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.

15 pages, 6 figures

References in corpus (6)

• Semiclassical Foundation of Universality in Quantum Chaos
• Periodic-Orbit Theory of Universality in Quantum Chaos
• Semiclassical form factor for spectral and matrix element fluctuations of multi-dimensional chaotic systems
• Analytical description of the structure of chaos
• Action differences between fixed points and accurate heteroclinic orbits
• Geometric determination of heteroclinic and unstable periodic orbit classical actions

Cited by in corpus (6)

• Semiclassical roots of universality in many-body quantum chaos
• A chaotic lattice field theory in one dimension
• Orbital carriers and inheritance in discrete-time quadratic dynamics
• Spin-relaxation anisotropy in a nanowire quantum dot with strong spin-orbit coupling
• Exact decomposition of homoclinic orbit actions in chaotic systems: Information reduction
• An asymptotic relationship between homoclinic points and periodic orbit stability exponents