This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and

Transcription

1 This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier s archiving and manuscript policies are encouraged to visit:

2 Physica A 389 (21) Contents lists available at ScienceDirect Physica A journal homepage: Suppressing Fermi acceleration in a two-dimensional non-integrable time-dependent oval-shaped billiard with inelastic collisions Diego F.M. Oliveira, Edson D. Leonel Departamento de Física, Instituto de Geociências e Ciências Exatas, UNESP - Univ Estadual Paulista, Av. 24A, 1515-Bela Vista, CEP: , Rio Claro, SP, Brazil Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, UNESP - Univ Estadual Paulista Av.24A, 1515-Bela Vista, CEP: , Rio Claro, SP, Brazil a r t i c l e i n f o a b s t r a c t Article history: Received 27 October 29 Available online 4 November 29 Keywords: Fermi acceleration Some dynamical properties of a classical particle confined inside a closed region with an oval-shaped boundary are studied. We have considered both the static and time-dependent boundaries. For the static case, the condition that destroys the invariant spanning curves in the phase space was obtained. For the time-dependent perturbation, two situations were considered: (i) non-dissipative and (ii) dissipative. For the non-dissipative case, our results show that Fermi acceleration is observed. When dissipation, via inelastic collisions, is introduced Fermi acceleration is suppressed. The behaviour of the average velocity for both the dissipative as well as the non-dissipative dynamics is described using the scaling approach. 29 Elsevier B.V. All rights reserved. 1. Introduction Billiards are dynamical systems where a point-like particle moves freely inside a closed region suffering collisions with the boundary. A billiard is defined by a connected region Q R D, with boundary Q R D 1 which separates Q from its complement. If the set Q is constant with respect to time, the particle does not change its energy upon collision with the boundary. In this case, the boundary is fixed. On the other hand, if Q = Q (t) the system has a time-dependent boundary and the particle may gain or lose energy upon collision with the boundary. Billiards with time-moving boundaries have been studied in recent years as an attempt to understand a phenomenon introduced by Enrico Fermi known as Fermi acceleration (FA) [1]. It is known in the literature that the phase space of a billiard system depends on the boundary configuration. Basically they are settled in three different classes including: (i) integrable, (ii) ergodic and (iii) mixed. A typical example of case (i) is the circular billiard. The integrability of such a case resembles the angular momentum conservation. Two examples of case (ii) are the Bunimovich stadium [2] and the Sinai billiard [3]. In case (iii), there is a representative number of billiards that present mixed phase space structure [4 1]. One important property in the mixed phase space is that chaotic seas are generally surrounding Kolmogorov Arnold Moser (KAM) islands which are confined by invariant spanning curves [11]. Particularly such curves cross the phase plane from one side to the other side separating different portions of the phase space. This mixed structure of phase space is indeed generic for non-degenerate Hamiltonian systems [12]. In this paper, we revisit the problem of an oval-shaped billiard considering both the static as well as time-moving boundaries. For the static case, we found analytically a condition at which invariant spanning curves in the phase space are destroyed. For time-dependent boundary we consider both the situations where the particle suffers elastic as well Corresponding author at: Departamento de Estatistica, Matematica Aplicada e Computacao, Instituto de Geociencias e Ciencias Exatas, Universidade Estadual Paulista, Av.24A, 1515-Bela Vista, CEP: , Rio Claro, SP, Brazil. address: edleonel@rc.unesp.br (E.D. Leonel) /$ see front matter 29 Elsevier B.V. All rights reserved. doi:1.116/j.physa

