Exact relations between homoclinic and periodic orbit actions in chaotic systems

Asymptotic relationship between homoclinic points and periodic orbit stability exponents

The magnitudes of the terms in periodic orbit semiclassical trace formulas are determined by the orbits' stability exponents. In this paper, we demonstrate a simple asymptotic relationship between those stability exponents and the phase-space positions of particular homoclinic points.

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

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 packets, coherent states, 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.