Mathematical Topic "Theory of dynamical systems"

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. 381-402, (chapter published).

B. Fiedler, C. Rocha, M. Wolfrum, A permutation characterization of Sturm global attractors of Hamiltonian type, J. Differential Equations, 252 (2012) pp. 588--623.
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. 617--659.
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 mode-locked semiconductor lasers subject to optical feedback, New J. Phys., 14 (2012) pp. 113033/1-113033/29.

T. Girnyk, M. Hasler, Y. Maistrenko, Multistability of twisted states in non-locally coupled Kuramoto-type models, Chaos Solitons Fractals, 22 (2012) pp. 013114/1--013114/10.
Abstract

A ring of N identical phase oscillators with interactions between L-nearest 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 free-running oscillators can be set to zero. The resulting system is of gradient type and therefore all its solutions converge to an equilibrium point. All so-called q-twisted 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 q-twisted states may become unstable and other q-twisted states may become stable. Finally, the existence of additional equilibrium states, called here multi-twisted 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 multi-twisted states grows exponentially as N -> inf. It is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discrete-time translational dynamical system where the space-variable (position on the ring) plays the role of time. The q-twisted states are then fixed points and the multi-twisted 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 Sakaguchi--Kuramoto model, Phys. Rev. Lett., 109 (2012) pp. 164101/1--164101/4.
Abstract

We investigate the transition to synchrony in a system of phase oscillators that are globally coupled with a phase lag (Sakaguchi-Kuramoto 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 Ott-Antonsen 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 two-dimensional arrays of non-locally coupled phase oscillators, Phys. Rev. E (3), 85 (2012) pp. 036210/1--036210/5.
Abstract

Recently it has been shown that large arrays of identical oscillators with non-local 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 two-dimensional 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, Phys. D, 241 (2012) pp. 1351--1357.
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 mean-field 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 codimension-two 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, Phys. Rev. A, 84 (2011) pp. 043848/1--043848/4.
Abstract

We study the relative stability of different one-dimensional (1D) and two-dimensional (2D) clusters of closely packed localized peaks of the Swift-Hohenberg 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 one-dimensional (1D) and two-dimensional (2D) clusters of closely packed localized peaks of the Swift-Hohenberg 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 small-area circular vertical cavity surface emitting lasers, Opt. Commun., 284 (2011) pp. 1299--1302.
Abstract

We investigate theoretically the dynamics of three low-order transverse modes in a small-area 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 J. Math. Anal., 43 (2011) pp. 788-802.

M. Wolfrum, O. Omel'chenko, Chimera states are chaotic transients, Phys. Rev. E (3), 84 (2011) pp. 015201/1-015201/4.
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 time-span 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 type-II supertransients in coupled map lattices.

M. Wolfrum, O. Omel'chenko, S. Yanchuk, Y. Maistrenko, Spectral properties of chimera states, Chaos, 21 (2011) pp. 0013112/1-013112/8.
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 mode-locked quantum dot laser, J. Opt. Soc. Amer. B Opt. Phys., 27 (2010) pp. 2102-2109.

P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, P. Mosiolek, T. Kapitaniak, Routes to complex dynamics in a ring of unidirectionally coupled systems, Chaos, 20 (2010) pp. 013111/1--013111/9.

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, Eur. Phys. J. D, 59 (2010) pp. 59-65.

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. 066201/1--066201/13.

O.E. Omel'chenko, M. Wolfrum, Y.L. Maistrenko, Chimera states as chaotic spatio-temporal patterns, Phys. Rev. E (3), 81 (2010) pp. 065201/1--065201/4.

M. Wolfrum, S. Yanchuk, A multiple time scale approach to the stability of external cavity modes in the Lang--Kobayashi system using the limit of large delay, SIAM J. Appl. Dyn. Syst., 9 (2010) pp. 519--535.

M. Wolfrum, S. Yanchuk, P. Hövel, E. Schöll, Complex dynamics in delay-differential equations with large delay, Eur. Phys. J. D., 191 (2010) pp. 91--103.

M. Tlidi, A.G. Vladimirov, D. Pieroux, D. Turaev, Spontaneous motion of cavity solitons induced by a delayed feedback, Phys. Rev. Lett., 103 (2009) pp. 103904/1--103904/4.

V.Z. Tronciu, Excitability and coherence resonance of DFB laser with passive dispersive reflector, Moldavian J. Phys. Sci., 7 (2008) pp. 218-223.