3 11 D.F.M. Oliveira, E.D. Leonel / Physica A 389 (21) as inelastic collisions with the boundary. For the case of elastic collisions, our results confirm the occurrence of Fermi acceleration. However, when inelastic collisions are taken into account, Fermi acceleration is suppressed. The paper is basically divided in two parts. In the first part, we study some dynamical properties for a static-boundary version of the oval-shaped billiard. We describe the system using a two-dimensional nonlinear map. Inside the boundary, we consider that the particle is moving freely and in the total absence of any external field. When the particle hits the boundary it changes the direction according to a specular reflection without suffering any loss of energy upon collision. The phase space of the static version is mixed. The shape of the boundary is controlled by two relevant control parameters. Depending on the combination of such control parameters, regions on the boundary might change from positive to negative curvature. The curvature κ(θ) can be set in three different classes, namely: (i) neutral, (ii) dispersing and (iii) focusing. Usually, they are denoted by Q [κ(θ) = ], Q + [κ(θ) > ] and Q [κ(θ) < ] respectively. We have found analytically a critical control parameter where no invariant spanning curves are observed in the phase space. This condition is basically related to the fact that the boundary may changes locally from positive to negative curvature, i.e., from dispersing to focusing regions. We explore the behaviour of the Lyapunov exponent as function of the number of collisions as well as the boundary deformation, therefore confirming the occurrence of chaotic portions on the phase space. In the second part of the paper we introduce a time-dependent perturbation on the boundary and therefore study some average quantities. We describe the dynamics of the particle by using a four-dimensional nonlinear map. Depending on the combination of the external control parameters, the boundary may assumes two different configurations: (a) breathing and; (b) non-breathing [13,14]. Our main goal in this part of the paper is introduce dissipation into the model using inelastic collisions. For such kind of dissipation, it is assumed that the particle experiences a fractional loss of energy upon collision with the moving boundary. Hence, both the normal and tangential components of the particle s velocity are affected by such kind of dissipation. We confirm that for the non-dissipative version of the time-moving boundary, the average velocity as well as its kinetic energy grows. However, the slope of growth for the non-breathing case is larger then for the breathing. Our results therefore confirm the validity of the LRA conjecture [15]. For the dissipative case, our main approach is to try to answer a very important question in systems where Fermi acceleration is present: Is dissipation, introduced via inelastic collisions, a sufficient condition to suppress the unlimited energy gain of the bouncing particle s in two-dimensional billiards? This question was addressed in Ref. [16] for one-dimensional stochastic system and confirmed for deterministic systems [17,18]. Our approach in this paper is original since we are studding, for the first time, a two-dimensional time-moving billiard close to the transition from conservative to dissipative. We describe such a phase transition, characterized via the behaviour of the average velocity, using scaling arguments [19,2]. The paper is organized as follows: in Section 2 we give all the details needed for the construction of the nonlinear mapping that describes the dynamics of the static oval-shaped billiard. Our numerical results for the static case are also shown in Section 2. Section 3 is devoted to discuss both the conservative and dissipative dynamics for the time-moving oval-shaped billiard. Scaling arguments are used to study the transition from limited energy gain to unlimited energy gain. Finally, in Section 4 we drawn our conclusions. 2. A static oval billiard and the mapping In this section we discuss the details for the construction of a nonlinear map that describes the dynamics of the problem. The model consists of a classical particle of mass m confined into a closed region of oval-like shape suffering elastic collisions with the boundary. When the particle hits the boundary, it is specularly reflected with the same velocity. The particle does not suffer influences of any external field along its linear trajectory. The dynamics of the particle is described in terms of a two-dimensional nonlinear mapping T(θ n, α n ) = (θ n+1, α n+1 ) where the dynamical variable θ n denotes the angular position of the particle when it hits the border while α n is the angle that the trajectory does with the tangent vector to the border at the angular position θ n (the illustrations of these angles are shown in Fig. 1). The index n denotes the nth collision of the particle with the boundary. The radius of the boundary is given in polar coordinates, by R(θ, p, ɛ) = 1 + ɛ cos(pθ). (1) The control parameter ɛ [, 1) controls the circle s boundary deformation. Therefore if ɛ =, a circular shape is recovered leading the system to be integrable. On the other hand, if ɛ an oval boundary is obtained and the system is nonintegrable. p is an integer number and θ [, 2π) is a counterclockwise angle measured with respect to the positive horizontal axis. It is shown in Fig. 2 the geometry of the boundary for different values of control parameters ɛ and p. The expressions for X(θ n ) and Y(θ n ) are X(θ n ) = [1 + ɛ cos(pθ n )] cos(θ n ), Y(θ n ) = [1 + ɛ cos(pθ n )] sin(θ n ). Given an initial condition (θ n, α n ), the angle between a tangent line at the boundary at the angular position θ n is [ Y ] (θ n ) φ n = arctan, X (θ n ) (2) (3) (4)

