Noise-Induced Phase Space Transport in Time-Periodic Hamiltonian Systems

Orbits in a three-dimensional potential subjected to periodic driving, V(x^i,t)=[1+m_0 sin(omega t) V_0(x^i), divide naturally into two types, regular and chaotic, between which transitions are seemingly impossible. The chaotic orbits divide in turn into two types, apparently separated by entropy barriers, namely sticky' orbit segments, which are locked’ to the driving frequency and exhibit little systematic energy diffusion, and wildly' chaotic segments, which are not so locked and can exhibit significant energy diffusion. Attention focuses on how the relative abundance of these different orbit types and the transition rate between sticky and wildly chaotic orbits depends on amplitude m_0, and on the extent to which these quantities can be altered by weak friction and/or noise and by pseudo-random variations in the driving frequency, idealized as an Ornstein-Uhlenbeck process with (in general) nonzero autocorrelation time t_c. When, in the absence of perturbations, there exist large measures of both regular and chaotic orbits, the primary effect of weak noise is to increase the relative measure of chaotic orbits. Alternatively, when almost all the orbits are already chaotic, noise serves primarily to accelerate transitions from sticky to wildly chaotic behavior. The presence or absence of friction is unimportant and the details of the noise seem largely immaterial. In particular, there is only a weak, roughly logarithmic dependence on amplitude, and additive and multiplicative noise typically have virtually identical effects. By contrast, allowing for high frequency noisy’ variations in omega tends to weaken effects of the driving, decreasing the relative measure of chaotic orbits and suppressing large scale energy diffusion.