On the viability of local criteria for chaos

TL;DR

This paper critically examines a geometric criterion for chaos based on local curvature, demonstrating that it is neither necessary nor sufficient, and presents examples of chaos in locally stable systems.

Contribution

The paper introduces chaotic systems with positive curvature and shows chaos can occur despite local stability, challenging the reliability of curvature-based chaos criteria.

Findings

Chaos can arise in systems with positive Gaussian curvature.

Local stability does not guarantee non-chaotic behavior.

Curvature-based criteria are insufficient to predict chaos.

Abstract

We consider here a recently proposed geometrical criterion for local instability based on the geodesic deviation equation. Although such a criterion can be useful in some cases, we show here that, in general, it is neither necessary nor sufficient for the occurrence of chaos. To this purpose, we introduce a class of chaotic two-dimensional systems with Gaussian curvature everywhere positive and, hence, locally stable. We show explicitly that chaotic behavior arises from some trajectories that reach certain non convex parts of the boundary of the effective Riemannian manifold. Our result questions, once more, the viability of local, curvature-based criteria to predict chaotic behavior.