Schneider, Complete synchronization of nearly identical systems, WIAS Workshop ``Dynamical Systems, Synchronization, Lasers'', February 26  27, 2003, Berlin, February 26, 2003.

K.R. Schneider, Immediate and delayed exchange of stabilities, Belarussian State University, Institute for Mathematics, Minsk, November 18, 2003.

K.R. Schneider, Slow invariant manifold for a random dynamical system with two timescales, EQUADIFF 2003, July 21  26, 2003, Hasselt, Belgium, July 25, 2003.

A. Vladimirov, Moving discrete solitons in multicore fibers and waveguide arrays, European Quantum Electronics Conference, June 22  27, 2003, München, June 25, 2003.

A. Vladimirov, Moving discrete solitons in multicore fibers and waveguide arrays, Conference dedicated to the 60th birthday of Prof. Paul Mandel, April 11  12, 2003, Université Libre de Bruxelles, Optique Nonlinéaire Théorique, Belgium, April 11, 2003.

M. Wolfrum, Attractors of semilinear parabolic equations on the circle, Dynamics of Structured Systems, December 14  20, 2003, Mathematisches Forschungszentrum Oberwolfach, December 16, 2003.

M. Wolfrum, Heteroclinic connections between rotating waves of scalar parabolic equations on the circle, EQUADIFF 2003, July 22  26, 2003, Hasselt, Belgium, July 23, 2003.

S. Yanchuk, Synchronization of two mutually coupled semiconductor lasers: Instantaneous coupling limit, WIAS Workshop ``Dynamics of Semiconductor Lasers'', September 15  17, 2003, Berlin, September 16, 2003.

S. Yanchuk, Synchronization phenomena in semiconductor laser, Sfb 555 Workshop ``Complex Nonlinear Processes'', September 11  13, 2003, Berlin, September 12, 2003.

S. Yanchuk, Complete synchronization of symmetrically coupled autonomous systems, EQUADIFF 2003, July 22  26, 2003, Hasselt, Belgium, July 25, 2003.

S. Yanchuk, Forced periodic frequency locking: Poincaré mapping approach, WIAS Workshop ``Dynamical Systems, Synchronization, Lasers'', February 26  27, 2003, Berlin, February 27, 2003.

S. Yanchuk, Synchronization of coupled autonomous systems, National Academy of Sciences of Ukraine, Institute of Mathematics, Kiev, April 21, 2003.

S. Yanchuk, Synchronization of two coupled LangKobayashi systems, National Institute of Applied Optics, Florence, Italy, May 7, 2003.

S. Yanchuk, Synchronization problem in twosection semiconductor lasers, Forschungsseminar ``Angewandte Analysis'', HumboldtUniversität zu Berlin, Institut für Mathematik, July 7, 2003.

M. Wolfrum, Heteroclinic connections and order structures for scalar parabolic PDE, Instituto Superior Tecnico, Lisbon, Portugal, June 11, 2002.
External Preprints

