Numerical study of the stochasticity of dynamical systems with more than two degrees of freedom

Abstract

Dynamical Systems with N degrees of freedom can be reduced, by the method of surfaces of section, to the study of a (2N - 2) dimensional mapping. We consider here, as a model problem, the mapping given, for N = 3, by the following equations: T x 1= x 0+a 1 sin(x 0+y 0)+b sin(x 0+y 0+z 0+t 0) y 1= x 0+y 0 ( mod2π) z 1= z 0+a 2 sin(z 0+t 0)+b sin(x 0+y 0+z 0+t 0) t 1= z 0+t 0 The purpose of the present paper is to test numerical methods for the study of the stochasticity of this dynamical system. Therefore, we study the two largest eigenvalue—in absolute magnitude—of the tangential linear mapping of T n . We then meet important precision problems on the computer, and use both a standard method with multiprecision computations and an entirely new one. The method enables us to show numerically that the mapping T is indeed close to a C-system, in the ergodic zone. This method can be generalized easily to systems with a small number N of degrees of freedom, and becomes then a numerical tool for the study of the stochasticity of such dynamical systems.