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Twisted chimera states and multi-core spiral chimera states on a two-dimension torus

  • Author(s): Xie, Jianbo
  • Advisor(s): Knobloch, Edgar
  • et al.
Abstract

This thesis studies systems of nonlocal phase-coupled oscillators. Various types of solutions have been discovered and analyzed with a combination of analytical and numerical methods. The work is motivated by recent interest in chimera states, in which domains of coherent and incoherent oscillators coexist. A major part focuses on the model

equation

\begin{equation}

\frac{\partial \theta(x,t)}{\partial t}=\omega -\int G(x-y)\sin[\theta(x,t)-\theta(y,t)+\alpha]\,dy, \label{eq:abstract_phase_eq}

\end{equation}

and its two-dimensional generalization.

In one-dimensional systems, the cases where $\omega$ is either a constant or space-dependent are investigated. When $\omega$ is a constant, we propose a class of coupling functions in which chimera states develop from random initial conditions. Several classes of chimera states have been found: (a) stationary multi-cluster states with evenly distributed coherent clusters, (b) stationary multi-cluster states with unevenly distributed clusters, and (c) a single cluster state traveling with a constant speed across the system. Traveling chimera states are also identified. A self-consistent continuum description of these states is provided and their stability properties analyzed through a combination of linear stability analysis and numerical simulation. When $\omega$ is space-dependent, two types of spatial inhomogeneity, localized and spatially periodic, are considered and their effects on the existence and properties of multi-cluster and traveling chimera states are explored. The inhomogeneity is found to break up splay states, to pin the chimera states to specific locations, and to trap traveling chimeras. Many of these states can be studied by constructing an evolution equation for a complex order parameter. Solutions of this equation are in good agreement with the results of numerical simulations.

The above mentioned solutions have counterparts in two-dimensional systems. However, some of these lose stability in two dimensions and hence cannot be obtained from numerical simulation with random initial conditions. In addition, some solutions which are unique in two dimensions have been found: (a) twisted chimera states with phase that varies uniformly across the coherent domain, and (b) multi-core spiral wave chimera states with evenly distributed phase-randomized cores. Similar stability analysis as for one-dimensional systems is provided.

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