4 D.F.M. Oliveira, E.D. Leonel / Physica A 389 (21) Fig. 1. Illustration of a particle s trajectory. where the expressions for X (θ n ) and Y (θ n ) are written as X (θ n ) = dr(θ n) dθ n cos(θ n ) R(θ n ) sin(θ n ), Y (θ n ) = dr(θ n) dθ n sin(θ n ) + R(θ n ) cos(θ n ). (5) The term dr(θ n )/dθ n is given by dr(θ n ) dθ n = ɛp sin(pθ n ). (6) To obtain the new angular position for the next collision of the particle with the boundary, one must solve numerically the following equation Y(θ n+1 ) Y(θ n ) = tan(α n + φ n )[X(θ n+1 ) X(θ n )], (7) where φ n is obtained from the slope between the tangent vector of the boundary at the angular position θ n and the positive horizontal axis. X(θ n+1 ) and Y(θ n+1 ) are the new positions of the particle at the angular position θ n+1, which is numerically obtained as solution of Eq. (7). The new angle α n+1, is obtained from geometrical considerations, as can be seen in Fig. 1. Hence, we conclude that α n+1 = φ n+1 (α n + φ n ). (8) The mapping that describes the dynamics of the particle in the oval-shaped billiard with statical boundary is given by { F(θn+1 ) = R(θ T : n+1 ) sin(θ n+1 ) Y(θ n ) tan(α n + φ n )[R(θ n+1 ) cos(θ n+1 ) X(θ n )], α n+1 = φ n+1 (α n + φ n ) (9) where θ n+1 is numerically obtained as the solution of F(θ n+1 ) =, R(θ n+1 ) = 1 + ɛ cos(pθ n+1 ) and φ n+1 = arctan[y (θ n+1 )/X (θ n+1 )]. In the next section we present and discuss some of our numerical results obtained for the static version of the boundary.

5 112 D.F.M. Oliveira, E.D. Leonel / Physica A 389 (21) Fig. 2. Shape of the boundary for different combinations of control parameters, as labelled in the figure Numerical results Let us now present and discuss some of our numerical results for the static-boundary version of the model. It is shown in Fig. 3 two typical phase space generated by iterating Eq. (9). One can see that a complex hierarchy of behaviours in the phase space is present including a large chaotic sea, KAM islands and the presence of invariant spanning curves for Fig. 3(a) and their absence for Fig. 3(b). The control parameters used in the construction of Fig. 3 were p = 2 and (a) ɛ =.8 and (b) ɛ =.21. The destruction of the invariant spanning curves are related to the length of the control parameter ɛ. Therefore, if one increase the control parameter slightly above the condition ɛ c = 1 1+p2 all the invariant spanning curves are destroyed, as shown in Fig. 3(b). The explanation for such kind of destruction is related to the shape of the boundary. For the case of ɛ < ɛ c, the boundary is convex, however if ɛ > ɛ c, one can observe non-convex pieces on the boundary (for more details see Appendix). The chaotic sea might be characterized in terms of the positive Lyapunov exponent. As discussed in [21,22], the Lyapunov exponents are defined as 1 λ j = lim n n ln Λ j, j = 1, 2, where Λ j are the eigenvalues of M = n i=1 J i(α i, θ i ) and J i is the Jacobian matrix evaluated over the orbit (α i, θ i ). It is shown in Fig. 4(a) a typical plot of the behaviour of the positive Lyapunov exponent as function of the number of collisions of the particle with the boundary for 1 different initial conditions, both randomly chosen along the chaotic sea. Each initial condition was iterated up to 1 8 times. The control parameters used in the construction of Fig. 4 were ɛ =.8 and p = 2. The average of the positive Lyapunov exponent for the ensemble of the 1 samples furnishes λ =.171 ±.2. The range ±.2 correspond to the standard deviation of the ten samples. The behaviour of the positive Lyapunov exponent as function of the control parameter ɛ is shown in Fig. 4(b) for a fixed p = 2. The ranges of values for ɛ used were ɛ [.6,.4]. The initial conditions used in the construction of Fig. 4(b) were α =.3 and θ uniformly distributed in the range of [, 2π]. Each point was obtained from the average of 5 different (1)

