# Transitions from chimeras to coherence: An analytical approach by means of the coherent stability function

## TL;DR

This paper develops an analytical method using the coherent stability function to study the transition from chimera states to coherence in coupled oscillators, validated through numerical simulations with leech neuron models.

## Contribution

It introduces the coherent stability function approach for analyzing transitions from chimeras to coherence, extending stability analysis tools to complex dynamical states.

## Key findings

- Analytical conditions for chimera to coherence transition derived.
- Numerical validation using leech neuron models confirms the approach.
- Observation of complete synchronization at high coupling strengths.

## Abstract

Chimera states have been a vibrant subject of research in the recent past, but the analytical treatment of transitions from chimeras to coherent states remains a challenge. Here we analytically derive the necessary conditions for this transition by means of the coherent stability function approach, which is akin to the master stability function approach that is traditionally used to study the stability of synchronization in coupled oscillators. In chimera states, there is typically at least one group of oscillators that evolves in a drifting, random manner, while other groups of oscillators follow a smoother, more coherent profile. In the coherent state, there thus exists a smooth functional relationship between the oscillators. This lays the foundation for the coherent stability function approach, where we determine the stability of the coherent state. We subsequently test the analytical prediction numerically by calculating the strength of incoherence during the transition point. We use leech neurons, which exhibit a coexistence of chaotic and periodic tonic spiking depending on initial conditions, coupled via non-local electrical synapses, to demonstrate our approach. We systematically explore various dynamical states with the focus on the transitions between chimeras and coherence, fully confirming the validity of the coherent stability function. We also observe complete synchronization for higher values of the coupling strength, which we verify by the master stability function approach.