S. Slepneva, B. O'Shaughnessy, A.G. Vladimirov, S. Rica, G. Huyet, Turbulent laser puffs, Preprint no. arXiv:1801.05509, Cornell University Library, 2018.
Abstract
The destabilisation of laminar flows and the development of turbulence has remained a central problem in fluid dynamics since Reynolds' studies in the 19th century. Turbulence is usually associated with complex fluid motions and most of the studies have so far been carried out using liquids or gases. Nevertheless, on a theoretical viewpoint, turbulence may also arise in a wide range of fields such as biology and optics. Here we report the results of experimental and theoretical investigation of the characteristic features of laminarturbulent transition in a long laser commonly used as a light source in medical imaging and sensing applications. This laminar to turbulence transition in the laser light is characterized by the appearance of turbulent puffs similar to those commonly observed in pipe flows and is accompanied by a loss of coherence and limits the range of applications. We present both experimental results and numerical simulations demonstrating that this transition is mediated by the appearance of a convective instability where localised structures develop into drifting bursts of turbulence, in complete analogy with spots, swirls and other structures in hydrodynamic turbulence 
D.R.M. Renger, P. Koltai , From large deviations to transport semidistances: Coherence analysis for finite Lagrangian data, Preprint no. arXiv:1709.02352, Cornell University Library, arXiv.org, 2017.
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic timereversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradientflow, steepestascent, or maximalentropyproduction equation. Onsager's original theorem is limited to closetoequilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic timereversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows. 
A.G. Vladimirov, S.V. Gurevich, M. Tlidi, Effect of Cherenkov radiation on localized states interaction, Preprint no. arXiv:1707.04458, Cornell University Library, arXiv.org, 2017.
Abstract
We study theoretically the interaction of temporal localized states in all fiber cavities and microresonatorbased optical frequency comb generators. We show that Cherenkov radiation emitted in the presence of third order dispersion breaks the symmetry of their interaction and greatly enlarges the interaction range thus facilitating the experimental observation of the soliton bound states. Analytical derivation of the reduced equations governing slow time evolution of the positions of two interacting localized states in the LugiatoLefever model with third order dispersion term is performed. Numerical solutions of the model equation are in close agreement with analytical predictions. 
P. Kravetc, D. Rachinskii, A.G. Vladimirov, Pulsating dynamics of slowfast population models with delay, Preprint no. arxiv.org:1601.06452, Cornell University Library, arXiv.org, 2016.
Abstract
We discuss a bifurcation scenario which creates periodic pulsating solutions in slowfast delayed systems through a cascade of almost simultaneous Hopf bifurcations. This scenario has been previously associated with formation of pulses in a delayed model of modelocked semiconductor lasers. In this work, through a case study of several examples, we establish that a cascade of Hopf bifurcations can produce periodic pulses, with a period close to the delay time, in population dynamics models and explore the conditions that ensure the realization of this scenario near a transcritical bifurcation threshold. We derive asymptotic approximations for the pulsating solution and consider scaling of the solution and its period with the small parameter that measures the ratio of the time scales. The role of competition for the realization of the bifurcation scenario is highlighted. 
I. Omelchenko, O. Omel'chenko, P. Hövel, E. Schöll, Multichimera states in FitzHughNagumo oscillators, Preprint no. arXiv:1212.3190, Cornell University Library, 2012.
Abstract
We demonstrate the existence of chimera states in a ring of identical oscillators described by FitzHughNagumo equations with nonlocal coupling. This class of elements serves as a paradigmatic model in neuroscience, chemical oscillations, and nonlinear electronic circuits. Applying a phasereduction technique we show that offdiagonal nonlocal coupling is a crucial factor for the appearance of chimera states, which consist of coexisting domains of coherent (phaselocked) and incoherent oscillators. Surprisingly, we find that for increasing coupling strength classical chimera states undergo transitions from one to multiple domains of incoherence. This additional spatial modulation is due to strong coupling interaction and thus cannot be observed in simple phaseoscillator models. 
P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, P. Mosiolek, T. Kapitaniak, Routes to complex dynamics in a ring of unidirectionally coupled systems, Preprint no. 667, DFG Research Center sc Matheon, 2009.
Abstract
We study the dynamics of a ring of unidirectionally coupled autonomous Duffing oscillators. Starting from a situation where the individual oscillator without coupling has only trivial equilibrium dynamics, the coupling induces complicated transitions to periodic, quasiperiodic, chaotic, and hyperchaotic behavior. We study these transitions in detail for small and large numbers of oscillators. Particular attention is paid to the role of unstable periodic solutions for the appearance of chaotic rotating waves, spatiotemporal structures and the Eckhaus effect for a large number of oscillators. Our analytical and numerical results are confirmed by a simple experiment based on the electronic implementation of coupled Duffing oscillators. 
M. Lichtner, M. Radziunas, Well posedness and smooth dependence for a semilinear hyperbolic system with nonsmooth data, Preprint no. 174, DFG Research Center sc Matheon, Technische Universität Berlin, 2004.