S. Yanchuk, M. Wolfrum, Destabilization patterns in large regular networks, Phys. Rev. E, 77 (2008) pp. 026212/1-026212/7.
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 well-known 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, J. Math. Anal. Appl., 348 (2008) pp. 508-515.
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 no-flux 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, Phys. Rev. E, 77 (2008) pp. 06520/1--06520/4.

M. Lichtner, Spectral mapping theorem for linear hyperbolic systems, Proc. Amer. Math. Soc., 136 (2008) pp. 2091-2101.

D. Turaev, A.G. Vladimirov, S. Zelik, Chaotic bound state of localized structures in the complex Ginzburg--Landau equation, Phys. Rev. E, 75 (2007) pp. 045601/1-045601/4.

M. Lichtner, M. Radziunas, L. Recke, Well-posedness, smooth dependence and center manifold reduction for a semilinear hyperbolic system from laser dynamics, Math. Methods Appl. Sci., 30 (2007) pp. 931--960.

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. 1--6.

S. Yanchuk, A. Stefanski, T. Kapitaniak, J. Wojewoda, Dynamics of an array of mutually coupled semiconductor lasers, Phys. Rev. E, 73 (2006) pp. 016209/1--016209/7.

S. Yanchuk, M. Wolfrum, P. Hövel, E. Schöll, Control of unstable steady states by strongly delayed feedback, Phys. Rev. E, 74 (2006) pp. 026201/1--026201/7.

M. Nizette, D. Rachinskii, A. Vladimirov, M. Wolfrum, Pulse interaction via gain and loss dynamics in passive mode-locking, Phys. D, 218 (2006) pp. 95--104.

A. Politi, F. Ginelli, S. Yanchuk, Y. Maistrenko, From synchronization to Lyapunov exponents and back, Phys. D, 224 (2006) pp. 90-101.

D.I. Rachinskii, A. Vladimirov, U. Bandelow, B. Hüttl, R. Kaiser, Q-switching instability in a mode-locked semiconductor laser, J. Opt. Soc. Amer. B Opt. Phys., 23 (2006) pp. 663--670.

A. Yulin, D. Skryabin, A.G. Vladimirov, Modulation instability of discrete solitons in coupled waveguides with group velocity dispersion, Optics Express, 14 (2006) pp. 12347--12352.

M. Wolfrum, S. Yanchuk, Eckhaus instability in systems with large delay, Phys. Rev. Lett., 96 (2006) pp. 220201/1--220201/4.

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, Appl. Phys. Lett., 87 (2005) pp. 181911/1--181911/3.

A. Vladimirov, D. Turaev, Model for passive mode locking in semiconductor lasers, Phys. Rev. A, 72 (2005) pp. 033808/1-033808/13.

S. Yanchuk, Discretization of frequencies in delay coupled oscillators, Phys. Rev. E, 72 (2005) pp. 036205/1-036205/5.

S. Yanchuk, Properties of stationary states of delay equations with large delay and applications to laser dynamics, Math. Methods Appl. Sci., 28 (2005) pp. 363--377.

K. Gatermann, M. Wolfrum, Bernstein's second theorem and Viro's method for sparse polynomial systems in chemistry, Adv. Appl. Math., 34 (2005) pp. 252--294.

D.I. Rachinskii, K.R. Schneider, Dynamic Hopf bifurcations generated by nonlinear terms, J. Differential Equations, 210 (2005) pp. 65--86.

M. Wolfrum, J. Härterich, Describing a class of global attractors via symbol sequences, Discrete Contin. Dyn. Syst., 12 (2005) pp. 531-554.

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. 026217/1--026217/11.

B. Fiedler, C. Rocha, M. Wolfrum, Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle, J. Differential Equations, 201 (2004) pp. 99--138.

K.R. Schneider, S. Yanchuk, L. Recke, Dynamics of two mutually coupled semiconductor lasers: Instantaneous coupling limit, Phys. Rev. E (3), 69 (2004) pp. 056221/1--056221/12.

S.V. Fedorov, N.N. Rosanov, A.N. Shatsev, N.A. Veretenov, A.G. Vladimirov, Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with saturable absorber, IEEE J. Quantum Electron., 39 (2003) pp. 216--226.

D. Rachinskii, K.R. Schneider, Delayed loss of stability in systems with degenerate linear parts, Z. Anal. Anwendungen, 22 (2003) pp. 433--453.

