Quantifying the transition from spiral waves to spiral wave chimeras in a lattice of self-sustained oscillators

TL;DR

This study investigates how increasing nonlocal coupling in a lattice of self-sustained oscillators causes a transition from regular spiral waves to complex spiral wave chimeras with chaotic and hyperchaotic dynamics, using sensitivity indices and Lyapunov analysis.

Contribution

It provides a detailed quantification of the transition from spiral waves to spiral wave chimeras in oscillator networks with nonlocal coupling, highlighting the role of sensitivity and chaos.

Findings

Transition from regular to chaotic oscillations at the spiral core

Formation of high-sensitivity regions expanding with coupling range

Emergence of hyperchaotic regimes with multiple positive Lyapunov exponents

Abstract

The present work is devoted to the detailed quantification of the transition from spiral waves to spiral wave chimeras in a network of self-sustained oscillators with two-dimensional geometry. The basic elements of the networks are the van der Pol oscillator and the FitzHugh-Nagumo neuron. Both models are in the regime of relaxation oscillations. We analyze the regime by using the indices of local sensitivity which enables us to evaluate the sensitivity of each individual oscillator at finite time. Spi-ral waves are observed in both lattices when the interaction between elements have the local character. The dynamics of all the elements is regular. There are no high-sensitive regions. We have discovered that when the coupling becomes nonlocal, the features of the systems significantly changes. The oscillation regime of the spiral wave center element switches to chaotic one. Besides this, a region with high sensitivity occurs around this oscillator. Moreover, we show that the latter expands in space with elongation of the coupling range. As a result, an incoherence cluster of the spiral wave chimera is formed exactly within this high-sensitive area. Formation of this cluster is accompanied by the sharp increase in values of the maximal Lyapunov exponent to the positive region. Furthermore, we explore that the system can even switch to hyperchaotic regime, when several Lyapunov exponents becomes positive.