Geometric investigation of chaos unfolding in Hamiltonian systems

TL;DR

This paper explores the geometric nature of chaos in Hamiltonian systems using the Jacobi-Levi-Civita equation, revealing that trajectory divergence occurs in discrete jumps at boundary interactions, influenced by boundary scattering and parametric resonance.

Contribution

It introduces a geometric perspective on chaos in Hamiltonian systems, emphasizing the discrete nature of trajectory divergence and its relation to boundary scattering and resonance effects.

Findings

Chaotic divergence occurs in discrete jumps at boundary turning points.

Boundary scattering significantly influences the chaotic behavior.

The divergence process relates to parametric resonance phenomena.

Abstract

In this work we revisit the geometric approach to chaos in Hamiltonian dynamics, by means of the Jacobi-Levi-Civita equation (JLCE). We inspect numerically two low-dimensional dynamical systems; show that, along chaotic orbits, the exponential divergence between nearby trajectories quantified by the JLCE does not unfold in a continuous manner, rather is closer to a multiplicative discrete process: in correspondence of each turning point, where the trajectory bounces away from the boundary of the energetically allowed region, the relative separation increases sharply and abruptly. We highlight through analytical and numerical arguments that the chaotic rather than regular nature of the trajectory is determined by the details of the scattering with the boundary, and interpret these results in terms of parametric resonance theory, and specifically the Mathieu equation.