Orbital complexity, short-time Lyapunov exponents, and phase space transport in time-independent Hamiltonian systems

This paper compares two alternative characterizations of chaotic orbit segments, one based on the complexity of their Fourier spectra, as probed by the number of frequencies n(k) required to capture a fixed fraction k of the total power, and the other based on the computed values of short-time Lyapunov exponents chi. An analysis of orbit ensembles evolved in several different two- and three-dimensional potentials reveals that there is a strong, roughly linear correlation between these alternative characterizations, and that computed distributions of complexities, N[n(k)] and short-time chi, N[chi] often assume similar shapes. This corroborates the intuition that chaotic segments which are especially unstable should have Fourier spectra with particularly broad-band power. It follows that orbital complexities can be used as probes of phase space transport and other related phenomena in the same manner as can short-time Lyapunov exponents.