Astron. Astrophys. 359, 386-396 (2000)

The power spectrum of geodesic divergences as an early detector of chaotic motion

Ch.L. Vozikis, H. Varvoglis and K. Tsiganis

Aristotle University of Thessaloniki, Department of Physics, Section of Astrophysics, Astronomy and Mechanics, 54006 Thessaloniki, Greece

Received 5 August 1999 / Accepted 19 April 2000

Abstract

We propose a new method for determining the stochastic or ordered nature of trajectories in non-integrable Hamiltonian dynamical systems. The method consists in constructing a time-series from the divergence of nearby trajectories and then performing a power spectrum analysis of the series. Ordered trajectories present a spectrum that consists of a few spikes while the spectrum of stochastic trajectories is continuous. A test of the method with three different systems, a 2-D mapping as well as a 2-D and a 3-D Hamiltonian, shows that the method is fast and efficient, even in the case of sticky trajectories. The method is also applied to the motion of asteroids in the Solar System.

Key words: chaos methods: numerical celestial mechanics, stellar dynamics

Send offprint requests to: Ch.L. Vozikis

Correspondence to: chriss@astro.auth.gr

This article contains no SIMBAD objects.

Contents

• 1. Introduction
• 2. The method
• 3. Evaluation of the method
  • 3.1. 2-D mapping
  • 3.2. An improved criterion
  • 3.3. Noise level
  • 3.4. 2-D Hamiltonian system
  • 3.5. 3-D Hamiltonian system
• 4. Comparison with other methods
  • 4.1. Single trajectories
    • 4.1.1. FLI
    • 4.1.2. Spectral distance (the 4D map)
    • 4.1.3. "Spectrum" of stretching numbers
  • 4.2. Sets of trajectories
• 5. Application to asteroidal motion
• 6. Conclusions and discussion
• Acknowledgements
• References

© European Southern Observatory (ESO) 2000

Online publication: June 30, 2000
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