Phase-flip bifurcation and synchronous transition in unidirectionally coupled parametrically excited pendula

TL;DR

This paper investigates how phase-flip bifurcation influences synchronization transitions in unidirectionally coupled parametrically excited pendula, highlighting differences between identical and non-identical systems and the role of Lyapunov exponents.

Contribution

It reveals the role of phase-flip bifurcation in synchronization and desynchronization, and clarifies the relationship between Lyapunov exponents and synchronization states.

Findings

Phase-flip bifurcation causes synchronization in identical systems.

In non-identical systems, it leads to desynchronization.

Complete synchronization involves equal magnitude and slope of the two largest Lyapunov exponents.

Abstract

Phase-flip bifurcation plays an important role in the transition to synchronization state in unidirectionally coupled parametrically excited pendula. In coupled identical system it is the cause of complete synchronization whereas in case of coupled non-identical system it causes desynchronization. In coupled identical systems negativity of conditional Lyapunov exponent is not always sufficient for complete synchronization. In complete synchronization state the largest conditional Lyapunov exponent and the second largest Lyapunov exponent are equal in magnitude and slope.