A Novel Criterion for Unpredictable Motions

TL;DR

This paper introduces a new criterion for identifying unpredictable motions in coupled autonomous systems with skew product structures, especially when generalized synchronization occurs, supported by theoretical analysis and examples.

Contribution

It provides sufficient conditions for unpredictable motions in response systems, extending understanding of complex dynamics in hybrid and Lorenz systems.

Findings

Unpredictable motions can exist under generalized synchronization.

Auxiliary system approach effectively detects synchronization.

Theoretical conditions are validated with hybrid and Lorenz system examples.

Abstract

We demonstrate the extension of unpredictable motions in coupled autonomous systems with skew product structure in the case that generalized synchronization takes place. Sufficient conditions for the existence of unpredictable motions in the dynamics of the response system are provided. The theoretical results are exemplified for coupled autonomous systems in which the drive is a hybrid dynamical system and the response is a Lorenz system. The auxiliary system approach and conditional Lyapunov exponents are utilized to detect the presence of generalized synchronization.