Hyperbolic Structure and Stickiness Effect: A case of a 2D Area-Preserving Twist Mapping

TL;DR

This paper investigates how hyperbolic structures influence the stickiness effect in chaotic orbits within a 2D area-preserving twist map, revealing a relationship between geometric properties and orbital diffusion.

Contribution

It introduces numerical algorithms to compute hyperbolic periodic orbits and their manifolds, linking geometric angles to the stickiness effect in a dynamical system.

Findings

The angle between stable and unstable manifolds affects the stickiness.

Hyperbolic structures significantly influence orbital diffusion speed.

Numerical methods effectively analyze hyperbolic structures in phase space.

Abstract

The stickiness effect suffered by chaotic orbits diffusing in the phase space of a dynamical system is studied in this paper. Previous works have shown that the hyperbolic structures in the phase space play an essential role in causing the stickiness effect. We present in this paper the relationship between the stickiness effect and the geometric property of hyperbolic structures. Using a two-dimensional area-preserving twist mapping as the model, we develop the numerical algorithms for computing the positions of the hyperbolic periodic orbits and for calculating the angle between the stable and unstable manifolds of the hyperbolic periodic orbit. We show how the stickiness effect and the orbital diffusion speed are related to the angle.