Time-dependent properties in two-dimensional and Hamiltonian mappings

Transcription

1 Eur. Phys. J. Special Topics 223, (2014) EDP Sciences, Springer-Verlag 2014 DOI: /epjst/e THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS Regular Article Time-dependent properties in two-dimensional and Hamiltonian mappings A.L.P. Livorati 1,a, J.A. de Oliveira 2, D.G. Ladeira 3, and E.D. Leonel 4 1 Instituto de Física da USP, Cidade Universitária, São Paulo, SP, Brazil 2 UNESP, Univ. Estadual Paulista, Câmpus São João da Boa Vista, São João da Boa Vista SP, Brazil 3 Departamento de Física e Matemática, Univ. Federal de São João del-rei, UFSJ, Rod. MG 443, Km 7, Fazenda do Cadete, Ouro Branco, MG, Brazil 4 UNESP, Univ. Estadual Paulista, câmpus de Rio Claro, IGCE, Departamento de Física, Av. 24A, 1515, Bela Vista, CEP: , Rio Claro, SP, Brazil Received 31 May 2014 / Received in final form 17 October 2014 Published online 10 December 2014 Abstract. Some scaling properties for chaotic orbits in a family of two-dimensional Hamiltonian mappings are studied. The phase space of the model exhibits chaos and may have mixed structure with periodic islands, chaotic seas and invariant spanning curves. Average properties of the action variable in the chaotic sea are obtained as a function of time (t). From scaling arguments, critical exponents for the ensemble average of the action variable are obtained. Scaling invariance is obtained as a function of the control parameter that controls the intensity of the nonlinearity. 1 Introduction Non-linear dynamics is a topic of study of several researchers along of decades. However, the problems that present Fermi acceleration have requested much attention. The phenomenon of Fermi acceleration is described when a classical particle acquires unbounded energy from collisions with a massive moving wall [1]. Applications of this phenomenon have acquired broad interest in different fields of physics including plasma physics, astrophysics, atomic physics, optics and even in the well known time dependent billiard problems [2 8]. In many cases, non-linear systems are described by two-dimensional mappings. Generally, the mappings are composed by control parameters that control the behaviour of the system including a possible transition from integrability to non integrability. In conservative systems, for specific choice of control parameters the phase space may exhibits a large chaotic sea or a mixed structure composed by periodic islands surrounded by a large chaotic sea and limited by invariant spanning curves. In this paper we revisit a family of two-dimensional and Hamiltonian systems described by action and angle variables and parametrized by control parameters ɛ and γ, a livorati@rc.unesp.br

2 2954 The European Physical Journal Special Topics as discussed previously in Ref. [9]. The control parameter ɛ controls the intensity of the nonlinearity of the system and γ is the exponent that gives us the family of maps. Using a transformation of action and angle variables and ranging the values of γ we recover different systems discussed in the literature including the bouncer [10] and Fermi-Ulam model [8]. We show that the ensemble average of the action variable is scaling invariant with respect to its control parameter ɛ and the time t. The paper is organized as follows. In Sect. 2 we present the model and discuss the variables and control parameters used. In Sect. 3 we use scaling properties as functions of time. A merger of all the curves of the average quantity is obtained. Finally in the Sect. 4 we present our concluding remarks. 2 The model In this section we present and discuss some dynamical properties for a set of twodimensional Hamiltonian mappings written as [9] { Jn+1 = J n ɛ sin(θ n ) T : θ n+1 =[θ n + J γ, (1) n+1 ] (mod 2π) where J and θ correspond to action and angle variables respectively, γ and ɛ are control parameters. The control parameter ɛ controls the transition from integrability to non-integrability. The appropriate choice of γ describes a large class of systems, including relevant applications for plasma physics, the dynamics of a wave packet, the Kepler map, the one-dimensional Fermi-Ulam accelerator model, the bouncer model, the periodically corrugated waveguide and some dynamical properties for a time-dependent potential well [11 18]. In this paper, we concentrate to discuss the results for γ = 1. The case of γ = 1 have been already carefully addressed in Ref. [19]. For γ = 1 and a change of the dynamical variables to J v and θ φ, the mapping (1) recovers the bouncer model [10,20,21]. If γ = 1 the one-dimensional Fermi-Ulam model [22] is recovered. In both cases v represents the velocity of the particle and φ denotes the phase of the moving wall. Given the physical applications of the mapping (1), we intend to explore the scaling properties for chaotic orbits as a function time (t) for the case of γ =1. The phase space generated from iteration of the mapping (1) is shown in Fig. 1. One sees that using the control parameters γ =1andɛ = 10, Fig. 1a shows a large chaotic sea while using γ = 1 andɛ =10 3. Figure 1b shows a set of periodic islands that are surrounded by a large chaotic sea that is limited by a set of invariant KAM curves. The size of the chaotic sea varies as the control parameters vary. A detailed study of the stability for the mapping (1) can be found in Refs. [9,23]. 3 Time-dependent scaling properties In this section, we investigate the behaviour of average quantities of the action variable along of chaotic orbits as functions of continuous dynamical variable time instead the discrete one, as the number of collision n, if we were considering the original model. Let us regard at first the trajectory of a single initial condition i. As we know, the variable J i (n) changes at each iteration according the map (1). Moreover, between two iterations the system is integrable and the variable J depends on variable t according to the expression J i (ɛ, τ) =Δt n+1, where Δt n+1 = J γ n+1. Therefore, adopting the

