Effective Lyapunov Numbers and Correlation Dimensions in a 3-D Hamiltonian System

Transport in Hamiltonian systems, in the case of strong perturbation, can be modeled as a diffusion process, with the diffusion coefficient being constant and related to the maximal Lyapunov number (Konishi 1989). In this respect the relation found by Lecar et al. (1992) between the escape time of asteroids, TE, and the Lyapunov time, TL, can be easilly recovered (Varvoglis & Anastasiadis 1996). However, for moderate perturbations, chaotic trajectories may have a peculiar evolution, owing to stickiness effects or migration to adjacent stochastic regions. As a result, the function χ(t), which measures the exponential divergence of nearby trajectories, changes behaviour within different time intervals. Therefore, trajectories may be divided into segments, i = 1,..., n, each one being assigned an “Effective” Lyapunov Number (ELN), λi = χ(ti).

Chaos and Its Impact on the Foundations of Statistical Mechanics

In this work we present a brief derivation of the periodic orbit expansion for simple dynamical systems, and then we apply it to the study of a classical statistical mechanical model, the Lorentz gas, both at equilibrium and in a nonequilibrium steady state. The results are compared with those obtained through standard molecular dynamics simulations, and they are found to be in good agreement. The form of the average using the periodic orbit expansion suggests the definition of a new dynamical partition function, which we test numerically. An analytic formula is obtained for the Lyapunov numbers of periodic orbits for the nonequilibrium Lorentz gas. Using this formula and other numerical techniques we study the nonequilibrium Lorentz gas as a dynamical system and obtain an estimate of the upper bound on the external field for which the system remains ergodic.

Pluto's Lyapunov numbers in different dynamical models

We present the results of numerical integrations of Pluto and some fictitious Plutos in three different models (the circular and the elliptic restricted three body problem and the outer solar system). We determined the “extension” of the stable region in these models by means of the Lyapunov Characteristic Numbers and by an analysis of the orbital elements.