Abstract
We study travelling chimera states in a ring of nonlocally coupled heterogeneous (with Lorentzian distribution of natural frequencies) phase oscillators. These states are coherence-incoherence patterns moving in the lateral direction because of the broken reflection symmetry of the coupling topology. To explain the results of direct numerical simulations we consider the continuum limit of the system. In this case travelling chimera states correspond to smooth travelling wave solutions of some integro-differential equation, called the Ott–Antonsen equation, which describes the long time coarse-grained dynamics of the oscillators. Using the Lyapunov–Schmidt reduction technique we suggest a numerical approach for the continuation of these travelling waves. Moreover, we perform their linear stability analysis and show that travelling chimera states can lose their stability via fold and Hopf bifurcations. Some of the Hopf bifurcations turn out to be supercritical resulting in the observation of modulated travelling chimera states.
Highlights
Introduction and main resultsMany living systems, including neurons, cardiac pacemaker cells, and fireflies are capable to produce rhythmic outputs and behave as self-sustained oscillators [1]
We considered a prototype model of nonlocally coupled heterogeneous phase oscillators and showed that in the case of a coupling topology with broken reflection symmetry this model can support travelling chimera states
As the coupling asymmetry grows these chimera states undergo a sequence of transformations, which can be adequately explained using the continuum limit Ott–Antonsen equation (5)
Summary
Many living systems, including neurons, cardiac pacemaker cells, and fireflies are capable to produce rhythmic outputs and behave as self-sustained oscillators [1]. This equation cannot be solved via the implicit function theorem because of two continuous symmetries, we apply the Lyapunov–Schmidt reduction technique and derive a new system of equations suitable for the continuation of travelling wave solutions to equation (5).
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