Chimera states through invariant manifold theory

In nonlinear dynamics, a chimera refers to a dynamical hybrid – a state in a network of identical coupled oscillators exhibiting a mix of both synchronous and asynchronous behaviour. Some oscillators synchronize in a persistent way, while others display unrelated asynchronous behaviour. The existence of such states was discovered only two decades ago, but they have since been observed in a wide variety of systems including coupled lasers and photoelectrochemical oscillators.
Theorists possess a well-developed theory for the nature of these states in the limit of infinitely many oscillators. Yet researchers know much less for large but finite networks. Chimeras can be generated in finite systems with careful design of coupling functions, but, in general, the field remains far from a comprehensive mathematical understanding of when and why such states arise.
In a recent paper, LML External Fellow Jeroen Lamb and colleagues present a new analysis of this issue, deriving rigorous results on chimera behaviour in a large but finite size network of identical oscillators. The network they consider consists of two symmetrically coupled star subnetworks, each subnetwork having a central hub connected to a large number N of peripheral nodes. All nodes are occupied by identical phase oscillators and interactions arise through a diffusive Kuramoto–Sakaguchi coupling with shear. The strength of the coupling between nodes in each of the star motif subnetworks is uniform, while another parameter ε gives the strength of the interaction between the peripheral nodes in the two subnetworks. An important feature is that when isolated, i.e., when ε=0, each of the subnetworks exhibits a bistable behaviour of its collective dynamics, reflecting the coexistence of a stable state with all peripheral nodes in synchrony and another stable state in which the motion is asynchronous.
In the paper, Lamb and colleagues study the dynamics in the presence of coupling (ε >0), and report numerical experiments showing that chimera states tend to persist (at least to the limit of the simulation time) when the coupling strength ε between the star subnetworks is small. In contrast, they collapse to a completely synchronized or asynchronous state when this coupling strength is not so small. The authors offer a mathematical explanation for these observations for sufficiently large N and sufficiently small ε. Their analysis relies on area dimensional reduction due to a Möbius group symmetry, averaging theory  and normally hyperbolic invariant manifold theory.
A preprint of the paper is available at

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