6 D.F.M. Oliveira, E.D. Leonel / Physica A 389 (21) a 3 b α α θ θ Fig. 3. Phase space for the oval-shaped billiard. The control parameters used in the construction of the figures were p = 2 and: (a) ɛ =.8; (b) ɛ =.21. In case (b), all the invariant spanning curves were destroyed. a.3 b λ.15 λ n ε Fig. 4. (Color online) Convergence of the positive Lyapunov exponent for the oval-shaped billiard. The control parameters used in the figure were p = 2 and (a) ɛ =.8. (b) Behaviour of the Lyapunov exponent as function of the control parameter ɛ. The arrow indicates the exact localization, in the parameter axis, of the invariant spanning curves destruction. initial conditions. Each initial condition was iterated 1 6 times. The error bars correspond to the standard deviation of the 5 samples. The arrow in the Fig. 4(b) indicates the value of the critical parameter ɛ c where all the invariant spanning curves are destroyed. 3. A time-dependent oval-shaped billiard and the map In this section we describe some dynamical properties for a time-dependent oval-shaped billiard considering both its dissipative and non-dissipative dynamics. The mapping is a generalization for time-dependent boundary of the mapping discussed in Section 2. Hence, when the particle hits the boundary both the normal and tangential components of the particle s velocity may change in module. The system is then described using a four-dimensional map T(θ n, α n, V n, t n ) = (θ n+1, α n+1, V n+1, t n+1 ) where the variables denote, respectively, the angular position of the particle; the angle that the trajectory of the particle does with the tangent line at the position of the collision; the absolute velocity of the particle and the instant of the hit with the boundary. The shape of the boundary, in polar coordinates, we consider in this section is given by R b (θ, p, ɛ, η 1, η 2, t) = 1 + η 1 cos(t) + ɛ [1 + η 2 cos(t)] cos(pθ). (11) The consideration of a time-dependent perturbation introduces two new dynamical variables, velocity and time, as well as two new control parameters η 1 and η 2. The control parameter η 1 defines the circle s time perturbation when η 2 characterizes the oval s time perturbation. The trivial case of η 1 = η 2 = recovers the static oval billiard. The condition where η 1 = η 2 defines the breathing case, while η 1 η 2 produces the non-breathing geometry [13,14]. Fig. 5 illustrates the geometry of six successive collisions of the particle with the time-dependent boundary. To obtain the map, we assume the

7 114 D.F.M. Oliveira, E.D. Leonel / Physica A 389 (21) Fig. 5. Illustration of 6 snapshots of a time-varying oval billiard. The corresponding angles that describe the dynamics are also illustrated for three collisions. following set of initial conditions is given (θ n, α n, V n, t n ). The particle s position is determined by X(θ n ) = R(θ n, t n ) cos(θ n ) and Y(θ n ) = R(θ n, t n ) sin(θ n ). The particle s velocity is written as V n = V n [cos(φ n + α n )î + sin(φ n + α n )ĵ], (12) where î and ĵ represent the unit vectors with respect to the X and Y axis, respectively. The auxiliary angle φ n is defined as [ Y ] (θ n, t n ) φ n = arctan, X (13) (θ n, t n ) where X (θ n, t n ) = dx(θ n, t n )/dθ n and Y (θ n, t n ) = dy(θ n, t n )/dθ n are given by Eq. (5) with dr(θ n, t n )/dθ n = pɛ[1 + η 2 cos(t n )] sin(pθ n ). The above expressions allow us to obtain the position of the particle as a function of time for t t n : X ρ (t) = X(θ n ) + V n cos(φ n + α n )(t t n ), (14) Y ρ (t) = Y(θ n ) + V n sin(φ n + α n )(t t n ). (15) The index ρ denotes that such coordinates correspond to the particle while the index b (see Eq. (11)) denotes the boundary. The distance of the particle measured with respect to the origin of the coordinate system is given by R ρ (θ n, t n ) = Xρ 2(t) + Y ρ 2(t) and θ ρ at X ρ (t), Y ρ (t) is θ ρ = arctan[y ρ (t)/x ρ (t)]. Therefore, the angular position at the next collision of the particle with the boundary, i.e. θ n+1, is numerically obtained by solving the following equation R ρ (θ n+1, t n ) = R b (θ n+1, t n ). The time at the next collision is obtained evaluating the expression Xρ 2 + Y ρ 2 t n+1 = t n +, V n where X = X(θ n+1 ) X(θ n ) and Y = Y(θ n+1 ) Y(θ n ). Since the referential frame of the boundary is moving, then, at the instant of the collision the following conditions must be matched (16) (17) V n+1 T n+1 = β V n T n+1, (18) V n+1 N n+1 = γ V n N n+1, where β [, 1] and γ [, 1] are the restitution coefficients. The completely inelastic collision happens when γ = β =. On the other hand, when γ = β = 1, corresponding to a complete elastic collision, all the results for the non-dissipative case are recovered. The upper prime indicates that the velocity of the particle is measured with respect to the moving wall referential frame. At the new angular position θ n+1, the unitary tangent and normal vectors are (19) T n+1 = cos(φ n+1 )î + sin(φ n+1 )ĵ, N n+1 = sin(φ n+1 )î + cos(φ n+1 )ĵ. Hence, one can easily find that (2) (21) V n+1 T n+1 = β V n T n+1 + (1 β) V b (t n+1 ) T n+1, (22) V n+1 N n+1 = γ V n N n+1 + (1 + γ ) V b (t n+1 ) N n+1, (23)

