Desynchronization bifurcation of coupled nonlinear dynamical systems

TL;DR

This paper investigates the desynchronization bifurcation in coupled nonlinear systems, specifically Rössler oscillators, revealing a square root dependence and a pitchfork bifurcation in the transverse manifold.

Contribution

It introduces a simple model for coupled integrable systems that captures the desynchronization phenomena as a pitchfork bifurcation.

Findings

Desynchronization occurs with a square root dependence on the parameter.

One Lyapunov exponent becomes positive, indicating divergence.

Desynchronization is characterized as a pitchfork bifurcation.

Abstract

We analyze the desynchronization bifurcation in the coupled R\"ossler oscillators. After the bifurcation the coupled oscillators move away from each other with a square root dependence on the parameter. We define system transverse Lyapunov exponents and in the desynchronized state one is positive while the other is negative implying that one oscillator is trying to fly away while the other is holding it. We give a simple model of coupled integrable systems that shows a similar phenomena and can be treated as the normal form for the desynchronization bifurcation. We conclude that the desynchronization is a pitchfork bifurcation of the transverse manifold.