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. 197--205.

K.R. Schneider, E. Shchetinina, One-parametric families of canard cycles: Two explicitly solvable examples, Math. Methods Appl. Sci., 2 (2003) pp. 74-75.

K.R. Schneider, S. Yanchuk, Complete synchronization of symmetrically coupled autonomous systems, Appl. Anal., 82 (2003) pp. 1127-1143.

K.R. Schneider, V.A. Sobolev, E. Shchepakina, New type of travelling wave solutions, Math. Methods Appl. Sci., 26 (2003) pp. 1349-1361.

A.G. Vladimirov, G. Kozyreff, P. Mandel, Synchronization of weakly stable oscillators and semiconductor laser arrays, Europhys. Lett., 61 (2003) pp. 613--619.

A.G. Vladimirov, G. Kozyreff, P. Mandel, Synchronization of weakly stable oscillators and semiconductor laser arrays, Europhys. Lett., 61 (2003) pp. 613--619.

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

S. Yanchuk, Y. Maistrenko, E. Mosekilde, Synchronization of time-continuous chaotic oscillators, Chaos, 13 (2003) pp. 388--400.

S. Yanchuk, G. Kristensen, I. Shushko, Dynamical approach to complex regional economic growth based on Keynesian model for China, Chaos Solitons Fractals, 18 (2003) pp. 937--952.

M. Wolfrum, A sequence of order relations: Encoding heteroclinic connections in scalar parabolic PDE, J. Differential Equations, 183 (2002) pp. 56--78.

M. Wolfrum, Geometry of heteroclinic cascades in scalar parabolic differential equations, J. Dynam. Differential Equations, 14 (2002) pp. 207--241.

P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, T. Kapitaniak, Dynamics of a large ring of unidirectionally coupled duffing oscillators, IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design, Aberdeen, UK, July 27 - 30, 2010, M. Wiercigroch, G. Rega, eds., VIII of Proceedings of the IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design, Springer, 2013, pp. 63--72.

C. Brée, S. Amiranashvili, U. Bandelow, Spatio-temporal 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. 131--132.

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. 1--1.

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. Schimansky-Geier, B. Fiedler, J. Kurths, E. Schöll, eds., World Scientific, New Jersey [et al.], 2007, pp. 185--212.

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 Electro-Optics and the European Quantum Electronics Conference (CLEO®/Europe-IQEC) Conference Digest (oral presentation IG-4-MON), IEEE, 2007, pp. 1--1.

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. 367--374.

M. Wolfrum, The concept of adjacency for stationary and non-stationary solutions of scalar semilinear parabolic PDE, in: EQUADIFF 2003. Proceedings of the International Conference on Differential Equations, Hasselt, Belgium, 22--26 July 2003, F. Dumortier, H. Broer, J. Mawhin, A. Vanderbauwhede, S. Verduyn Lunel, eds., World Scientific, Singapore, 2005, pp. 678--684.

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, 22--26 July 2003, F. Dumortier, H. Broer, J. Mawhin, A. Vanderbauwhede, S. Verduyn Lunel, eds., World Scientific, Singapore, 2005, pp. 494--496.

S. Yanchuk, M. Wolfrum, Instabilities of equilibria of delay-differential equations with large delay, in: Proceedings of the ENOC-2005 Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven, The Netherlands, August 7--12, 2005, D.H. VAN Campen, M.D. Lazurko, W.P.J.M. caps">caps">van der Oever, eds., 2005, pp. 1060--1065.

M. Wolfrum, S. Yanchuk, Synchronous and asynchronous instabilities of two lasers with a long delayed coupling, in: Proceedings of the ENOC-2005 Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven, The Netherlands, August 7--12, 2005, D.H. VAN Campen, M.D. Lazurko, W.P.J.M. caps">caps">van der Oever, eds., 2005, pp. 2069--2073.

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. 51--62.

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. 101--138.

O. Omel'chenko, Coherence-incoherence patterns in a ring of non-locally coupled phase oscillators, Preprint no. 1777, WIAS, Berlin, 2013.
Abstract, Postscript (8980 kByte), PDF (5295 kByte)

We consider a paradigmatic spatially extended model of non-locally coupled phase oscillators which are uniformly distributed within a one-dimensional interval and interact depending on the distance between their sites modulo periodic boundary conditions. This model can display peculiar spatio-temporal patterns consisting of alternating patches with synchronized (coherent) or irregular (incoherent) oscillator dynamics, hence the name coherence-incoherence 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 coherence-incoherence patterns and to suggest directions for searching new ones.

