Multistability, Phase Diagrams and Statistical Properties of the Kicked Rotor: A Map With Many Coexisting Attractors

Martins, Luciano Camargo; Gallas, Jason A. C.

doi:10.1142/s0218127408021294

Cited by 26 publications

(25 citation statements)

References 40 publications

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“…Such a region of the parameter space indeed agrees with Ref. 44. The procedure used to construct the figure was to divide both Fig.…”

Section: -3supporting

confidence: 84%

“…We specifically take into account its two-dimensional parameter space, namely, the dissipation parameter c and the amplitude of the kicks K. We revisit the parameter space of the dissipative standard map where the existence of self-similar structures called shrimps was shown. 44 According to Ref. 43, "Shrimps are formed by a regular set of adjacent windows centered around the main pair of intersecting superstable parabolic arcs.…”

Section: Introductionmentioning

confidence: 99%

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Shrimp-shape domains in a dissipative kicked rotator

Oliveira

Robnik

Leonel

2011

Chaos

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Some dynamical properties for a dissipative kicked rotator are studied. Our results show that when dissipation is taken into account a drastic change happens in the structure of the phase space in the sense that the mixed structure is modified and attracting fixed points and chaotic attractors are observed. A detailed numerical investigation in a two-dimensional parameter space based on the behavior of the Lyapunov exponent is considered. Our results show the existence of infinite self-similar shrimp-shaped structures corresponding to periodic attractors, embedded in a large region corresponding to the chaotic regime. A strongly dissipative kicked rotator is studied. The investigation is based mainly on the asymptotic behavior of the Lyapunov exponents. Our results show that by reducing the two-dimensional map to a one-dimensional map in the limit of infinite kicks, the Feigenbaum's d is recovered. An investigation in the parameter space (the dissipation parameter and the kicking parameter) reveals the existence of infinite families of self-similar structures of shrimp-shape embedded in a large region corresponding to the chaotic regime. The organization of stability shrimps reported here for the discrete-time kicked rotator agrees well with the parameter space organization reported recently in the literature for several flows (continuous-time systems).

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“…Such a region of the parameter space indeed agrees with Ref. 44. The procedure used to construct the figure was to divide both Fig.…”

Section: -3supporting

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Shrimp-shape domains in a dissipative kicked rotator

Oliveira

Robnik

Leonel

2011

Chaos

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“…For that purpose we calculate the participation ratio defined by η = ( i P (p i ) 2 ) −1 /N . P (p i ) is a discretized limiting momentum (p) distribution, taken after evolving 10000 time steps a bunch of 10000 random initial conditions in the p = [−k/(1 − γ); k/(1 − γ)] band of the cylindrical phase space (i.e., the trapping region defined in [23]). We have taken a number of bins given by a Hilbert space dimension N = 1000.…”

Section: Properties Of the Leading Eigenstates: The Decay Towardmentioning

confidence: 99%

“…This measure is a good indicator of the complexity of the asymptotic distribution. It is worth mentioning that the DSM contains multistability regions with a great number of coexisting attractors [23]. However, we are interested in clearly identifying those regions where just regular behavior is found and those where a chaotic attractor is present.…”

Section: Properties Of the Leading Eigenstates: The Decay Towardmentioning

confidence: 99%

Signatures of classical structures in the leading eigenstates of quantum dissipative systems

Carlo¹,

Ermann²,

Rivas³

et al. 2017

Phys. Rev. E

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By analyzing a paradigmatic example of the theory of dissipative systems-the classical and quantum dissipative standard map-we are able to explain the main features of the decay to the quantum equilibrium state. The classical isoperiodic stable structures typically present in the parameter space of these kinds of systems play a fundamental role. In fact, we have found that the period of stable structures that are near in this space determines the phase of the leading eigenstates of the corresponding quantum superoperator. Moreover, the eigenvectors show a strong localization on the corresponding periodic orbits (limit cycles). We show that this sort of scarring phenomenon (an established property of Hamiltonian and projectively open systems) is present in the dissipative case and it is of extreme simplicity.

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“…First, while for strong dissipation the dynamics is characterized by only one attractor, the coexistence of a multitude of attractors for a given set of parameters is the norm for weakly dissipative systems [37]. As the conservative limit is approached, more and more coexisting attractors appear, as it has been shown for several model systems such as the standard map [37,38], the Hénon map [39,40], and in more realistic systems like a suspension bridge model [41] (see also the review in [42]). Second, of particular interest is the crossover from dissipative to conservative dynamics [43][44][45].…”

Section: Introductionmentioning

confidence: 98%

Different types of critical behavior in conservatively coupled Hénon maps

et al. 2015

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We study the dynamics of two conservatively coupled Hénon maps at different levels of dissipation. It is shown that the decrease of dissipation leads to changes in the structure of the parameter plane and the scenarios of transition to chaos compared to the case of infinitely strong dissipation. Particularly, the Feigenbaum line becomes divided into several fragments. Some of these fragments have critical points of different types, namely, of C and H type, as their terminal points. Also the mechanisms of formation of these Feigenbaum line ruptures are described.

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Multistability, Phase Diagrams and Statistical Properties of the Kicked Rotor: A Map With Many Coexisting Attractors

Cited by 26 publications

References 40 publications

Shrimp-shape domains in a dissipative kicked rotator

Shrimp-shape domains in a dissipative kicked rotator

Signatures of classical structures in the leading eigenstates of quantum dissipative systems

Different types of critical behavior in conservatively coupled Hénon maps

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