Resonant driving of chaotic orbits.

Abstract

Finite time segments of chaotic orbits in strongly nonintegrable potentials often exhibit complicated power spectra, which, despite being broadband, are dominated by frequencies \ensuremath{\omega} appropriate for ``nearby'' regular orbits. This implies that even low amplitude periodic driving can trigger complicated resonant couplings, evidencing a sensitive dependence on the driving frequency \ensuremath{\Omega}. Numerical experiments involving individual chaotic orbits indicate that the response to a low amplitude time-periodic perturbation, as measured, e.g., by the maximum excursion in energy arising within a given time interval, can exhibit a sensitive dependence on \ensuremath{\Omega}, with substantial structure even on scales \ensuremath{\delta}\ensuremath{\Omega}${10}^{\mathrm{\ensuremath{-}}4}$ times a typical natural frequency \ensuremath{\omega}. Ensembles of chaotic initial conditions driven with a frequency \ensuremath{\Omega} comparable to the natural frequencies of the unperturbed orbits typically display diffusive behavior: The distribution of energy changes, N(\ensuremath{\delta}E(t)), at any given time t is Gaussian and the rms value of the change in energy \ensuremath{\delta}${\mathit{E}}_{\mathrm{rms}}$=A(\ensuremath{\Omega},E)\ensuremath{\alpha}${\mathit{t}}^{1/2}$, where \ensuremath{\alpha} denotes the driving amplitude. For fixed energy E, the proportionality constant A is independent of the detailed choice of initial conditions, but can exhibit a complicated dependence on \ensuremath{\Omega}. Potential implications for galactic dynamics are discussed.