Exact relations between homoclinic and periodic orbit actions in chaotic systems

Abstract

Homoclinic and unstable periodic orbits in chaotic systems play central roles in various semiclassical sum rules. The interferences between terms are governed by the action functions and Maslov indices. In this article, we identify geometric relations between homoclinic and unstable periodic orbits, and derive exact formulae expressing the periodic orbit classical actions in terms of corresponding homoclinic orbit actions plus certain phase space areas. The exact relations provide a basis for approximations of the periodic orbit actions as action differences between homoclinic orbits with well-estimated errors. This make possible the explicit study of relations between periodic orbits, which results in an analytic expression for the action differences between long periodic orbits and their shadowing decomposed orbits in the cycle expansion.

A chaotic lattice field theory in one dimension

“…But that is Newton's Second Law: "acceleration equals force," so Percival and Vivaldi [138] refer to this formulation as 'Newtonian'. Here we follow Allroth [2], Mackay, Meiss, Percival, Kook & Dullin [61,109,[122][123][124], and Li and Tomsovic [113] in referring to it as 'Lagrangian'.…”

“…In preparing this section we have found expositions of Lagrangian dynamics for discrete time systems by MacKay, Meiss and Percival [123,124], and Li and Tomsovic [113] particulary helpful. Hill's formula as derived by Mackay and Meiss [122] and Allroth [2] (see Allroth eq.…”

“…11.2), but only a few key things to understand. For a 1-dimensional lattice, there are only two kinds of qualitatively different symmetry transformations, (i) translations (113) and reflections (115), which reverse the direction of translation.…”

“…For example, the 'temporal Bernoulli' defining equation ( 28) retains its form under conjugation by any C ∞ translation (113),…”

Orbital carriers and inheritance in discrete-time quadratic dynamics

“…With respect to applications beyond the scope of dynamical systems, we mention briefly that the concept of inheritance is potentially attractive for atomic physics, where it seems to imply the interesting and unsuspected possibility of rearranging certain orbit-dependent contributions in cycle expansions and semiclassical sums needed for calculating energy spectra and density of states using, e.g., Gutzwiller's trace formula 15,16,17,18,19,20,21,22 .…”