8 D.F.M. Oliveira, E.D. Leonel / Physica A 389 (21) Fig. 6. Behaviour of the average velocity as function of the number of collisions of the particle with the boundary. The control parameters used were p = 2, ɛ =.4 and two combinations of η 1 = η 2 =.1 and η 1 = η 2 =.5. The initial velocity used was V = 1 3. where V b (t n+1 ) is the velocity of the boundary which, written as where V b (t n+1 ) = dr b(t n+1 ) dt n+1 [cos(θ n+1 )î + sin(θ n+1 )ĵ], (24) dr b (t n+1 ) dt n+1 = η 1 sin(t n+1 ) η 2 sin(t n+1 ) cos(pθ n+1 ). (25) Then we have V n+1 = (V n+1 T n+1 ) 2 + (V n+1 N n+1 ) 2. (26) Finally, the angle α n+1 is written as [ V n+1 ] N n+1 α n+1 = arctan V n+1. T n+1 (27) Our numerical results for the time-dependent version discuss basically the behaviour of the average velocity of the particle. It was used two different procedures to obtain the average velocity. Firstly, we evaluate the average velocity over the orbit for a single initial condition V i = 1 n + 1 n V i,j, j= where the index i corresponds to a sample of an ensemble of initial conditions. Hence, the average velocity is written as V = 1 M M V i, i=1 where M denotes the number of different initial conditions. We have considered M = 5 in our simulations. Before discuss the results for the dissipative case, let us show the behaviour of the average velocity for the situation where γ = β = 1, i.e., the non-dissipative case. Fig. 6 shows the behaviour of the average velocity as function of the number of collisions of the particle with the boundary for the breathing case. The control parameters used were p = 2, ɛ =.4, two different values for η 1 = η 2 =.1 and η 1 = η 2 =.5 and the initial velocity V = 1 3. After a short transient, the slope of growth is of the order of.16. The numerical simulations were obtained using double precision with accuracy of 1 13 and evolved up to 1 9 collisions with the boundary. Such result allow us to confirm that for the non-dissipative timedependent oval-shaped billiard, the phenomenon of Fermi acceleration is indeed observed thus confirming the validity of the LRA conjecture. In the next sections, we shall discuss a scaling observed for the average velocity Scaling results for the non-dissipative case We begin discussing a scaling observed for the average velocity of the particle as function of V and n. It is shown in Fig. 7 the behaviour of the V n for different initial velocities. We have chosen the non-breathing geometry. Hence, the control (28) (29)

9 116 D.F.M. Oliveira, E.D. Leonel / Physica A 389 (21) Fig. 7. Behaviour of V n for different initial velocities. The control parameters used were p = 2, ɛ =.2, η1 =.1 and η 2 =.2. parameters used in Fig. 7 were p = 2, ɛ =.2, η 1 =.1 and η 2 =.2. Additionally, eighteen different values of V were chosen. We can see that all curves of the average velocity behave quite similarly. For short n, the average velocity remains constant. Then after a changeover, all the curves start growing with the same growing exponent. Based on the behaviour shown in Fig. 7, we propose the following: 1. For short n, say n n x, V behaves according to V sat V ζ, where ζ is a critical exponent; 2. For n n x, the average velocity is given by V n ν, where ν is also a critical exponent; 3. The crossover iteration number that marks the change from constant velocity to growth is written as (3) (31) n x V ξ, where ξ is a dynamic exponent. With these three initial suppositions, we suppose that the average velocity is described in terms of a scaling function of the type V(V, n) = l V(l a V, l b n), where l is the scaling factor, a and b are scaling exponents. If we chose l a V = 1, then l = V 1/a. Hence Eq. (33) is given by V(V, n) = V 1/a V1 (V b/a n), where V1 (V b/a n) = V(1, V b/a n) is assumed to be constant for n n x. Comparing Eqs. (34) and (3), we obtain ζ = 1/a. On the other hand, if we chose l = n 1/b, Eq. (33) is rewritten as V(V, n) = n 1/b V2 (n a/b V ), where the function V2 is defined as V2 (n a/b V ) = V(n a/b V, 1). It is also assumed to be constant for n n x. Comparing Eqs. (35) and (31) we find ν = 1/b. Considering the two different expressions for the scaling factor l obtained for n n x and n n x, we obtain n = V ζ ν. Comparing Eqs. (32) and (36) we find ξ = ζ /ν. The exponent ν is obtained from a power law fitting for the average velocity when n n x. Thus, an average of these values gives that ν =.786(9). Fig. 8 shows the behaviour of (a) Vsat V (32) (33) (34) (35) (36)

