Abstract
About two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence–incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott–Antonsen reduction technique.
Highlights
Many living organisms, chemical and physical systems can behave as self-sustained oscillators (Winfree 1980)
We consider a more complicated and less explored type of chimera states, called breathing chimera states, which is characterized by nonstationary macroscopic dynamics
We demonstrate that ansatz (8) does describe breathing chimera states and can be used for their continuation as well as for their stability analysis
Summary
Chemical and physical systems can behave as self-sustained oscillators (Winfree 1980). When two or more self-sustained oscillators interact with each other, their rhythms tend to adjust in a certain order resulting in their partial or complete synchronization (Pikovsky et al 2001; Arenas et al 2008). Such phenomena have been observed in many real-world systems and laboratory experiments, including Josephson junction arrays (Wiesenfeld et al 1996), populations of fireflies (Buck and Buck 1968) and yeast cells (De Monte et al 2008), chemical (Taylor et al 2009) and electrochemical oscillators (Kiss et al 2002).
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