Time Correlations and Diffusion of a Conservative Forced Pendulum

Abstract

The time correlations and diffusion of chaotic orbits in a periodically forced pendulum without friction are studied. The pendulum exhibits a Poincare section with period T at times t = jT (j =0 , 1, 2, ··· ). The time-correlation function C(t) ≡� p(t)p(0)� of the angular velocity p(t) oscillates with period T ,e ven ast →∞ , since the average quantities of the system have a periodicity with period T , due to the periodic external force. Studying the approach to asymptotic oscillation, we find that the time-correlation function C(t) exhibits an inverse power decay t −(β−1) (1 <β <2), where there exist islands of accelerator-mode tori. Then, it is also shown that the power spectrum Ip(ω )o fp(t) obeys an inverse power law ω −(2−β) for small frequency ω � 2π/T. We also calculate the mean square displacement σ 2 (n) of the angular variable qn ≡ q(nT ) on the Poincare section, and show that σ 2 (n) ∝ n ζ (ζ =3 − β) for n →∞ , leading to anomalous diffusion with 1 <ζ< 2.