Ivan I. Shevchenko

LYAPUNOV AND DIFFUSION TIMESCALES IN THE SOLAR NEIGHBORHOOD

We estimate the Lyapunov times (characteristic times of predictability of motion) in Quillens models for the dynamics in the solar neighborhood. These models take into account perturbations due to the Galactic bar and spiral arms. For estimating the Lyapunov times, an approach based on the separatrix map theory is used. The Lyapunov times turn out to be typically of the order of 10 Galactic years. We show that only in a narrow range of possible values of the problem parameters the Galactic chaos is adiabatic; usually it is not slow. We also estimate the characteristic diffusion times in the chaotic domain. In a number of models, the diffusion times turn out to be small enough to permit migration of the Sun from the inner regions of the Milky Way to its current location. Moreover, due to the possibility of ballistic flights inside the chaotic layer, the chaotic mixing might be even far more effective and quicker than in the case of normal diffusion. This confirms the dynamical possibility of Minchev and Famaeys migration concept.

Marginal Resonances and Intermittent Behaviour in the Motion in the Vicinity of a Separatrix

A condition upon which sporadic bursts (intermittent behaviour) of the relative energy become possible is derived for the motion in the chaotic layer around the separatrix of a non-linear resonance. This is a condition for the existence of a marginal resonance, i.e. a resonance located at the border of the layer. A separatrix map in Chirikovs form [Chirikov, B. V., Phys. Reports 52, 263 (1979)] is used to describe the motion. In order to provide a straightforward comparison with numeric integrations, the separatrix map is synchronized to the surface of the section farthest from the saddle point. The condition of intermittency is applied to clear out the nature of the phenomenon of bursts of the eccentricity of chaotic asteroidal trajectories in the 3/1 mean motion commensurability with Jupiter. On the basis of the condition, a new intermittent regime of resonant asteroidal motion is predicted and then identified in numeric simulations.

On the recurrence and Lyapunov time scales of the motion near the chaos border

Conditions for the emergence of a statistical relationship between

Maximum Lyapunov Exponents for Chaotic Rotation of Natural Planetary Satellites

T_r

The rotation states predominant among the planetary satellites

, the chaotic transport (recurrence) time, and

Observations and Theoretical Analysis of Lightcurves of Natural Satellites of Planets

T_L

Lyapunov exponents in resonance multiplets

, the local Lyapunov time (the inverse of the numerically measured largest Lyapunov characteristic exponent), are considered for the motion inside the chaotic layer around the separatrix of a nonlinear resonance. When numerical values of the Lyapunov exponents are measured on a time interval not greater than

Lyapunov exponents in the Hénon-Heiles problem

T_r

The Kepler map in the three-body problem

, the relationship is shown to resemble the quadratic one. This tentatively explains numerical results presented in the literature.

The width of a chaotic layer

Based on the Chirikov approach [1, 2] within the context of the theory of separatrix mappings, we suggest and substantiate a simple method for estimating the maximum Lyapunov characteristic exponent (MLCE) of motion in a chaotic layer in the neighborhood of nonlinear resonance separatrix of a Hamiltonian system under an asymmetric periodic perturbation. For a number of natural planetary satellites, using this method, the estimates are made of the MLCE of chaotic rotation (relative to the center of mass) in the main chaotic layer, i.e., in the chaotic layer in the neighborhood of a synchronous resonance separatrix. The value of the MLCE is determined by an orbital eccentricity and by the parameter of the satellites dynamic asymmetry. The quantity inverse to the MLCE gives a typical time of predictable rotational motion.