Basins of Attraction for Chimera States

Chimera states---curious symmetry-broken states in systems of identical coupled oscillators---typically occur only for certain initial conditions. Here we analyze their basins of attraction in a simple system comprised of two populations. Using perturbative analysis and numerical simulation we evaluate asymptotic states and associated destination maps, and demonstrate that basins form a complex twisting structure in phase space. Understanding the basins' precise nature may help in the development of control methods to switch between chimera patterns, with possible technological and neural system applications.Comment: Please see Ancillary files for the 4 supplementary videos including description (PDF

A globally stable attractor that is locally unstable everywhere

We construct two examples of invariant manifolds that despite being locally unstable at every point in the transverse direction are globally stable. Using numerical simulations we show that these invariant manifolds temporarily repel nearby trajectories but act as global attractors. We formulate an explanation for such global stability in terms of the `rate of rotation' of the stable and unstable eigenvectors spanning the normal subspace associated with each point of the invariant manifold. We discuss the role of this rate of rotation on the transitions between the stable and unstable regimes

Heteroclinic Dynamics of Localized Frequency Synchrony: Stability of Heteroclinic Cycles and Networks

In the first part of this paper, we showed that three coupled populations of identical phase oscillators give rise to heteroclinic cycles between invariant sets where populations show distinct frequencies. Here, we now give explicit stability results for these heteroclinic cycles for populations consisting of two oscillators each. In systems with four coupled phase oscillator populations, different heteroclinic cycles can form a heteroclinic network. While such networks cannot be asymptotically stable, the local attraction properties of each cycle in the network can be quantified by stability indices. We calculate these stability indices in terms of the coupling parameters between oscillator populations. Hence, our results elucidate how oscillator coupling influences sequential transitions along a heteroclinic network where individual oscillator populations switch sequentially between a high and a low frequency regime; such dynamics appear relevant for the functionality of neural oscillators