Scaling features of a breathing circular billiard

Ladeira, Denis Gouvêa; Silva, J. Kamphorst Leal da

doi:10.1088/1751-8113/41/36/365101

Cited by 14 publications

(11 citation statements)

References 26 publications

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“…Since the idea was used with success in the Fermi-Ulam model [12], different authors extended the formalism and hence applied with great success the scaling approach in other mappings [25][26][27][28][29][30][31].…”

Section: A Phenomenological Description For the Critical Exponents Anmentioning

confidence: 99%

“…Indeed, in statistical mechanics, phase transitions are linked to abrupt changes in spatial structure of the system [10,11] and mainly due to variations of control parameters. In a dynamical system however, a phase transition is particularly related to modifications in the structure of the phase space of the system [12,13]. Therefore near a phase transition, the dynamics of the system is described by the use of a scaling function [14,15] where critical exponents characterise the dynamics near the criticality.…”

Section: Introductionmentioning

confidence: 99%

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A dynamical phase transition for a family of Hamiltonian mappings: A phenomenological investigation to obtain the critical exponents

et al. 2015

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A dynamical phase transition from integrability to non-integrability for a family of 2-D Hamiltonian mappings whose angle, θ , diverges in the limit of vanishingly action, I, is characterised. The mappings are described by two parameters: (i) , controlling the transition from integrable ( = 0) to non-integrable ( = 0); and (ii) γ , denoting the power of the action in the equation which defines the angle. We prove the average action is scaling invariant with respect to either or n and obtain a scaling law for the three critical exponents.

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Section: A Phenomenological Description For the Critical Exponents Anmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A dynamical phase transition for a family of Hamiltonian mappings: A phenomenological investigation to obtain the critical exponents

et al. 2015

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“…The scaling (3) has also been validated for several dynamical systems represented by the standard map, such as the Fermi-Ulam model [14][15][16][17][18], time-dependent potential wells [19], and waveguide billiards [18,20] among others [21,22].…”

Section: Introduction and Modelmentioning

confidence: 99%

Scaling properties of discontinuous maps

Méndez-Bermúdez

Aguilar-Sánchez

2012

Phys. Rev. E

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We study the scaling properties of discontinuous maps by analyzing the average value of the squared action variable I(2). We focus our study on two dynamical regimes separated by the critical value K(c) of the control parameter K: the slow diffusion (K < K(c)) and the quasilinear diffusion (K > K(c)) regimes. We found that the scaling of I(2) for discontinuous maps when K ≪ K(c) and K ≫ K(c) obeys the same scaling laws, in the appropriate limits, as Chirikov's standard map in the regimes of weak and strong nonlinearity, respectively. However, due to the absence of Kolmogorov-Arnold-Moser tori, we observed in both regimes that I(2) ∝ nK(β) for n ≫ 1 (n being the nth iteration of the map) with β ≈ 5/2 when K ≪ K(c) and β ≈ 2 for K ≫ K(c).

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“…Near a phase transition, the dynamics of a system is frequently described using scaling functions [7][8][9][10][11][12][13] where critical exponents characterize the dynamics around criticality. In recent years, much effort has been devoted to describe the dynamics, and hence the scaling properties, of nonlinear dissipative and non-dissipative mappings [14,15]. One of the main results of those phenomenological studies is the definition of universality classes gathering maps whose average dynamics is characterized by the same critical exponents [16].…”

Section: Introduction and Modelmentioning

confidence: 99%

Analytical description of critical dynamics for two-dimensional dissipative nonlinear maps

2016

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The critical dynamics near the transition from unlimited to limited action diffusion for two families of well known dissipative nonlinear maps, namely the dissipative standard and dissipative discontinuous maps, is characterized by the use of an analytical approach. The approach is applied to explicitly obtain the average squared action as a function of the (discrete) time and the parameters controlling nonlinearity and dissipation. This allows to obtain a set of critical exponents so far obtained numerically in the literature. The theoretical predictions are verified by extensive numerical simulations. We conclude that all possible dynamical cases, independently on the map parameter values and initial conditions, collapse into the universal exponential decay of the properly normalized average squared action as a function of a normalized time. The formalism developed here can be extended to many other different types of mappings therefore making the methodology generic and robust.

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Scaling features of a breathing circular billiard

Cited by 14 publications

References 26 publications

A dynamical phase transition for a family of Hamiltonian mappings: A phenomenological investigation to obtain the critical exponents

A dynamical phase transition for a family of Hamiltonian mappings: A phenomenological investigation to obtain the critical exponents

Scaling properties of discontinuous maps

Analytical description of critical dynamics for two-dimensional dissipative nonlinear maps

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