Three-dimensional billiards: Visualization of regular structures and trapping of chaotic trajectories

The dynamics in three-dimensional (3D) billiards leads, using a Poincaré section, to a four-dimensional map, which is challenging to visualize. By means of the recently introduced 3D phase-space slices, an intuitive representation of the organization of the mixed phase space with regular and chaotic dynamics is obtained. Of particular interest for applications are constraints to classical transport between different regions of phase space which manifest in the statistics of Poincaré recurrence times. For a 3D paraboloid billiard we observe a slow power-law decay caused by long-trapped trajectories, which we analyze in phase space and in frequency space. Consistent with previous results for 4D maps, we find that (i) trapping takes place close to regular structures outside the Arnold web, (ii) trapping is not due to a generalized island-around-island hierarchy, and (iii) the dynamics of sticky orbits is governed by resonance channels which extend far into the chaotic sea. We find clear signatures of partial transport barriers. Moreover, we visualize the geometry of stochastic layers in resonance channels explored by sticky orbits.

Figure 1

Poincaré recurrence statistics $P\left(t\right)$ in the 3D billiard, defined in Eq. (3), for the real flight time (red dashed line) and the number of mappings (blue line). The dotted line indicates a power-law decay $\sim t^{-\gamma }$ with $\gamma =1.2$. The upper inset shows the 3D paraboloid billiard with part of the boundary removed for visual reasons. Inside a sticky trajectory (blue line) is shown, which starts and returns to region $\mathrm{\Lambda }$ (yellow ring). The lower inset shows the trapped orbit (blue dots) and regular phase-space structures (gray) in a phase-space representation.

Figure 2

Phase space for the two 2D billiards embedded in the 3D paraboloid billiard. Regular tori (red rings) are arranged around an elliptic fixed point (black dot) in the center ${\mathbf{u}}_{\mathrm{fp}}^{\text{2D}}=(0,0)$. The regular island is embedded in the chaotic region, for which one chaotic orbit is shown (blue dots). (a) For $\alpha =1.04$ two resonances 8:3 and 3:1 within the regular region and one 10:3 resonance at the edge of the regular island are shown. (b) For $b=1.12$ two resonances 3:1 and 10:3 within the regular region and one 14:4 resonance at the edge of the island are shown.

Figure 3

Phase space of the billiard defined in Eq. (3) with $a=1.04$ and $b=1.12$ for $|p_y|\le \varepsilon ={10}^{-4}$. Regular tori appear as 1D lines (gray). For the labeled tori ($\begin{array}{|c|}\hline \mathrm{a}\\ \hline\end{array}$–$\begin{array}{|c|}\hline \mathrm{e}\\ \hline\end{array}$) the corresponding trajectories in configuration space are shown in Fig. 4. For a rotating view see the Supplemental Material [115].

Figure 4

Three-dimensional paraboloid billiard with boundaries $\partial {\mathrm{\Omega }}_2$ (green) and $\partial {\mathrm{\Omega }}_1$ for only the part with $y<0$ and trajectories in configuration space ($\begin{array}{|c|}\hline \mathrm{a}\\ \hline\end{array}$–$\begin{array}{|c|}\hline \mathrm{e}\\ \hline\end{array}$) for the highlighted tori shown in Fig. 3: (a) trajectory (red) close to ${\mathcal{M}}_x^{\mathrm{fp}}$, (b) trajectory (blue) close to ${\mathcal{M}}_y^{\mathrm{fp}}$, (c) trajectory (orange) of an uncoupled resonance related to the billiard system with $a=1.04$, (d) trajectory (cyan) of a coupled resonance, and (e) periodic trajectory (pink) of a double resonance with period 35.

