The Feigenbaum's delta for a high dissipative bouncing ball model

Oliveira, Diego F. M.; Leonel, Edson D.

doi:10.1590/s0103-97332008000100012

Cited by 9 publications

(9 citation statements)

References 22 publications

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“…1. Such a universal constant has been found numerically in other systems, namely, the logistic map 70 and the dissipative Fermi-Ulam model 71 just to mention two of them. On the other hand, if this "two-dimensional character is preserved," the universal constant may be different.…”

Section: -3mentioning

confidence: 72%

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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Section: -3mentioning

confidence: 72%

Shrimp-shape domains in a dissipative kicked rotator

Oliveira

Robnik

Leonel

2011

Chaos

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“…Following our quest, we now demonstrate that is it possible to construct a sort of effective stroboscopic Hamiltonian for this time-dependent periodic problem, as it is done for the classical particle with the oscillating wall 9 , 10 . This operator emerges naturally when we study the time evolution operator that can give us the oscillatory response of the system, so that in essence we are constructing a simile of Bloch’s Theorem 13 , but for the time domain, in the case of oscillating boundary conditions.…”

Section: Introductionmentioning

confidence: 94%

“… 8 also reported a similar behavior for the infinite square-well potential but they didn’t found the rule to do so since they focused their study on low, intermediate, and high frequencies of the wall motion; finding adiabatic, chaotic, and periodic behavior respectively. We will see, as a by-product of our quest, that the classical macroscopic effects of the Fermi acceleration process 9 and chaotic dynamics for large amplitude oscillating walls 10 could also be associated with the respective dynamics in the quantum domain. Our problem of interest is a quantum variant of the famous Fermi acceleration problem 11 , but with impenetrable walls instead of inhomogeneous magnetic fields.…”

Section: Introductionmentioning

confidence: 95%

Controlling the Quantum State with a time varying potential

Carrasco

Rogan

Valdivia

2017

Sci Rep

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The problem of controlling the quantum state of a system is investigated using a time varying potential. As a concrete example we study the problem of a particle in a box with a periodically oscillating infinite square-well potential, from which we obtain results that can be applied to systems with periodically oscillating boundary conditions. We derive an analytic expression for the frequencies of resonance between states, and against standard intuition, we show how to use this behavior to control the quantum state of the system at will. In particular, we offer as an example the transition from the ground state to the first excited state of the square well potential. At first sight, it may be counter intuitive that we can control the state of such a quantum Hamiltonian, as the Schrödinger equation conserves the norm of the wave function. In this manuscript, we show how that can be achieved.

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“…Therefore, for n ≫ 1 and v ≫ ǫ the PDF of particle velocities tends to a Gaussian distribution [Eq. (14)].…”

Section: Long Transientsmentioning

confidence: 99%

“…(13), on the basis of an ensemble of 1.2 × 10 6 particles initially distributed as ρ(v, 0) = δ(v − ǫ). The analytical result derived through the application of the CLT [Eq (14)…”

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confidence: 99%

A consistent approach for the treatment of Fermi acceleration in time-dependent billiards

Karlis

Diakonos

Constantoudis

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The standard description of Fermi acceleration, developing in a class of time-dependent billiards, is given in terms of a diffusion process taking place in momentum space. Within this framework, the evolution of the probability density function (PDF) of the magnitude of particle velocities as a function of the number of collisions n is determined by the Fokker-Planck equation (FPE). In the literature, the FPE is constructed by identifying the transport coefficients with the ensemble averages of the change of the magnitude of particle velocity and its square in the course of one collision. Although this treatment leads to the correct solution after a sufficiently large number of collisions have been reached, the transient part of the evolution of the PDF is not described. Moreover, in the case of the Fermi-Ulam model (FUM), if a standard simplification is employed, the solution of the FPE is even inconsistent with the values of the transport coefficients used for its derivation. The goal of our work is to provide a self-consistent methodology for the treatment of Fermi acceleration in time-dependent billiards. The proposed approach obviates any assumptions for the continuity of the random process and the existence of the limits formally defining the transport coefficients of the FPE. Specifically, we suggest, instead of the calculation of ensemble averages, the derivation of the one-step transition probability function and the use of the Chapman-Kolmogorov forward equation. This approach is generic and can be applied to any time-dependent billiard for the treatment of Fermi-acceleration. As a first step, we apply this methodology to the FUM, being the archetype of time-dependent billiards to exhibit Fermi acceleration.

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The Feigenbaum's delta for a high dissipative bouncing ball model

Cited by 9 publications

References 22 publications

Shrimp-shape domains in a dissipative kicked rotator

Shrimp-shape domains in a dissipative kicked rotator

Controlling the Quantum State with a time varying potential

A consistent approach for the treatment of Fermi acceleration in time-dependent billiards

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