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Astron. Astrophys. 320, 65-73 (1997)

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

Henry E. Kandrup 1, 2, Barbara L. Eckstein 1 and Brendan O. Bradley 3

1 Department of Astronomy, University of Florida, Gainesville, FL 32611, USA
2 Department of Physics and Institute for Fundamental Theory, University of Florida, Gainesville, FL 32611, USA
3 Department of Mathematics, University of New Mexico, Albuquerque, NM 87131, USA (kandrup@astro.ufl.edu; eck@astro.ufl.edu; brendan@math.unm.edu)

Received 20 May 1996 / Accepted 17 September 1996


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 [FORMULA] 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, [FORMULA], associated with an ensemble of chaotic segments of length [FORMULA] can be used as a diagnostic for phase space transport in much the same way as the distribution of maximum short time Lyapunov exponents, [FORMULA], associated with the same ensemble.

Key words: galaxies: kinematics and dynamics – chaos

Send offprint requests to: H.E. Kandrup (Florida Astronomy)

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998