### Abstract

The detailed phase space structure near the smooth outermost KAM surface is studied carrying out with the mushroom billiard. When the rotation number of the outermost surface α is the rational Fibonacci ratio (α_{k} = F_{k+2}/F_{k+4}, k ≥ 0), some singular points always exist on the outermost surface, but the stagnant layers sharply disappear when k increases. On the other hand, when the rotation number approaches to the limit irrational number (k → ∞), some singular periodic points remain in chaotic sea far from the outermost KAM surface, and that the stagnant layers are formed arround each singular point. The phase volume of the stagnant layers is theoretically evaluated, and it is shown that the trapping time in each stagnant layer obeys the inverse power distribution. This explains the universal aspect of the slow dynamics in the mushroom billiard, where the power spectral density reveals log ω (ω < 1) for 0.5 ≲ r/R indifferent from the rotation number α of the outermost KAM surface. Indeed, even in other cases for irrational α, for instance, algebraic irrationals and transcendental irrationals, the power spectrum reveals the same scaling. But in other conditions for r/R ∼ 0:5, the power spectrum obeys the ω^{-v}-scaling originated from the bouncing motion in the stark area, and the spectral transition is numerically determined.

Original language | English |
---|---|

Article number | 024002 |

Journal | Journal of the Physical Society of Japan |

Volume | 83 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2014 Feb 15 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Journal of the Physical Society of Japan*,

*83*(2), [024002]. https://doi.org/10.7566/JPSJ.83.024002

**Stagnant motion in hamiltonian dynamics mushroom billiard case with smooth outermost KAM tori.** / Tsugawa, Satoru; Aizawa, Yoji.

Research output: Contribution to journal › Article

*Journal of the Physical Society of Japan*, vol. 83, no. 2, 024002. https://doi.org/10.7566/JPSJ.83.024002

}

TY - JOUR

T1 - Stagnant motion in hamiltonian dynamics mushroom billiard case with smooth outermost KAM tori

AU - Tsugawa, Satoru

AU - Aizawa, Yoji

PY - 2014/2/15

Y1 - 2014/2/15

N2 - The detailed phase space structure near the smooth outermost KAM surface is studied carrying out with the mushroom billiard. When the rotation number of the outermost surface α is the rational Fibonacci ratio (αk = Fk+2/Fk+4, k ≥ 0), some singular points always exist on the outermost surface, but the stagnant layers sharply disappear when k increases. On the other hand, when the rotation number approaches to the limit irrational number (k → ∞), some singular periodic points remain in chaotic sea far from the outermost KAM surface, and that the stagnant layers are formed arround each singular point. The phase volume of the stagnant layers is theoretically evaluated, and it is shown that the trapping time in each stagnant layer obeys the inverse power distribution. This explains the universal aspect of the slow dynamics in the mushroom billiard, where the power spectral density reveals log ω (ω < 1) for 0.5 ≲ r/R indifferent from the rotation number α of the outermost KAM surface. Indeed, even in other cases for irrational α, for instance, algebraic irrationals and transcendental irrationals, the power spectrum reveals the same scaling. But in other conditions for r/R ∼ 0:5, the power spectrum obeys the ω-v-scaling originated from the bouncing motion in the stark area, and the spectral transition is numerically determined.

AB - The detailed phase space structure near the smooth outermost KAM surface is studied carrying out with the mushroom billiard. When the rotation number of the outermost surface α is the rational Fibonacci ratio (αk = Fk+2/Fk+4, k ≥ 0), some singular points always exist on the outermost surface, but the stagnant layers sharply disappear when k increases. On the other hand, when the rotation number approaches to the limit irrational number (k → ∞), some singular periodic points remain in chaotic sea far from the outermost KAM surface, and that the stagnant layers are formed arround each singular point. The phase volume of the stagnant layers is theoretically evaluated, and it is shown that the trapping time in each stagnant layer obeys the inverse power distribution. This explains the universal aspect of the slow dynamics in the mushroom billiard, where the power spectral density reveals log ω (ω < 1) for 0.5 ≲ r/R indifferent from the rotation number α of the outermost KAM surface. Indeed, even in other cases for irrational α, for instance, algebraic irrationals and transcendental irrationals, the power spectrum reveals the same scaling. But in other conditions for r/R ∼ 0:5, the power spectrum obeys the ω-v-scaling originated from the bouncing motion in the stark area, and the spectral transition is numerically determined.

UR - http://www.scopus.com/inward/record.url?scp=84894127862&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84894127862&partnerID=8YFLogxK

U2 - 10.7566/JPSJ.83.024002

DO - 10.7566/JPSJ.83.024002

M3 - Article

VL - 83

JO - Journal of the Physical Society of Japan

JF - Journal of the Physical Society of Japan

SN - 0031-9015

IS - 2

M1 - 024002

ER -