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

This paper summarizes an investigation of the statistical properties of orbits escaping from three different two-degrees-of-freedom Hamiltonian systems which exhibit global stochasticity. Each time-independent ${\mathrm{H=H}}_0{\mathrm{+\epsilon H}}^{\prime },$ with $H_0$ an integrable Hamiltonian and ${\mathrm{\epsilon H}}^{\prime }$ a nonintegrable correction, not necessarily small. Despite possessing very different symmetries, ensembles of orbits in all three potentials exhibit similar behavior. For ε below a critical ${\epsilon }_0,$ escapes are impossible energetically. For somewhat higher values, escape is allowed energetically but still many orbits never escape. The escape probability P computed for an arbitrary orbit ensemble decays toward zero exponentially. At or near a critical value ${\epsilon }_1{\mathrm{>\epsilon }}_0$ there is a rather abrupt qualitative change in behavior. Above ${\epsilon }_1,$ P typically exhibits (1) an initial rapid evolution toward a nonzero $P_0\mathrm{(\epsilon ),}$ the value of which is independent of the detailed choice of initial conditions, followed by (2) a much slower subsequent decay toward zero which, in at least one case, is well fit by a power law ${\mathrm{P(t)\propto t}}^{\mathrm{-\mu }},$ with $\text{μ≈0.35–0.40.}$ In all three cases, $P_0$ and the time T required to converge toward $P_0$ scale as powers of ${\mathrm{\epsilon -\epsilon }}_1,$ i.e., $P_0{\mathrm{\propto (\epsilon -\epsilon }}_1{)}^{\alpha }$ and ${\mathrm{T\propto (\epsilon -\epsilon }}_1{)}^{\beta },$ and T also scales in the linear size r of the region sampled for initial conditions, i.e., ${\mathrm{T\propto r}}^{\mathrm{-\delta }}.$ To within statistical uncertainties, the best fit values of the critical exponents α, β, and δ appear to be the same for all three potentials, namely $\text{α≈0.5,}$ $\text{β≈0.4,}$ and $\text{δ≈0.1,}$ and satisfy $\text{α−β−δ≈0.}$ The transitional behavior observed near ${\epsilon }_1$ is attributed to the breakdown of some especially significant KAM tori or cantori. The power law behavior at late times is interpreted as reflecting intrinsic diffusion of chaotic orbits through cantori surrounding islands of regular orbits.