Synchronization of Coupled Nonlinear Oscillators with Applications to Photonic Arrays

Master of Science Thesis in Applied mathematics Department of Mathematics, School of Science and Technology Nazarbayev University 2019In recent years, the study of synchronization of coupled oscillators have been the subject of intense research interest, leading to many new and unexpected phenomena. Our research is first focused on the analysis of a network of coupled nonlinear oscillators exhibiting the breakdown of synchronization into fascinating “chimera states” exhibiting the coexistence of synchronized and unsynchronized parts. We then apply these ideas to laser arrays of photonic “oscillators”, which have numerous applications in optical communications, sensing and imaging. First of all, we demonstrate the occurrence of synchronization and chimera states in a simpler problem, consisting of a ring of coupled 4D simplified Lorenz systems, in which each oscillator is described by a Li-Sprott oscillator [1]. An interesting feature of each oscillator is the coexistence of a limit cycle and two symmetric strange attractors for some specific range of parameters, which influences the global synchronization dynamics and leads to the formation of chimera states. Inspired by this model, we study some fascinating oscillatory phenomena of coupled photonic oscillators consisting of dimers of semiconductor lasers, each of which is capable of performing limit cycle oscillations. Coupling in an appropriate way a large number of dimers in long arrays we find that they can exhibit combinations of oscillatory patterns involving long amplitude oscillations (LAO) and also localized oscillations of very small amplitude close to the fixed points (LOCFP). As preliminary results of this investigation, we show the coexistence of LOA and LOCFP patterns reminiscent of “chimera–like” states and LOCFP “breather– like” phenomena. Both of these behaviors are shown to be spatially robust, when we calculate the Discrete Laplacian of their amplitudes for long times