A mushroom billiard with triangular foot. (a) Configuration space. (b) Phase-space representation of the semicircular hat, where is the normalized position and is the normalized reflection angle. The parameters are and , which implies .
RTS of the mushroom billiard for various choices of the control parameter . The results are consistent with a power-law tail with exponent . The distribution for was shifted vertically upward by one decade for clarity. Inset: the distribution in the interval for and .
(a) The open circular billiard for two different lengths of the hole: and . Diamonds (◆) correspond to a periodic orbit and squares (∎) to a periodic orbit . (b) Phase-space representation of (a). Circles (●) correspond to a periodic orbit , which is studied in detail in Fig. 5 , and the other symbols are the same as in (a). The four horizontal lines represent the border between the chaotic and regular regions. The first escape regions for and are the areas limited by the dashed and dotted curves, respectively.
(Color) Intervals of the control parameter for which orbits are MUPOs. All orbits with are shown.
Detailed analysis of orbits for . (a) Configuration space, where two orbits that cross the circle with radius are shown. The orbit represented by circles (●) does not hit the hole, while the orbit represented by triangles (▾) hits the hole on the right-hand side (see amplification). (b) Phase-space representation of the orbits in (a), where it is shown that they are, respectively, outside and inside the first escape region. A small perturbation in the reflection angle of the first orbit leads to a continuous rotation of the orbit in (a) [horizontal drift in (b)] until the trajectory hits the hole [enters the first escape region in (b)]. (c) Time evolution of the distance from the regular region to a chaotic trajectory that approaches the family of MUPOs in the original mushroom billiard.
(Color online) Escape regions in the phase space of the open circular billiard for . In each panel, the escape region is shown in black and the escape regions for all are shown in gray. In the first row, we show the cases , , and , while in the second row we show the case and successive amplifications for this case. Different symbols correspond to the different orbits , , and , from top to bottom, respectively. The figures at the bottom right show that only the first and the last of these orbits are MUPOs.
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