Biased diffusion inside regular islands under random symplectic perturbations

We study the random concatenation of slightly different two-dimensional Hamiltonian maps with a mixed phase space. We consider a regular island whose fixed point is identical for all maps. Trajectories of the concatenated maps near this fixed point are no longer confined to invariant tori. We derive a stochastic model for the distance from the fixed point, which turns out to be a biased random walk with multiplicative noise. We give an analytical prediction of the survival probability of trajectories inside the regular island, which asymptotically is the product of a power law and an exponential. We confirm these results numerically for the parametrically perturbed standard map.

Figure 1

Demonstration of torus hopping. In the unperturbed case, an initial point ${\mathbf{x}}_0$ near an elliptic fixed point ${\mathbf{x}}_f$ remains on a specific invariant torus (dashed lines). In the perturbed case, ${\mathbf{x}}_0$ is mapped by some random map $F_{K_0}$ along one of its invariant tori to ${\mathbf{x}}_1$. Then ${\mathbf{x}}_1$ moves along a different torus belonging to the map $F_{K_1}$ and so on. This leads to a motion transversal to the invariant tori of the unperturbed map.

Figure 2

(a) Five trajectories of the deterministic standard map at $K_n=1$. (b) Evolution of the elliptical radius $r_n$ along these trajectories and its average over ten iterations (black lines). The dashed red line approximates the average of the elliptical radius of the boundary between the regular island and the chaotic sea.

Figure 3

(a) Two trajectories, started at ${\mathbf{x}}_0=(\pi ,0.5)$ with initial elliptical radius $r_0=0.537$, of the perturbed standard map with noise strengths $\sigma =0.01$ [yellow (light gray)] and $\sigma =0.1$ [blue (dark gray)]. (b) Evolution of the elliptical radius $r_n$. The dashed red line approximates the average of the elliptical radius of the boundary of the regular island.

Figure 4

(a) Evolution of the probability distribution of the elliptical radius $r_n$ for 10 000 realizations of the perturbed standard map with $\sigma =0.02$ for the initial condition ${\mathbf{x}}_0=(\pi ,0.5)$ on a torus with average elliptical radius $r_0=0.542$. (b) Time dependence of the mean value of $r_n$ compared to prediction (19) (dashed lines) for various noise amplitudes $\sigma =0.016,0.012,0.008,0.004,0.002$ from top to bottom. (c) Variance compared to prediction (20) (dashed lines) for the same values of $\sigma$, from top to bottom. Note that for the chosen initial point ${\mathbf{x}}_0$ and noise amplitudes $\sigma$ almost none of the trajectories have escaped the regular island during the observed time of 1000 iterations.

Figure 5

(a) Survival probability $P\left(t\right)$ inside an ellipse with radius $r_{\max }=0.5$ [see inset of (b)] for different pairs $(\sigma ,r_0)$ of the noise amplitude and the initial average elliptical radius, $(0.01,0.054),(0.01,0.215),(0.02,0.054),(0.02,0.215)$ from right to left, compared to prediction Eq. (33) (dashed lines). (b) Same data divided by the exponential $e^{-\gamma t}$ plotted vs $\gamma t$. It is additionally compared to the asymptotic prediction Eq. (34) (dotted lines). Data for different $\sigma$ are almost indistinguishable after scaling.

Figure 6

(a) Survival probability $P\left(t\right)$ for escape defined by $q\notin [0,2\pi ]$ [see inset of (b)] for different pairs $(\sigma ,r_0)$ of the noise amplitude and the initial average elliptical radius, $(0.01,0.542),(0.01,1.109),(0.02,0.542),(0.02,1.109)$ from right to left, compared to prediction Eq. (33) (dashed lines) with $r_{\max }=2.036$ from last KAM torus. (b) Same data divided by the exponential $e^{-\gamma t}$ plotted vs $\gamma t$. It is additionally compared to the asymptotic prediction Eq. (34) (dotted lines).