Resonant driving of chaotic orbits

Finite time segments of chaotic orbits in strongly nonintegrable potentials often exhibit complicated power spectra, which, despite being broadband, are dominated by frequencies ω 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 Ω. 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 Ω, with substantial structure even on scales δΩ<${10}^{\mathrm{-}4}$ times a typical natural frequency ω. Ensembles of chaotic initial conditions driven with a frequency Ω comparable to the natural frequencies of the unperturbed orbits typically display diffusive behavior: The distribution of energy changes, N(δE(t)), at any given time t is Gaussian and the rms value of the change in energy δ${\mathit{E}}_{\mathrm{rms}}$=A(Ω,E)α${\mathit{t}}^{1/2}$, where α 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 Ω. Potential implications for galactic dynamics are discussed.