Exact decomposition of homoclinic orbit actions in chaotic systems: Information reduction

TL;DR

This paper develops exact formulas to decompose complex homoclinic orbit actions into simpler components using phase-space cell areas, potentially simplifying semiclassical calculations in chaotic systems.

Contribution

It introduces a method to express complex homoclinic orbit actions as linear combinations of simpler ones plus phase-space cell areas, reducing the complexity of semiclassical sums.

Findings

Exact relations for homoclinic orbit actions derived

Complex orbit actions expressed via simpler orbit actions and cell areas

Potential for simplified semiclassical formulas in chaotic dynamics

Abstract

Homoclinic and heteroclinic orbits provide a skeleton of the full dynamics of a chaotic dynamical system and are the foundation of semiclassical sums for quantum wave packet, coherent state, and transport quantities. Here, the homoclinic orbits are organized according to the complexity of their phase-space excursions, and exact relations are derived expressing the relative classical actions of complicated orbits as linear combinations of those with simpler excursions plus phase-space cell areas bounded by stable and unstable manifolds. The total number of homoclinic orbits increases exponentially with excursion complexity, and the corresponding cell areas decrease exponentially in size as well. With the specification of a desired precision, the exponentially proliferating set of homoclinic orbit actions is expressible by a slower-than-exponentially increasing set of cell areas, which may present a means for developing greatly simplified semiclassical formulas.