Effects of periodic forcing in chaotic scattering

The effects of a periodic forcing on chaotic scattering are relevant in certain situations of physical interest. We investigate the effects of the forcing amplitude and the external frequency in both the survival probability of the particles in the scattering region and the exit basins associated to phase space. We have found an exponential decay law for the survival probability of the particles in the scattering region. A resonant-like behavior is uncovered where the critical values of the frequencies $\omega \simeq 1$ and $\omega \simeq 2$ permit the particles to escape faster than for other different values. On the other hand, the computation of the exit basins in phase space reveals the existence of Wada basins depending of the frequency values. We provide some heuristic arguments that are in good agreement with the numerical results. Our results are expected to be relevant for physical phenomena such as the effect of companion galaxies, among others.

Figure 1

Level curves of the Hénon-Heiles potential, exits, limit of bounded motion, and the eight ${\Pi }_i$ nonlinear normal modes and three Lyapunov orbits $L_i$ on the $(x,y)$ plane for (a) $E<E_e$ and (b) $E>E_e$.

Figure 2

Plot of the exit basins of the unperturbed case. Blue (dark gray), red (gray), and yellow (light gray) denote the set of initial conditions that escape through exits 1, 2, or 3, respectively. The green (lower) color on the right panel denotes the initial conditions that are trapped in the scattering region. Panels (a) and (b) [respectively, (c) and (d)] show the basins in physical space $(x,y)$ [in phase space $(y,\dot{y})$] for two different values of the energy, $E=0.2$ and $E=0.25$. Panel (e) represents the $(y,E)$ diagram in which we can observe that $E_e=1/6$ is the threshold value for which the particles can escape. Initial conditions in panels (a) and (b) are chosen to maintain the symmetry of the problem. In panels (c) and (d) the initial conditions are such that $x\left(0\right)=0$ and $\dot{x}\left(0\right)$ is obtained from the energy $E$. Finally, in panel (e) we take $x\left(0\right)=\dot{y}\left(0\right)=0$, and $\dot{x}\left(0\right)$ is computed from $E$.

Figure 3

Typical trajectories of the forced Hénon-Heiles system with initial conditions $(x_0,{\dot{x}}_0,y_0,{\dot{y}}_0)=(0,0,0,0)$ and parameter values $A=0.2$ and (a) $\omega =0$, (b) $\omega =1$, (c) $\omega =10$, and (d) $\omega =0.1$, respectively. We can observe different dynamical behaviors depending on the frequency values: escaping dynamics [(b) and (c)] or quasiperiodic motions [(a) and (d)].

Figure 4

Plots of exit basins for the system with forcing with $\omega =0.1,1,10$. (Left) Plot of the exit basins and the basins of attraction of the Hénon-Heiles Hamiltonian in the physical space $(x,y)$ and the phase space $(y,\dot{y})$ for the energy values indicated in each figure. The color code is as follows: green (lower) corresponds to bounded orbits and blue (dark gray), yellow (light gray), and red (gray) correspond to the initial condition that leads to exits numbers 1, 2, and 3, respectively. (Right) Plot of the exit basins in the $(y,E)$ diagram. Initial conditions are chosen as in Fig. 2.

Figure 5

Plot of the $(y,\omega )$ diagram for $E=0.01,0.05,0.15,0.2$. The color code is as described in the captions to previous figures. Green (upper) corresponds to bounded orbits. Notice that for $E=0.2$ there are escapes for all values of the frequency $\omega$ and resonant values also lead to escaping behavior in all the energies. Wada basins are present in presence of forcing as can be observed in the basins in both physical space $(x,y)$ and phase space $(y,Y)$ for $\omega =1$, $\omega =0,1$ and $\omega =10$.

Figure 6

Typical exponential decay law for the particles remaining in the scattering region at time $t$. $R$ denotes the fraction of particles remaining in the scattering region. Here, we shoot $5\times {10}^3$ particles with energy $E=0.2$ from $(x_0,y_0)=(0,-0.5)$ and $\theta \in (0,2\pi )$. The forcing amplitude is $A=0.1$. We observe the optimal value of the frequency $\omega =1$ for which the escapes take place faster than for other values of $\omega$.

Figure 7

(a) Plot of $\alpha =1/\tau$ versus $ln$ of the external frequency $ln\left(\omega \right)$, for $E=0.15<E_e$ and $A=0.1$, showing a resonant-like behavior in which the main resonant frequency is ${\omega }_1\simeq 1$. (b) A plot for the case $E=0.20>E_e$, where again this resonant phenomenon is observed very clearly.

Figure 8

Plot of the escape times $T_e$ versus the amplitude $A$ of the external forcing, for a single trajectory, for $\omega =1$ (black), $\omega =10$ [green (light gray)], and $\omega =0.1$ [red (dark gray)], respectively. When the value of the amplitude of the external forcing is increased, the escape times of the particles decreases as expected. In all simulations $E=0.12$, $\theta =0$, and $(x_0,y_0)=(0,-0.5)$.

Figure 9

Plots of escape times $T_e$ vs frequency $\omega$ for several values of the energy up to a maximum propagation time of $10000$. From top to bottom: $E=0.01,0.05,0.15,0.2$. The mean values of the escape times for all values of coordinate $y$, as shown in Fig. 5, are depicted in red (gray). The mean plus-minus $3\sigma$ error are depicted in blue (dark gray) and the values in between in green (light gray). Notice that for $E=0.2$ all the orbits escape before the maximum time. Forcing amplitude is $A=0.1$.

Figure 10

Plot of the orbit period $T_i$ versus the energy of the nonlinear normal modes ${\Pi }_{4-7}$ and the Lyapunov orbits $L_i$ for the unperturbed Hénon-Heiles Hamiltonian. Small inset: Plot of the nonlinear normal modes as the energy grows.

Figure 11

Plot of the final kinetic energy $K$ versus the frequency $\omega$ for $E=0.05$, forcing amplitude $A=0.1$, and propagation time $t=3T_1\approx 3\times 2\pi$. We clearly observe the local extrema around the resonant frequencies $\omega \simeq 1$ and $\omega \simeq 2$, as previously mentioned.