Classical dynamics of the time-dependent elliptical billiard

In this work we study the nonlinear dynamics of the static and the driven ellipse. In the static case, we find numerically an asymptotical algebraic decay for the escape of an ensemble of noninteracting particles through a small hole due to the integrable structure of the phase space of the system. Furthermore, for a certain hole position, a saturation value in the decay that can be tuned arbitrarily by varying the eccentricity of the ellipse is observed and explained. When harmonic boundary oscillations are applied, this saturation value, caused by librator-type orbits, is gradually destroyed via two fundamental processes which are discussed in detail. As a result, an amplitude-dependent emission rate is obtained in the long-time behavior of the decay, suggesting that the driven elliptical billiard can be used as a controllable source of particles.

Figure 1

PSS of the ellipse (upper part) and typical trajectories (lower part), $A=2$, $B=1$. The rotator orbit repeatedly touches a confocal ellipse, the librator orbit a confocal hyperbola.

Figure 2

Semilogarithmic plot of the escape rate; two different hole positions are shown (double-logarithmic scale in the inset).

Figure 3

Dependence of the saturation value $N_s$ on $\epsilon$.

Figure 4

Fraction of remaining particles in the IVE as a function of time for different values of the amplitude $C$ (semilogarithmic plot in the inset).

Figure 5

Same as Fig. 4 for the HVE; additionally $2\pi$-periodic oscillations of the decay are shown in the inset.

Figure 6

Vertical process in coordinate space.

Figure 7

Vertical process in phase space.

Figure 8

Horizontal process in coordinate space.

Figure 9

Horizontal process in phase space.

Figure 10

Escape time versus initial conditions in phase space for $C=0.10$ (IVE). The escape time is largest for initial conditions close to the elliptic fixed points.

Figure 11

Escape time versus initial angular momentum $F({\phi }_0,p_0)$ ($F=0$ corresponds to the separatrix in the static ellipse and $F=-3.1$ to the elliptic fixed points) in the IVE for $C=0.10$ and 0.01 (inset).

Figure 12

Same as Fig. 11 for the HVE, $C=0.25$ and $C=0.05$ (inset).

Figure 13

Average escape time and standard deviation in the IVE as a function of the initial value of $F({\phi }_0,p_0)$ for $C=0.10$.

Figure 14

Density distribution of $F({\phi }_0,p_0)$ in the IVE of the whole, the escaped, and the remaining ensemble for $C=0.01$ and 0.10 (inset).

Figure 15

Same as Fig. 14 for the HVE; $C=0.05$ and 0.25 (inset).

Figure 16

Escape time versus the absolute value of the escape velocity in the IVE for $C=0.01$ and 0.10 (inset); each point corresponds to a pair $\mathbf{(}\mid \mathit{v}\left(t_{\mathit{esc}}\right)\mid ,t_{\mathit{esc}}\mathbf{)}$.

Figure 17

Same as Fig. 16 for the HVE; $C=0.05$.

Figure 18

Distribution of the escape velocity in the IVE for different values of $C$.

Figure 19

Distribution of the escape velocity for the HVE; $C=0.05$.