Towards a Theory for the Formation of Chimera Patterns in Complex Networks

TL;DR

This paper introduces a systematic, theory-based approach to understanding the formation of chimera patterns in complex networks, emphasizing the role of network topology and spectral properties in their emergence.

Contribution

It formalizes a method using pattern formation theory and spectral analysis to explain chimera states, linking network structure to pattern formation in coupled oscillators.

Findings

Network topology randomness influences chimera emergence.

Amplitude and phase chimeras arise through different mechanisms.

The approach is validated with simulations and empirical data.

Abstract

Chimera states, marked by the coexistence of order and disorder in systems of coupled oscillators, have captivated researchers with their existence and intricate patterns. Despite ongoing advances, a fully understanding of the genesis of chimera states remains challenging. This work formalizes a systematic method by evoking pattern formation theory to explain the emergence of chimera states in complex networks, in a similar way to how Turing patterns are produced. Employing linear stability analysis and the spectral properties of complex networks, we show that the randomness of network topology, as reflected in the localization of the graph Laplacian eigenvectors, determines the emergence of chimera patterns, underscoring the critical role of network structure. In particular, this approach explains how amplitude and phase chimeras arise separately and explores whether phase chimeras can be chaotic or not. Our findings suggest that chimeras result from the interplay between local and global dynamics at different time scales. Validated through simulations and empirical network analyses, our method enriches the understanding of coupled oscillator dynamics.