Chaos, complexity, and short time Lyapunov exponents: Two alternative characterisations of chaotic orbit segments

This paper compares two tools useful in characterising ensembles of chaotic orbit segments in a time-independent galactic potential, namely Fourier spectra and short time Lyapunov exponents. Motivated by the observation that nearly regular orbit segments have simpler spectra than do wildly chaotic segments, the complexity n(k) of a discrete Fourier spectrum, defined as the number of frequencies that contain a fraction k of the total power, is identified as a robust quantitative diagnostic in terms of which to classify different chaotic segments. Comparing results derived from such a classification scheme with the computed values of short time Lyapunov exponents shows that there is a strong, often nearly linear, correlation between the complexity of an orbit and its sensitive dependence on initial conditions. Chaotic segments characterised by complex Fourier spectra tend systematically to have a larger maximum short time Lyapunov exponent than do segments with simpler spectra. It follows that the distribution of complexities, N[n(k)], associated with an ensemble of chaotic segments of length at can be used as a diagnostic for phase space transport in much the same way as the distribution of maximum short time Lyapunov exponents, N[chi], associated with the same ensemble.