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

Abstract

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) = of the coordinate y(t), this leads to an integrable linear evolution equation. Then, the memory function γ(t) enables us to define a frequency-dependent chaos-induced friction coefficient of the system, γ(iω). We show that this friction coefficient is related to the time correlation function Φ(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 γ r (ω) consists of three sharp peaks at frequencies ω = 0, 0.859 and 1.891. This leads to a three-term approximation of the memory function, γ(t), with a correlation time τ r ∼ 5T (with T = 2π). It is also shown that the structure of the power spectrum I y (ω) of y(t) consists of four sharp peaks at frequencies ω = 0, 0.500, 0.797 and 1.000. This leads to a four-term approximation of the time correlation function C y (t) with a correlation time τ M ∼ 6T. The frequencies and line widths of the sharp peaks of I y (ω) are given by the friction coefficient γ(iω).