Dynamical Properties of a Generalized Collision Rule for Multi-Particle Systems

Abstract

The theoretical basis for the Lyapunov exponents of continuous- and discrete-time dynamical systems is developed, with the inclusion of the statement and proof of the Multiplicative Ergodic Theorem of Oseledec. The numerical challenges and algorithms to approximate Lyapunov exponents and vectors are described, with multiple illustrative examples. A novel generalized impulsive collision rule is derived for particle systems interacting pairwise. This collision rule is constructed to address the question of whether or not the quantitative measures of chaos (e.g. Lyapunov exponents and Kolmogorov-Sinai entropy) can be reduced in these systems. Major results from previous studies of hard-disk systems, which interact via elastic collisions, are summarized and used as a framework for the study of the generalized collision rule. Numerical comparisons between the elastic and new generalized rules reveal many qualitatively different features between the two rules. Chaos reduction in the new rule through appropriate parameter choice is demonstrated, but not without affecting the structural properties of the Lyapunov spectra (e.g. symmetry and conjugate-pairing) and of the tangent space decomposition (e.g. hyperbolicity and domination of the Oseledec splitting). A novel measure of the degree of domination of the Oseledec splitting is developed for assessing the impact of fluctuations in the local Lyapunov exponents on the observation of coherent structures in perturbation vectors corresponding to slowly growing (or contracting) modes. The qualitatively different features observed between the dynamics of generalized and elastic collisions are discussed in the context of numerical simulations. Source code and complete descriptions for the simulation models used are provided.