3 Advanced Computational and Experimental Techniques in Nonlinear Dynamics 2955 Fig. 1. Phase space for mapping (1) using the control parameters: (a) ɛ =10andγ =1 and (b) ɛ =10 3 and γ = 1. procedure used in Ref. [19], we define the average in time J i (t) by the equation J i (ɛ, t) = 1 t t 0 J i (ɛ, τ)dτ. (2) Note that to evaluate the above integral we need to take into account the time elapsed in the iterations of map (1) between the instants t and t +Δt for a given Δt. The details of such approach are furnished in Ref. [19]. The main interest in this average, is that it takes into account the time-dependence of the full trajectory. Now we consider an ensemble of M chaotic trajectories and obtain the average value for the deviation of the action variable ω(ε, t) = 1 M M i=1 J 2 i (ε, t) J i2 (ε, t). (3) Figures 2a, b show the behaviour of ω as a function of time t for different values of the control parameters ɛ and γ, as labelled in the figure. We see that scaling description is more straightforward when we replace ω by the new variable ω/ɛ as shown in Fig. 2b. Let us now discuss the results obtained by numerical simulations. Figure 2b shows the behaviour of ω(ɛ, t). Initial conditions used to calculate ω were set up in an equally split range of 5000 bins in the angle variable θ 0 [0, 2π], and the initial value for J 0 was kept constant equal to the value of ɛ, during this range variation. One can see that ω is constant up to time t x and then it grows for large times.

4 2956 The European Physical Journal Special Topics Fig. 2. Plot of: (a) ω vs. t and (b) ω/ɛ vs. t for different values of ɛ (as labelled in figure), considering a fixed γ =1. Based on the behaviour seen in Fig. 2 we can suppose that: (i) For short times, say t t x, the behaviour of ω is constant written as ω J 0 ɛ α1 ; (4) (ii) For large times, t t x, ω grows according to a power law as ω ( tɛ 2) β ; (5) (iii) The crossover t x, that characterizes the transition from the constant plateau to the growing regime is where α 1, β, α 2 are called the critical exponents. t x ɛ α2, (6)

5 Advanced Computational and Experimental Techniques in Nonlinear Dynamics 2957 Fig. 3. Plot of t x vs. ɛ for γ =1. Fig. 4. Overlap onto a single and universal plot using the same control parameters used in Fig. 2. The critical exponents α 1 and α 2 can be obtained from extensive numerical simulations. Firstly, fitting the initial constant regime the critical α 1 = 1. The regime of growth, we obtain that the critical β = /3. The critical exponent α 2 is obtained from specific plots. It is shown in Fig. 3 a plot t x ɛ. All the results shown in Fig. 3 were obtained using very long simulations of iterations. The ensemble average used was M = A power law fitting to the curves plotted in Fig. 3 furnishes that α 2 =0.98(7) 1. Since we have now obtained the critical exponents, finally the scaling hypotheses can be verified. One can see that in Fig. 4 there is a merge of three different curves of ω generated from different values of the control parameter ɛ into a single and universal plot. 4 Conclusion As a summary, we studied the dynamics of two-dimensional and Hamiltonian mappings. The average properties of the action variable in the chaotic sea were characterized using scaling arguments. Critical exponents were obtained as function of

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