Figure 5

Frequency space of the 3D paraboloid billiard defined in Eq. (3) for $a=1.04$ and $b=1.12$. In total $9.6\times {10}^6$ frequencies $({\nu }_x,{\nu }_y)$ for the 2D tori are displayed (gray points). The rightmost tip $({\nu }_x^{\mathrm{fp}},{\nu }_y^{\mathrm{fp}})=(0.41143,0.35130)$ corresponds to the elliptic-elliptic fixed point ${\mathbf{u}}_{\mathrm{fp}}$. Two Lyapunov families of elliptic 1D tori ${\mathcal{M}}_x^{\mathrm{fp}}$ (red) and ${\mathcal{M}}_y^{\mathrm{fp}}$ (blue) emanate from this point. Some important resonance lines are shown as magenta dashed lines. The insets show magnifications of the frequency space. Colored points marked by $\begin{array}{|c|}\hline \mathrm{a}\\ \hline\end{array}$–$\begin{array}{|c|}\hline \mathrm{e}\\ \hline\end{array}$ correspond to the examples shown in Figs. 3 and 4.

Figure 6

Trapped orbit with recurrence time $t_{\text{rec}}\simeq 1.6\times {10}^9$ in the phase-space representation with time encoded in color. The box indicates the magnification shown in Fig. 7. For a rotating view, see the Supplemental Material [115].

Figure 7

Magnification (rotated) of the trapped orbit of Fig. 6 with recurrence time $t_{\text{rec}}\simeq 1.6\times {10}^9$ in the phase space with time $[0,t_{\text{rec}})$ encoded in color. Regular tori of some important resonance channels $22\text{:}0\text{:}7, 14\text{:}2\text{:}5, 24\text{:}-2\text{:}7$, and $15\text{:}1\text{:}5$ are shown in black. The sticky orbit approaches the regular structures by going across several resonance channels. For the time-resolved animation, see the Supplemental Material [115].

Figure 8

Frequency-space representation of the trapped orbit (see Fig. 7), with $t_{\text{rec}}=1.6\times {10}^9$ and time encoded in color: (a) magnification of Fig. 5 with regular tori (gray dots) and selected resonance lines, (b) and (c) frequency-time signals ${\nu }_y\left(t\right)$ and ${\nu }_x\left(t\right)$, respectively, and (d) magnification of frequency space [see the inset in (a)] in local coordinates.

Figure 9

Poincaré recurrence statistics $P\left(t\right)$ for the 2D billiard with $a=1.04$ for ${10}^{13}$ trajectories started in region $\mathrm{\Lambda }$ for the number of mappings (blue line) and real flight time (red dashed line). The dotted line indicates a power-law decay $\sim t^{-\gamma }$ with $\gamma =1.5$. The upper inset shows the opened 2D billiard with the parabola as a boundary and a short sticky trajectory with $t_{\text{rec}}=131$. The lower inset shows the Poincaré section $(x,p_x)$ with regular tori (red curves), opening $\mathrm{\Lambda }$ (yellow rectangle), and corresponding trapped orbit (blue dots).

Figure 10

(a) Trapped orbit of the 2D billiard with $a=1.04$ with recurrence time $t_{\text{rec}}\simeq 1.52\times {10}^7$ in a magnification of phase space with time $[0,t_{\text{rec}})$ encoded in color. Additionally shown is the boundary circle (solid black curve) and important surrounding resonances are labeled as the fraction $n/m$. For the time-resolved animation, see the Supplemental Material [115]. (b) Frequency-time representation $\nu \left(t\right)$ of the sticky orbit. Frequencies of important resonances are shown as magenta dashed lines and the frequency ${\nu }_{\text{BC}}$ of the boundary circle as a solid horizontal line.

Figure 11

Trapped orbit with recurrence time $t_{\text{rec}}\simeq 7.6\times {10}^7$ and time $t$ encoded in color: (a) magnification of Fig. 5 with regular tori (gray dots) and selected resonance lines, (b) and (c) frequency-time signals ${\nu }_y\left(t\right)$ and ${\nu }_x\left(t\right)$, and (d) phase-space representation; for a rotating view, see the Supplemental Material [115].