Supreme Local Lyapunov Exponents and Chaotic Impulsive Synchronization

TL;DR

This paper introduces the supreme local Lyapunov exponent (SLLE) to better predict local instability in impulsively synchronized chaos, enabling more reliable synchronization by ensuring negative SLLE.

Contribution

The paper proposes the concept of SLLE, a new measure that characterizes local instability and improves the prediction and stability of impulsive chaos synchronization.

Findings

Negative SLLE guarantees perpetual synchronization.

SLLE allows for larger impulsive intervals in synchronization.

SLLE effectively predicts local instability in chaotic systems.

Abstract

Impulsively synchronized chaos with criterion from conditional Lyapunov exponent is often interrupted by desynchronized bursts. This is because the Lyapunov exponent cannot characterize local instability of synchronized attractor. To predict the possibility of the local instability, we introduce a concept of supreme local Lyapunov exponent (SLLE), which is defined as supremum of local Lyapunov exponents over the attractor. The SLLE is independent of the system trajectories and therefore, can characterize the extreme expansion behavior in all local regions with prescribed finite-time interval. It is shown that the impulsively synchronized chaos can be kept forever if the largest SLLE of error dynamical systems is negative and then the burst behavior will not appear. In addition, the impulsive synchronization with negative SLLE allows large synchronizable impulsive interval, which is significant for applications.