Weak transient chaos

TL;DR

The paper investigates weak transient chaos caused by sub-exponential divergence near non-chaotic attractors, proposing methods to detect it through perturbations and a new Lyapunov exponent definition.

Contribution

It introduces a novel approach to detect weak transient chaos in deterministic systems using an alternative Lyapunov exponent measure.

Findings

Weak transient chaos can be revealed with small random perturbations.

A new Lyapunov exponent definition detects transient chaos in unperturbed systems.

Demonstrated on a master-slave system with heteroclinic cycle and Van-der-Pol-Duffing oscillator.

Abstract

A phenomenon of weak transient chaos is discussed that is caused by sub-exponential divergence of trajectories in the basin of a non-chaotic attractor. Such a regime is not easy to detect, because conventional characteristics, such as the largest Lyapunov exponent is non-positive. Here we study, how such a divergence can be exposed and detected. First, we show that weak transient chaos can be exposed if a small random perturbation is added to the system, leading to positive values of the largest Lyapunov exponent. Second, we introduce an alternative definition of the Lyapunov exponent, which allows us to detect weak transient chaos in the deterministic unperturbed system. We show that this novel characteristic becomes positive, reflecting transient chaos. We demonstrate this phenomenon and its detection using a master-slave system where the master possesses a heteroclinic cycle attractor, while the slave is the Van-der-Pol-Duffing oscillator possessing a stable limit cycle.