Stochastic Web Map: Survival probability and escape frequency

Abstract

We study transport and escape in the Stochastic Web Map (SWM), an area-preserving system with phase-space structure controlled by a symmetry parameter $q$ and nonlinearity $K$. By analyzing the survival probability $P_{\text{S}}(n)$ and escape frequency $P_{\text{E}}(\ln n)$, we show that in the chaotic regime escape dynamics is governed by a single time scale $n_{\text{typ}}\propto K^{-2}h^{2}$; here $h$ is the size of the escape horizon. Deviations at large $K$ and small $h$ indicate a breakdown of the quasilinear approximation. Then, upon rescaling the time by $n_{\text{typ}}$, escape statistics becomes universal, independent of $q$. These results demonstrate that escape is controlled by global transport rather than symmetry.

Figures (6)

• Figure 1: Poincaré surfaces of section for the Stochastic Web Map of Eq. \ref{['Eq:SWM']} with $K<1$ (left panels), $K=10$ (right panels), and several values of $q$. A random initial condition with $\theta_{0}\in (-15,15)$ and $I\in(-15,15)$ was iterated $10^4$ times (black dots). The red, cyan and orange dashed lines in panels (b, d, f, h, j) indicate the position of holes in the plane $(\theta,I)$ with radii $h = 500$, $300$ and $100$, respectively.
• Figure 2: (a, b, d, e, g, h, j, k, m, n) Survival probability $P_{\text{S}}(n)$ as a function of $n$ for the Stochastic Web Map for several combinations of $K$ and $h$, and $q = 3$, $4$, $5$, $6$ and $7$. Full lines correspond to fits of the data using Eq. \ref{['Eq:Ps(n)']}, from which $\mu$ is extracted. Curves were computed from an ensemble of $10^7$ trajectories up to $n = 10^{5}$. (c, f, i, l, o). $P_{\text{S}}(n)$ as a function of $n/n_{\text{typ}}$. The relation $n_{\text{typ}} \approx \mu$ (insets) is shown as a reference.
• Figure 3: Histograms for the frequency of particle escape $P_{\text{E}}(\text{ln }n)$ when (a, d, g, j, m) $K=10$ and (b, e, h, k, n) $K=50$ for several combinations of $h$, and $q = 3$, $4$, $5$, $6$ and $7$ for the Stochastic Web Map. (c, f, i, l, o) $P_{\text{E}}(\text{ln }n)$ as a function of $n/n_{\text{typ}}$. Each histogram was constructed from an ensemble of $10^{7}$ trajectories.
• Figure 4: (a) Typical iteration time $n_{\text{typ}}$ as a function of $K$ for the Stochastic Web Map. Here $h =100$ (red symbols), $h=300$ (green symbols), and $h=500$ (blue symbols); different values of $q$ are shown: $q = 3$ ($\circ$), $q = 4$ ($\Box$), $q = 5$ ($\diamond$), $q = 6$ ($\triangle$) and $q = 7$ ($\lhd$). The scaling $n_{\text{typ}}\propto K^{-2}$ (dashed lines) is shown as a reference. (b) $n_{\text{typ}}$ as a function of $h$ for $K=10$ (red symbols), $K=30$ (green symbols), and $K=50$ (blue symbols), for the values of $q$ reported in (a). The scaling $n_{\text{typ}}\propto h^{2}$ (dashed lines) is shown as a reference. (c) Typical escape time $n_{\text{typ}}$ normalized by $h^{2}$, as a function of $K$. Same data as in panel (a).
• Figure 5: (a, b, d, e, g, h, j, k, m, n) Survival probability $P_{\text{S}}(n)$ as a function of $n$ for the Stochastic Web Map in the non-ergodic regime, with several combinations of $K$ and $h$, and $q = 3$, $4$, $5$, $6$ and $7$. Curves were computed from an ensemble of $10^7$ trajectories up to $n = 10^{5}$. (c, f, i, l, o) $P_{\text{S}}(n)$ as a function of $n/n_{\text{typ}}$. The values of $K$ in (a,d,g,j,m) are ($1.6$, $0.8$, $1.4$, $0.8$, $1.7$), respectively.