Diffusion and scaling in escapes from two-degrees-of-freedom Hamiltonian systems

This paper summarizes an investigation of the statistical properties of orb its escaping from three different two-degrees-of-freedom Hamiltonian system s which exhibit global stochasticity. Each time-independent H = H-0 + epsil on H', with H-0 an integrable Hamiltonian and epsilon H' a nonintegrable co rrection, not necessarily small. Despite possessing very different symmetri es, ensembles of orbits in all three potentials exhibit similar behavior. F or epsilon below a critical epsilon(0), escapes are impossible energeticall y. For somewhat higher values, escape is allowed energetically but still ma ny orbits never escape. The escape probability P computed for an arbitrary orbit ensemble decays toward zero exponentially. At or near a critical valu e epsilon(1) > epsilon(0) there is a rather abrupt qualitative change in be havior. Above epsilon(1), P typically exhibits (1) an initial rapid evoluti on toward a nonzero P-0 (epsilon), the value of which is independent of the detailed choice of initial conditions, followed by (2) a much slower subse quent decay toward zero which, in at least one case, is well fit by a power law P(t)proportional to t(-mu), with mu approximate to 0.35-0.40. In all t hree cases, P-0 and the time T required to converge toward P-0 scale as pow ers of epsilon-epsilon(1), i.e., P(0)proportional to(epsilon-epsilon(1))(al pha) and T proportional to(epsilon-epsilon(1))(beta), and T also scales in the linear size r of the region sampled for initial conditions, i. e., T pr oportional to r(-delta). To within statistical uncertainties, the best fit values of the critical exponents alpha, beta, and delta appear to be the sa me for all three potentials, namely alpha approximate to 0.5, beta approxim ate to 0.4, and delta approximate to 0.1, and satisfy alpha-beta-delta appr oximate to 0. The transitional behavior observed near epsilon(1) is attribu ted to the breakdown of some especially significant KAM tori or cantori. Th e power law behavior at late times is interpreted as reflecting intrinsic d iffusion of chaotic orbits through cantori surrounding islands of regular o rbits. (C) 1999 American Institute of Physics. [S1054-1500(99)02302-2].