Physics
Scientific paper
Jan 1995
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995phdt........19w&link_type=abstract
Thesis (PH.D.)--UNIVERSITY OF FLORIDA, 1995.Source: Dissertation Abstracts International, Volume: 57-02, Section: B, page: 1143
Physics
Scientific paper
The fact that 'realistic' galactic potentials are not necessarily integrable implies that a large number of stars should be expected to follow stochastic trajectories. Orbits of stars in an elliptical galaxy are assumed to be affected by both the smooth mean field and internal fluctuations associated with discreteness effects. The interplay of these two effects is studied in the context of the gravitational N-body problem. These studies indicate that both the mean field and discreteness effects have an effect on the instability of the trajectories. These discreteness effects are also modelled as noise, perturbing trajectories in a nonintegrable potential. The conclusion here is that perturbations in the phase space variables grow exponentially on a time scale associated with the instability of the deterministic trajectories, even though the time scale associated with changes in collisionless invariants remains the conventional relaxation time. Because physically relevant nonintegrable potentials are typically non-hyperbolic, the standard argument that shadowing implies that noisy orbits behave like deterministic orbits does not hold. A non-hyperbolic shadowing theorem is used to determine bounds on the effect that noise has in such a potential. This noise is interpreted to be due to discreteness effects caused by close encounters. Shadowing times are calculated for both maps and differential equations, yielding the following conclusions. First, shadowing times for such systems tend towards a roughly log-normal distribution. These distributions appear to exhibit a fractal dependence on initial conditions. It has been conjectured previously that the shadowing times are expected to be proportional to 1/sqrtδ, where delta is the amplitude of the noise. It is found that, for low noise levels in Hamiltonian differential equations, the shadowing times are considerably shorter than this conjecture would indicate. Finally, the locations where shadowing breaks down appear to be correlated with the location of KAM tori and cantori of Hamiltonian systems.
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