R. Arkhipov, A. Pimenov, M. Radziunas, A.G. Vladimirov, D. Arsenjević, D. Rachinskii, H. Schmeckebier, D. Bimberg, Hybrid mode-locking in edge-emitting semiconductor lasers: Simulations, analysis and experiments, Preprint no. 1734, WIAS, Berlin, 2012.
Abstract, Postscript (2221 kByte), PDF (830 kByte)

Hybrid mode-locking in a two section edge-emitting 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 mode-locking 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 mode-locked laser fP . Finally, we provide an experimental demonstration of hybrid mode-locking in a 20 GHz quantum-dot laser with the modulation frequency of the reverse bias applied to the absorber section close to fP =2.

N. Nefedov, L. Recke, K. Schneider, On existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations, Preprint no. 1683, WIAS, Berlin, 2012.
Abstract, Postscript (910 kByte), PDF (169 kByte)

We consider a singularly perturbed parabolic periodic boundary value problem for a reaction-advection-diffusion equation. We construct the interior layer type formal asymptotics and propose a modified procedure to get asymptotic lower and upper solutions. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution with an interior layer and estimate the accuracy of its asymptotics. Moreover, we are able to establish the asymptotic stability of this solution with interior layer

V.F. Butuzov, N.N. Nefedov, L. Recke, K. Schneider, On a singularly perturbed initial value problem in case of a double root of the degenerate equation, Preprint no. 1672, WIAS, Berlin, 2011.
Abstract, Postscript (272 kByte), PDF (123 kByte)

We study the initial value problem of a singularly perturbed first order ordinary differential equation in case that the degenerate equation has a double root. We construct the formal asymptotic expansion of the solution such that the boundary layer functions decay exponentially. This requires a modification of the standard procedure. The asymptotic solution will be used to construct lower and upper solutions guaranteeing the existence of a unique solution and justifying its asymptotic expansion.

D. Turaev, A.G. Vladimirov, S. Zelik, Strong synchronization of weakly interacting oscillons, Preprint no. 1659, WIAS, Berlin, 2011.
Abstract, Postscript (23 MByte), PDF (7332 kByte)

We study interaction of well-separated oscillating localized structures (oscillons). We show that oscillons emit weakly decaying dispersive waves, which leads to formation of bound states due to subharmonic synchronization. We also show that in optical applications the Andronov-Hopf bifurcation of stationary localized structures leads to a drastic increase in their interaction strength.

J. Sieber, M. Wolfrum, M. Lichtner, S. Yanchuk, On the stability of periodic orbits in delay equations with large delay, Preprint no. 1586, WIAS, Berlin, 2011.
Abstract, Postscript (1285 kByte), PDF (349 kByte)

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, Thermodynamic limit approach for bifurcation analysis of chimera states, Internes Seminar des Fachbereiches "Nonlinear Dynamics", Technische Universitát Berlin, January 10, 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, 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 des 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 mode-locked quantum dot semiconductor lasers, International Conference ``Laser Optics 2012'', St. Petersburg, Russia, June 25 - 29, 2012.

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

R. Arkhipov, Hybrid mode-locking in semiconductor quantum dot lasers: Simulation, analysis and comparison with experiments, ITN PROPHET Mid-Term Review Meeting, October 9 - 11, 2012, Paris, France, October 11, 2012.

R. Arkhipov, Numerical analysis of hybrid mode-locking in semiconductor quantum dot lasers, XIV All-Russian Scientific School-Seminar ``Wave Phenomena in Inhomogeneous Media'' (Waves-2012), Zvenigorod, Russia, 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 all-optical streak camera based on ultrafast laser beam deflection by light-induced 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 mode-locking in two-section 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 broad-band THz radiation using plasma-generating femtosecond light pulses, IPHT-Kolloquium, Institut für Photonische Technologien (IPHT), Jena, November 20, 2012.

O. Omel'chenko, Coherence-incoherence patterns in systems of non-locally 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 Sakaguchi--Kuramoto model, 7th Crimean School and Workshop ``Emergent Dynamics of Oscillatory Networks'', May 20 - 27, 2012, Mellas, Crimea, Ukraine, May 22, 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, Coherence-incoherence patterns in systems of non-locally coupled phase oscillators, Statistical Physics and Nonlinear Dynamics & Stochastic Processes, Humboldt-Universität zu Berlin, Institut für Physik, Berlin, June 18, 2012.

