Intrinsic stickiness and chaos in open integrable billiards: Tiny border effects

Abstract

Rounding border effects at the escape point of open integrable billiards are analyzed via the escape times statistics and emission angles. The model is the rectangular billiard and the shape of the escape point is assumed to have a semicircular form. Stickiness and self-similar structures for the escape times and emission angles are generated inside "backgammon" like stripes of initial conditions. These stripes are born at the boundary between two different emission angles but same escape times. As the rounding effects increase, backgammon stripes start to overlap and the escape times statistics obeys the power law decay and anomalous diffusion is expected. Tiny rounded borders (around 0.1% from the whole billiard size) are shown to be sufficient to generate the sticky motion, while borders larger than 10% are enough to produce escape times with chaotic decay.

Stickiness in a bouncer model: A slowing mechanism for Fermi acceleration

“…For our simulations, most of the slower decay was characterized as a power law. Indeed, in the literature it is known that the power law decay, for such cumulative recurrence time distributions for other dynamical systems [24,25], which includes also billiards systems [22,[26][27][28][29], is set in a range of −γ ∈ [1.5,2.5] and that our results match this range. We stress, however, that the total understanding and this behavior is still an open problem, and extensive theoretical and numerical simulations are required to describe its behavior properly.…”

Chaotic and Arnold stripes in weakly chaotic Hamiltonian systems

“…ICs which start inside the chaotic stripes collide, at least once, with the rounded corner so that distinct emission angles can be observed. Different from what is observed from border effects [3], here the sequence of isoemissions stripes in Fig. 3(b), and the corresponding multicolor chaotic stripes from Fig.…”

“…Usually it is assumed to have stickiness when a power-law decay with γ esc > 1.0 is observed for two decades in time. Power-law decays with γ esc = 1.0 [3,29] appear in integrable systems, which is the case of our model when R/L = 0.0. Figure 4 displays in a semi-log plot the For a detailed discussion of a similar behavior we refer the reader to [3].…”

“…Power-law decays with γ esc = 1.0 [3,29] appear in integrable systems, which is the case of our model when R/L = 0.0. Figure 4 displays in a semi-log plot the For a detailed discussion of a similar behavior we refer the reader to [3]. However, for R/L = 1.0 × 10 −2 [log(R/L) = −4.6], 5.0 × 10 −2 [log(R/L) = −3.0] (curves b, a), which correspond to values from Fig.…”

Influência do fenômeno de stickiness em alguns sistemas dinâmicos clássicos

“…Se umaórbita passa próximo de uma ilha de estabilidade, ela pode ficar aprisionada por um determinado intervalo de tempo finito ao redor dessa ilha, escapar e seguir seu caminho normalmente pelo mar de caos. Se a ilha de estabilidade possuir um conjunto de cantori, esse aprisionamento temporárioé ainda mais forte [45,46,47,60,65,134]. Esse processo dinâmico de aprisionamento deórbitas influencia a dinâmica do sistema, de maneira a ser um mecanismo para retardar o fenômeno de aceleração de Fermi [114].…”