Critical exponents for a transition from integrability to non-integrability via localization of invariant tori in the Hamiltonian system

Leonel, Edson D.; Oliveira, Juliano A. de; Saif, Farhan

doi:10.1088/1751-8113/44/30/302001

Cited by 28 publications

(37 citation statements)

References 33 publications

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“…All along this time, the average velocity is a constant producing a plateau in velocity vs. n (number of collisions) plot which is still unexplained. Such plateaus are also observed in many other different types of mappings making the subject of broad interest; (ii) the scaling present in the phase space for conservative 2-D mappings was clearly made [33] using a transition from local to globally chaotic behavior [12]. Similar scaling is also observed for dissipative dynamics however so far there is no theoretical explanation for the origin of the results.…”

Section: Introductionmentioning

confidence: 54%

A symmetry break in energy distribution and a biased random walk behavior causing unlimited diffusion in a two dimensional mapping

Oliveira

Silva

Leonel

2015

Physica A: Statistical Mechanics and its Applications

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Section: Introductionmentioning

confidence: 54%

A symmetry break in energy distribution and a biased random walk behavior causing unlimited diffusion in a two dimensional mapping

Oliveira

Silva

Leonel

2015

Physica A: Statistical Mechanics and its Applications

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“…The position of the FISC varies with ϵ, and an analytical estimation for its position, by using a connection with well known standard mapping [2] can be found in Refs. [46]. Considering the results obtained in the above mentioned papers, the position of FISC is estimated as…”

Section: The Model and The Mappingmentioning

confidence: 99%

On the statistical and transport properties of a non-dissipative Fermi-Ulam model

Livorati

Dettmann

Caldas

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The transport and diffusion properties for the velocity of a Fermi-Ulam model were characterized using the decay rate of the survival probability. The system consists of an ensemble of non interacting particles confined to move along and experience elastic collisions with two infinitely heavy walls. One is fixed, working as a returning mechanism of the colliding particles while the other one moves periodically in time. The diffusion equation is solved and the diffusion coefficient is numerically estimated by means of the averaged square velocity. Our results show remarkably good agreement of the theory and simulation for the chaotic sea below the first elliptic island in the phase space. From the decay rates of the survival probability, we obtained transport properties that can be extended to other nonlinear mappings, as well to billiard problems. We study the dynamics of an ensemble of non interacting particles moving constrained by two infinitely heavy walls, where one of then is moving periodically in time and the other is fixed. This problem, also known as Fermi-Ulam model, has application in many areas, including astrophysics, atom-optics, quantum mechanics, among others. The diffusive behaviour of the velocity, here set as the way the transport of orbits occurs in the phase space, is investigated considering transport properties obtained from the decay rate of the survival probability, defined by means of escape formalism. Since the system present mixed dynamics, stickiness phenomenon may influence the transport causing anomalous diffusion. In this study we developed an analytical approach for the diffusion coefficient along the transport through the chaotic sea considering escape rate formalism and survival probability analysis. The numerical results we obtained are in good agreement with the theory, and confirm the robustness of the formalism. The results obtained here can be extended to other similar dynamical systems.

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“…In recent years, the dynamics of nonlinear maps, dissipative and non-dissipative, has been satisfactorily investigated by means of scaling studies of quantities like ⟨I⟩,  I 2  , and ω; see for example Refs. [7][8][9][10][11][12]20,21,23]. In particular, since I is proportional to momentum and I 2 to energy, the corresponding average values of these two quantities have physical relevance because depending on the system described by our discontinuous map they may become experimentally accessible to measurements.…”

Section: Characterization Of the Discontinuous Map M γmentioning

confidence: 99%

“…As examples, we can mention some well known mappings having in common the choice of f (θ n , I n+1 ) = sin(θ n ) and h(θ n , I n+1 ) = 0: Chirikov's standard map [2,3], g(I n+1 ) = I n+1 , also known as Taylor-Chirikov's map; the bouncer model [4], g(I n+1 ) = ξ I n+1 ; the logistic twist map [5], g(I n+1 ) = I n+1 + ξ I 2 n+1 ; the Fermi-Ulam accelerator model [6,7], g(I n+1 ) = 2/I n+1 ; a generalized Fermi-Ulam accelerator (FU) model [8][9][10][11],…”

Section: Introduction and Modelmentioning

confidence: 99%

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Scaling properties for a family of discontinuous mappings

Méndez-Bermúdez

Oliveira

Aguilar-Sánchez

et al. 2015

Physica A: Statistical Mechanics and its Applications

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h i g h l i g h t s• We study scaling properties of a family of nonlinear discontinuous maps. • This family is the discontinuous-map representation of well-known nonlinear systems. • The exponent characterizing the family of maps defines universality classes. a b s t r a c tScaling exponents that describe a transition from integrability to non-integrability in a family of two-dimensional, nonlinear, and discontinuous mappings are obtained. The mapping considered is parameterized by the exponent γ in the action variable. The scaling exponents describing the behavior of the average square action along the chaotic orbits are obtained for different values of γ ; therefore classes of universality can be defined. For specific values of γ our mapping acts as the discontinuous-map representation of well-known nonlinear systems, thus making our study broadly applicable. Also, the formalism used is general and the procedure can be extended to characterize many other dynamical systems.

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Critical exponents for a transition from integrability to non-integrability via localization of invariant tori in the Hamiltonian system

Cited by 28 publications

References 33 publications

A symmetry break in energy distribution and a biased random walk behavior causing unlimited diffusion in a two dimensional mapping

A symmetry break in energy distribution and a biased random walk behavior causing unlimited diffusion in a two dimensional mapping

On the statistical and transport properties of a non-dissipative Fermi-Ulam model

Scaling properties for a family of discontinuous mappings

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