Chaos in Co-operative Dynamics of Alternately Synchronized Ensembles of Globally Coupled Self-oscillators
In many cases dynamics of spatially extended systems can be thought of as cooperative action of a large number of relatively simple elements, like interacting limit-cycle oscillators, and treated as low-dimensional dynamics in terms of appropriate collective modes (Cross and Hohenberg, 1993), In this context, a paradigm concept is the Kuramoto model of ensemble of globally coupled phase oscillators distributed in some range of the natural frequencies, and the synchronizationdesynchronization transition in such ensemble (Kuramolo, 1984; Strogalz, 2000; Pikovsky et al., 2002; Acebrón, 2005). Now, an interesting question arises about a possibility of occurrence of hyperbolic attractors on the level of the collective modes. This problem may be of interest in various fields, e.g. in hydrodynamics, in laser physics and nonlinear optics, electronics, neurodynamics. In this chapter we start from a short introduction to dynamics of a globally coupled ensemble of limitcycle oscillators and discuss different levels of reduction, including slow complex amplitude equations and phase equations; the last represents exactly the Kuramoto model. Then, we turn to some artificial, but feasible and analytically convenient model of two alternately synchronized and desynchronized interacting ensembles of globally coupled oscillators, which demonstrate transformation of the phase of collective excitation in the course of dynamic evolution in accordance with the expanding circle map at successive stages of synchronization of the subsystems. The consideration is based on a joint work with Pikovsky and Rosenblum (Kuznetsov et al., 2010).
KeywordsCollective Mode Collective Excitation Phase Oscillator Fast Oscillation Frequency Detuning
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