Simple and complex chimera states in a nonlinearly coupled oscillatory medium

Area of Science:

• Nonlinear dynamics
• Complex systems theory
• Statistical physics

Background:

• Chimera states, a unique phenomenon in coupled oscillator systems, exhibit coexisting domains of synchronized and desynchronized behavior.
• Understanding the formation and evolution of chimera states is crucial for various fields, including neuroscience and network dynamics.

Purpose of the Study:

• To investigate chimera states in a one-dimensional medium of nonlinear, nonlocally coupled phase oscillators.
• To identify and characterize different types of chimera states and their stability.
• To explore the transition pathways from stable chimera states to more complex dynamical regimes.

Main Methods:

• Formulation of a reduced third-order ordinary differential equation for stationary rotating nonhomogeneous solutions using a local coarse-grained complex order parameter.
• Analysis of the ordinary differential equation to find periodic orbits representing chimera-type and other inhomogeneous states.
• Stability analysis of the identified solutions to determine their dynamical behavior.

Main Results:

• Stationary rotating nonhomogeneous solutions, including chimera states, were found as periodic orbits of the derived ordinary differential equation.
• Stability calculations indicated that only a subset of these states are dynamically stable.
• An oscillatory instability was identified, leading to a 'breathing chimera' where synchronous domains split into subdomains with distinct mean frequencies.

Conclusions:

• The study successfully identified and characterized chimera states and related inhomogeneous solutions in a nonlinear oscillator system.
• The findings reveal a progression of instabilities, from stable chimeras to breathing and ultimately turbulent chimeras.
• The developed mathematical framework provides a powerful tool for analyzing complex dynamics in extended nonlinear systems.