The Generalization of the Periodic Orbit Dividing Surface in Hamiltonian Systems with three or more degrees of freedom -- I

TL;DR

This paper introduces a novel method to construct dividing surfaces in high-dimensional Hamiltonian systems using periodic orbits, bypassing the complex computation of normally hyperbolic invariant manifolds.

Contribution

The authors generalize the periodic orbit dividing surface construction to systems with three or more degrees of freedom, simplifying the analysis of complex Hamiltonian dynamics.

Findings

The method accurately reproduces dynamical information similar to that obtained from normally hyperbolic invariant manifolds.

It avoids complex computations by requiring only the location of a single periodic orbit.

Validated using benchmark examples for systems with two and three degrees of freedom.

Abstract

We present a method that generalizes the periodic orbit dividing surface construction for Hamiltonian systems with three or more degrees of freedom. We construct a torus using as a basis a periodic orbit and we extend this to a $2n-2$ dimensional object in the $2n-1$ dimensional energy surface. We present our methods using benchmark examples for two and three degree of freedom Hamiltonian systems to illustrate the corresponding algorithm for this construction. Towards this end we use the normal form quadratic Hamiltonian system with two and three degrees of freedom. We found that the periodic orbit dividing surface can provide us the same dynamical information as the dividing surface constructed using normally hyperbolic invariant manifolds. This is significant because, in general, computations of normally hyperbolic invariant manifolds are very difficult in Hamiltonian systems with three or more degrees of freedom. However, our method avoids this computation and the only information that we need is the location of one periodic orbit.