Multifractality, stickiness, and recurrence-time statistics

Abud, C. Vieira; Carvalho, R. Egydio de

doi:10.1103/physreve.88.042922

Cited by 20 publications

(11 citation statements)

References 37 publications

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“…We remark that our Poincaré-time (PT) analysis is different from the one studied in [33,35], although the two share the same philosophy. There, several conditional measures are constructed by selecting only those Poincaré cycles that have RT falling in some specific time-range, which is picked empirically from the RT statistics.…”

Section: Weak-chaos Transitionmentioning

confidence: 88%

Anomalous dynamics and the choice of Poincaré recurrence set

Sala

Artuso²,

Manchein

2016

Phys. Rev. E

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We investigate the dependence of Poincaré recurrence-time statistics on the choice of recurrence set by sampling the dynamics of two- and four-dimensional Hamiltonian maps. We derive a method that allows us to visualize the direct relation between the shape of a recurrence set and the values of its return probability distribution in arbitrary phase-space dimensions. Such a procedure, which is shown to be quite effective in the detection of tiny regions of regular motion, allows us to explain why similar recurrence sets have very different distributions and how to modify them in order to enhance their return probabilities. Applied to data, this enables us to understand the coexistence of extremely long, transient powerlike decays whose anomalous exponent depends on the chosen recurrence set.

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Section: Weak-chaos Transitionmentioning

confidence: 88%

Anomalous dynamics and the choice of Poincaré recurrence set

Sala

Artuso²,

Manchein

2016

Phys. Rev. E

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“…"External" stickiness however, arises due to the existence of the boundaries between regular and chaotic regions. Stickiness results in non-exponential decays of both the time-correlation functions and Poincaré recurrence distributions of the system's chaotic dynamics [37][38][39][40] .…”

Section: Introductionmentioning

confidence: 99%

The Iris billiard: Critical geometries for global chaos

Page

Antoine

Dettmann

et al. 2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We introduce the Iris Billiard, consisting of a point particle enclosed by a unit circle enclosing a central scattering ellipse of fixed elongation (defined as the ratio of the semi-major to the semi-minor axes). When the ellipse degenerates to a circle, the system is integrable, otherwise it displays mixed dynamics. Poincaré sections are displayed for different elongations. Recurrence plots are then applied to the long-term chaotic dynamics of trajectories launched from the unstable period-2 orbit along the semi-major axis i.e., one that initially alternately collides with the ellipse and the circle. We obtain numerical evidence of a set of critical elongations at which the system transitions to global chaos. The transition is characterized by an endogenous escape event, E , which is the first time a trajectory launched from the unstable period-2 orbit misses the ellipse. The angle of escape, θ esc and distance of closest approach, d min of the escape event are studied, and are shown to be exquisitely sensitive to the elongation. The survival probability that E has not occurred after n collisions is shown to follow an exponential distribution.

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“…13 Moreover, the presence of stickiness of trajectories a ects the transport of particles in the phase space. 5 Di erent methods have been proposed to identify sticky orbits, such as recurrence analysis, 14 recurrence time statistics, 15 and nitetime Lyapunov exponent (FTLE). 16 These methods require a large number of iterations of the map, as well as it is necessary to know the position of the islands in the phase space.…”

Section: Introductionmentioning

confidence: 99%

Using rotation number to detect sticky orbits in Hamiltonian systems

Santos

Mugnaine

Szezech

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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In Hamiltonian systems, depending on the control parameter, orbits can stay for very long times around islands, the so-called stickiness e ect caused by a temporary trapping mechanism. Di erent methods have been used to identify sticky orbits, such as recurrence analysis, recurrence time statistics, and nite-time Lyapunov exponent. However, these methods require a large number of map iterations and to know the island positions in the phase space. Here, we show how to use the small divergence of bursts in the rotation number calculation as a tool to identify stickiness without knowing the island positions. This new procedure is applied to the standard map, a map that has been used to describe the dynamic behavior of several nonlinear systems. Moreover, our procedure uses a small number of map iterations and is proper to identify the presence of stickiness phenomenon for di erent values of the control parameter.

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Multifractality, stickiness, and recurrence-time statistics

Cited by 20 publications

References 37 publications

Anomalous dynamics and the choice of Poincaré recurrence set

Anomalous dynamics and the choice of Poincaré recurrence set

The Iris billiard: Critical geometries for global chaos

Using rotation number to detect sticky orbits in Hamiltonian systems

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