O. Omel'chenko, Nonuniversal transitions to synchrony in the Sakaguchi--Kuramoto model, Seminar Applied Analysis, Humboldt-Universität zu Berlin, October 29, 2012.

O. Omel'chenko, What are chimera states, Westfälische Wilhelms-Universität Münster, Center for Nonlinear Science, November 6, 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 Mathematiker-Vereinigung (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 semi-strong 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, Russia, 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 non-locally coupled Kuramoto-type models, Universität Potsdam, Institut für Physik und Astronomie, October 25, 2010.

T. Girnyk, Multistability of twisted states in non-locally coupled Kuramoto-type models, École Polytechnique Fédérale de Lausanne, Laboratory of Nonlinear Systems (EPFL-LANOS), 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 semi-strong 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 semi-strong 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 self-pulsing 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 Electro-Optics 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), Republic of Korea, 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, The Netherlands, August 22, 2008.

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

S. Yanchuk, Bifurcations in lattices of unidirectionally coupled oscillators, Jour fixe, Graduiertenkolleg ``Analysis, Numerics, and Optimization of Multiphase Problems'', Humboldt-Universität 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 master-oscillator power-amplifier 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'', Humboldt-Universität zu Berlin, Institut für Physik, June 19, 2008.

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

M. Wolfrum, The Eckhaus scenario in delay-differential 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, Humboldt-Universität 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, Slow-motion of multi-pulse solutions in reaction-diffusion 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, Humboldt-Universität 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, Kick-off Workshop ``Materials in New Light'', Humboldt-Universität zu Berlin, Institut für Physik, Berlin, January 6, 2006.

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

U. Bandelow, Simulation and analysis of spatio-temporal effects in complex laser structures, Kick-off Workshop ``Materials in New Light'', Humboldt-Universität 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 mode-locked 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, Russia, June 20, 2006.

A. Vladimirov, Nonlinear dynamics in multimode and spatially extended laser systems, Moscow State University, Physics Faculty, Russia, 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, Russia, 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 mode-locked 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, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, May 27, 2005.

S. Yanchuk, Instabilities of equililbria of delay-differential equations with large delay, ENOC 2005 (EUROMECH Nonlinear Oscillations Conference), August 7 - 12, 2005, Eindhoven, The 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 Lang-Kobayashi 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 mode-locking, International Quantum Electronics Conference, München, June 12 - 17, 2005.

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

U. Bandelow, Analyse dynamischer Effekte in Optoelektronik und Photonik, Institutsseminar, Ferdinand-Braun-Institut für Höchstfrequenztechnik, Berlin, December 9, 2005.

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

M. Wolfrum, Systems of delay differential equations with large delay, Otto-von-Guericke-Universität 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 High-dimensional 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 delay-differential 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 mode-locked 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, Russia, September 16, 2004.

K.R. Schneider, Invariante Mannigfaltigkeiten für zufällige dynamische Systeme mit schnellen und langsamen Variablen, Workshop GAMM-Fachausschuss "`Dynamik und Regelungstheorie"' und VDI/VDE-GMA-Ausschuss 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, Russia, June 16, 2004.

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

U. Bandelow, Simulation of mode-locked lasers based on a distributed time-domain model, WIAS Workshop ``Dynamics of Semiconductor Lasers'', September 15 - 17, 2003, Berlin, September 17, 2003.

K.R. Schneider, Canard solutions of finite and infinite-dimensional dynamical systems, Moscow State University, Faculty of Physics, Russia, 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 time-scales, 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 Lang-Kobayashi systems, National Institute of Applied Optics, Florence, Italy, May 7, 2003.

S. Yanchuk, Synchronization problem in two-section semiconductor lasers, Forschungsseminar ``Angewandte Analysis'', Humboldt-Universität 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.

I. Omelchenko, O. Omel'chenko, P. Hövel, E. Schöll, Multi-chimera states in FitzHugh--Nagumo 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 FitzHugh-Nagumo equations with nonlocal coupling. This class of elements serves as a paradigmatic model in neuroscience, chemical oscillations, and nonlinear electronic circuits. Applying a phase-reduction technique we show that off-diagonal nonlocal coupling is a crucial factor for the appearance of chimera states, which consist of coexisting domains of coherent (phase-locked) 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 phase-oscillator 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.