Stable Chimera States: A Geometric Singular Perturbation Approach

TL;DR

This paper investigates the emergence and stability of chimera states in multilayer networks of heterogeneous oscillators using geometric singular perturbation theory, revealing conditions for stable and breathing chimeras.

Contribution

It introduces a geometric singular perturbation approach to analyze co-evolving coupling strengths, deriving conditions for stable chimera states and exploring complex patterns like breathing chimeras.

Findings

Derived conditions for stable chimera states with co-evolutionary coupling.

Generated persistent breathing chimera patterns through adaptive laws.

Numerically demonstrated relaxation oscillations and canard cycles related to chimeras.

Abstract

Over the past decades chimera states have attracted considerable attention given their unexpected symmetry-breaking spatio-temporal nature, simultaneously exhibiting synchronous and incoherent behaviours under specific conditions. Despite relevant precursory results of such unforeseen states for diverse physical and topological configurations, there remain structures and mechanisms yet to be unveiled. In this work, using mean-field techniques, we analyze a multilayer network composed by two populations of heterogeneous Kuramoto phase oscillators with coevolutive coupling strengths. Moreover, we employ Geometric Singular Perturbation Theory (GSPT) with the inclusion of a time-scale separation between the dynamics of the network elements and the adaptive coupling strength connecting them, gaining a better insight into the behaviour of the system from a fast-slow dynamics perspective. Consequently, we derive the necessary and sufficient condition to produce stable chimera states when considering a co-evolutionary intercoupling strength. Additionally, under the aforementioned constraint and with a suitable adaptive law election, it is possible to generate intriguing patterns, such as persistent breathing chimera states. Thereafter, we analyze the geometric properties of the mean-field system with a co-evolutionary intracoupling strength and demonstrate the production of stable chimera states which depend on the associated network. Finally, relaxation oscillations and canard cycles, also related to breathing chimeras, are numerically produced under identified conditions due to the geometry of our system.