Transport of chaotic trajectories from regions distant from or near to structures of regular motion of the Fermi-Ulam model

The chaotic portion of phase space of the simplified Fermi-Ulam model is studied under the context of transport of trajectories in two scenarios: (i) the trajectories are originated from a region distant from the islands of regular motion and are transported to a region located at a high portion of phase space and (ii) the trajectories are originated from chaotic regions around the islands of regular motion and are transported to other regions around islands of regular motion. The transport is investigated in terms of the observables histogram of transport and survival probability. We show that the histogram curves are scaling invariant and we organize the survival probability curves in four kinds of behavior, namely (a) transition from exponential decay to power law decay, (b) transition from exponential decay to stretched exponential decay, (c) transition from an initial fast exponential decay to a slower exponential decay, and (d) a single exponential decay. We show that, depending on choice of the regions of origin and destination, the transport process is weakly affected by the stickiness of trajectories around islands of regular motion.

Figure 1

Phase space of Fermi-Ulam model for (a) $\varepsilon ={10}^{-4}$, (b) $\varepsilon ={10}^{-3}$, and (c) $\varepsilon ={10}^{-2}$.

Figure 2

Histograms of transport for different values of parameter $\varepsilon$.

Figure 3

(a) The crossover $n_p$ as a function of $\varepsilon$ and (b) the maximum value of each histogram versus $\varepsilon$.

Figure 4

The appropriate scale transformations overlap the different curves of the histograms into a single and hence universal curve.

Figure 5

(a) Log-linear plot of survival probability as function of $n$ illustrates the exponential decay observed for small values of $n$. (b) Log-log plot of survival probability illustrates power-law decays for big values of $n$.

Figure 6

(a) Log-log plot of crossover value $n_c$ of survival probability as function of $\varepsilon$. (b) Collapse of survival probability curves in a single curve after a rescaling.

Figure 7

(a) Phase space of model in gray (violet); ICs chosen in the chaotic sea inside $W_5$ and $W_{10}$ in black. (b) Amplification of region around window $W_{10}$.

Figure 8

(a) Log-log plot of survival probability of transport between different windows for different values of $\varepsilon$. (b) Log-linear plots of the initial regime of decay for the curves displayed in (a).

Figure 9

Plot of survival probability as function of $n$.

Figure 10

Log-linear plot of survival probability for different values of parameter and different windows.

Figure 11

Survival probability as function $n$ for the transport between different windows.