Stable chimera states: A geometric singular perturbation approach

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Network Science

Background:

• Chimera states exhibit unique spatiotemporal symmetry-breaking, displaying both synchronous and incoherent behaviors.
• Previous research explored chimera states in various configurations, but underlying mechanisms require further elucidation.
• Understanding these states is crucial for diverse fields, including neuroscience and engineering.

Purpose of the Study:

• To analyze chimera states in multilayer networks of heterogeneous Kuramoto phase oscillators.
• To investigate the role of coevolutive coupling strengths in the emergence and stability of chimera states.
• To explore novel chimera patterns, such as breathing chimeras, using fast-slow dynamics.

Main Methods:

• Utilized mean-field techniques to model coupled populations of Kuramoto phase oscillators.
• Employed geometric singular perturbation theory to analyze fast-slow dynamics of adaptive coupling strengths.
• Derived analytical conditions for stable chimera states and explored their geometric properties.

Main Results:

• Established the necessary and sufficient condition for stable chimera states with coevolutionary intercoupling strength.
• Generated persistent breathing chimera states by selecting appropriate adaptive laws.
• Demonstrated stable chimera states with coevolutionary intracoupling strength and analyzed their geometric properties.
• Numerically produced relaxation oscillations and canard cycles related to breathing chimeras.

Conclusions:

• Coevolutive coupling strengths are key to generating and stabilizing chimera states in heterogeneous oscillator networks.
• The fast-slow dynamics perspective provides deeper insights into the mechanisms underlying complex emergent behaviors.
• This work offers a framework for understanding and potentially controlling chimera states and related phenomena in complex systems.