^{1,a)}, Adilson E. Motter

^{2}and Holger Kantz

^{3}

### Abstract

We investigate the dynamical properties of chaotic trajectories in mushroom billiards. These billiards present a well-defined simple border between a single regular region and a single chaotic component. We find that the stickiness of chaotic trajectories near the border of the regular region occurs through an infinite number of marginally unstable periodic orbits. These orbits have zero measure, thus not affecting the ergodicity of the chaotic region. Notwithstanding, they govern the main dynamical properties of the system. In particular, we show that the marginally unstable periodic orbits explain the periodicity and the power-law behavior with exponent observed in the distribution of recurrence times.

The stickiness of chaotic trajectories in Hamiltonian systems, characterized by long tails in the recurrence time statistics, is usually associated with the presence of partial barriers to the transport in the neighborhood of hierarchies of Kolmogorov-Arnold-Moser (KAM) islands. However, as we show, these hierarchical structures are not necessary for the occurrence of stickiness. Here we study mushroom billiards,

^{1}which are analytically solvable systems without hierarchies of KAM islands, and we show that these systems present stickiness due to the presence of one-parameter families of marginally unstable periodic orbits within the chaotic region.

E.G.A. is supported by CAPES (Brazil) and DAAD (Germany). A.E.M. is supported by the U.S. Department of Energy under Contract No. W-7405-ENG-36.

I. INTRODUCTION

II. NUMERICAL OBSERVATION OF STICKINESS

III. MUPOS INSIDE THE CHAOTIC REGION

A. Open circular billiard

B. Stability of periodic orbits

C. Distribution of families of MUPOs

IV. MUPOS AND STICKINESS

V. CONCLUSIONS

### Key Topics

- Statistical properties
- 4.0
- Thermodynamic properties
- 4.0
- Kinematics
- 2.0
- Atom optics
- 1.0
- Cavitation
- 1.0

## Figures

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 .

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 .

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.

(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.

(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.

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.

(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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