Uncertainty Relation for Chaos

TL;DR

This paper establishes a new necessary condition for chaos emergence, stating that the product of the maximal positive Lyapunov exponent and the time spent in the unstable region must exceed one, supported by theoretical analysis and examples.

Contribution

It introduces a novel criterion involving both the Lyapunov exponent and the duration in unstable regions for chaos emergence.

Findings

The product of the maximal positive exponent and unstable duration must exceed one for chaos.

Theoretical analysis justifies the new chaos criterion.

Examples demonstrate the criterion's applicability.

Abstract

A necessary condition for the emergence of chaos is given. It is well known that the emergence of chaos requires a positive exponent which entails diverging trajectories. Here we show that this is not enough. An additional necessary condition for the emergence of chaos in the region where the trajectory of the system goes through, is that the product of the maximal positive exponent times the duration in which the system configuration point stays in the unstable region should exceed unity. We give a theoretical analysis justifying this result and a few examples.