Dynamics of Delay Equations, Theory and Applications - Abstract
Faggian, Marco
It is well known that complex dynamical evolution may originate from simple low dimensional dynamical systems when a time-delayed feedback mechanism is considered. These may typically happen in systems where the propagation time of a signal is not negligible with respect to the typical timescale of the local dynamics. Examples include many biological systems and laser physics, where a long delayed feedback may be easily obtained by optical or electronic device. A deep analogy exists between delayed feedback and spatially extended dynamical systems. In particular, it is well known that deterministic systems with long time-delay $tau$ may be interpreted in terms of a suitable spatiotemporal dynamics citeuno of spatial size $tau$. In this work we extend this interpretation to stochastic differential equations, considering a simple bistable system with long delayed feedback and multiplicative noise, introduced in a way to preserve the lowest energy state. This dynamics can be described with the following prototype equation: $$ d x_t =-U'_a(x_t)+g x_t-tau+x_t dW_t $$ where $dW$ is a Wiener process, $U'_a(x_t)=x_t(x_t-1)(x_t+1+a)$, $a$ is the asymmetry coefficient of the potential $U(x_t)$ and $g>0$ is the feedback coefficient. Our numerical analysis shows that -- as the asymmetry in the bi-stable potential is carefully changed -- our system undergoes a transition into an absorbing state of the effective spatio-temporal dynamics , i.e. the lowest energy state, with critical exponents compatible with the celebrated Directed Percolation class. This simple model is believed to qualitatively capture the behavior of a class of laser systems such as a bistable semiconductor laser with long delayed feedback [2-3]. While extending the validity of the space-time analogy and verifying several features of front dynamics, we expect our results to possibly trigger new directions of theoretical and experimental research. The possibility to independently generate and erase localized states in our setup as in spatially-extended systems, enable their use as otpical information bits in a fast, all-optical setup. beginthebibliography9 bibitemunoG. Giacomelli and A. Politi, Phys. Rev. Lett. 76, 2686 (1996). bibitemdueG. Giacomelli, F. Marino, M.A. Zaks and S. Yanchuk, Europ. Phys. Lett. 99 58005 (2012). bibitemtreGiacomelli, F. Marino, M.A. Zaks and S. Yanchuk, Phys. Rev. E 88 062920 (2013). endthebibliography"