The Memory Function and Chaos-Induced Friction in the Chaotic Hénon-Heiles System

A non-Markovian linear stochastic equation for the momentum p(y)(t) is derived for the purpose of clarifying transport processes in the chaotic Henon-Heiles system with the aid of the Mori projection operator formalism. For the time correlation function C-y(t) < y(t)y(O)> of the coordinate y(t), this leads to an integrable linear evolution equation. Then, the memory function gamma(t) enables us to define a frequency-dependent chaos-induced friction coefficient of the system, gamma(iw). We show that this friction coefficient is related to the time correlation function phi(t) of a nonlinear force f (t), which can be computed numerically. Thus, in the case that the total energy is E = 1/6, it turns out that the structure of the frequency-dependent friction coefficient gamma(r)(w) consists of three sharp peaks at frequencies w = 0, 0.859 and 1.891. This leads to a three-term approximation of the memory function, gamma(t), with a correlation time tau(r) similar to 5T (with T = 2 pi). It is also shown that the structure of the power spectrum I-y(w) of y(t) consists of four sharp peaks at frequencies w = 0, 0.500, 0.797 and 1.000. This leads to a four-term approximation of the time correlation function Cy(t) with a, correlation time tau(M) similar to 6T. The frequencies and line widths of the sharp peaks of I-y (w) are given by the friction coefficient gamma(iw).