Statistics – Applications
Scientific paper
May 2000
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2000dda....31.0105s&link_type=abstract
American Astronomical Society, DDA Meeting #31, #01.05; Bulletin of the American Astronomical Society, Vol. 32, p.856
Statistics
Applications
Scientific paper
When a nonlinear oscillator is perturbed by a small amplitude, rapid frequency force, a narrow chaotic layer forms in the local neighborhood of the oscillator's separatrix. Alternatively, if the oscillator is perturbed by a large amplitude, slow frequency force, a broad chaotic sea fills the resonant region between the oscillator's pulsating separatrices. The two limiting cases are often called "fast chaos" and "slow chaos" respectively, because the rate of chaotic diffusion is very different in these two situations. In applications to celestial mechanics, fast chaos has been associated with overlapping mean motion resonances in the asteroid belt and slow chaos has been used to explain the chaotic evolution of planetary obliquities. I will present a model of resonant interaction between two planets that appears to simultaneously exhibit both fast and slow chaos. The model is a four-dimensional symplectic map which includes both a mean motion resonance and a secular resonant interaction. After an appropriate coordinate transformation, the model can be divided into a fast oscillator and a slow oscillator that are dynamically coupled. The fast oscillator remains in the thin chaotic layer near its separatrix and slowly forces the slow oscillator. The slow oscillator exhibits an adiabatic "invariant" that occasionally jumps as the trajectory crosses a slowly pulsating separatrix. Interestingly, one can derive analytic expressions for the rates of fast and slow diffusion for this model.
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