10 D.F.M. Oliveira, E.D. Leonel / Physica A 389 (21) a b Fig. 8. (a) Plot of V sat V. (b) Behaviour of n x as function of V. a b / / Fig. 9. (a) Behaviour of average velocity for different values of V. (b) Their collapse onto a single and universal plot. and (b) n x V. Applying power law fittings we obtain ζ =.988(2) = 1 and ξ = 1.261(2) 5/4. Considering the previous values of both ζ and ν and using ξ = ζ /ν, we find that ξ = 1.25(1) 5/4. Such result is in good accordance with our numerical data. A confirmation of the initial hypotheses comes from a collapse different curves of V n onto a single and universal plot, as demonstrated in Fig. 9(b) Scaling results for the dissipative case In this section we concentrate to characterize the behaviour of the average velocity in terms of the number of collisions with the boundary and as a function of the control parameter γ. Note we are considering only dissipation acting along the normal component of the particle s velocity. The main goal of this section is to confirm that the introduction of inelastic collisions of the particle with the boundary is sufficient condition to suppress Fermi acceleration. Indeed, this confirmation would reinforce a conjecture proposed for an stochastic one-dimensional model [16] and confirmed for deterministic systems [18]. The conjecture says basically: For one-dimensional billiard problems that show unlimited energy growth for both their deterministic and stochastic dynamics, the introduction of inelastic collision in the boundaries is a sufficient condition to break down the phenomenon of Fermi acceleration. Such phenomenon is also expected to be observed in two-dimensional, timevarying billiard problems since the FA mechanism is the same as that of the one-dimensional case. We study then a dissipative version of the oval-shaped billiard close to the transition from unlimited to limited energy growth. Such a transition happens when the control parameter γ 1. However the transition is better characterized if the following variable transformation is made: γ (1 γ ). To obtain the average velocity, each initial condition has a fixed initial velocity, V = 1 3 and randomly chosen α [, π], θ [, 2π], t [, 2π]. The control parameters used were η 1 = η 2 =.1 and β = 1. However, other combinations of control parameters η 1 and η 2 lead to similar results. The behaviour of V n for different values of γ is shown in Fig. 1. It is easy to see in Fig. 1 two different kinds of behaviours. For short n, the average velocity grows according to a power law and suddenly it bends towards a regime of

11 118 D.F.M. Oliveira, E.D. Leonel / Physica A 389 (21) Fig. 1. (a) Behaviour of V n for different values of γ, as labelled in the figure. The control parameters used in the construction of the figure were p = 2, ɛ =.4 and η 1 = η 2 =.1. a b Fig. 11. (a) Behaviour of V sat (1 γ ). (b) Behaviour of the crossover number n x against (1 γ ). A power law fitting in (a) furnishes α =.37(5) while in (b) z =.78(1). saturation for long enough values of n. The changeover from growth to the saturation is marked by a crossover iteration number n x. For such a behaviour, we propose the following scaling hypotheses: 1. When n n x the average velocity is V n δ, 2. As the iteration number increases, n n x, the average velocity approaches a regime of saturation, that is described as (37) V sat (1 γ ) α, 3. The crossover iteration number that marks the regime of growth to the constant velocity is written as n x (1 γ ) z, where α, δ and z are critical exponents. (38) (39)

12 D.F.M. Oliveira, E.D. Leonel / Physica A 389 (21) a b Fig. 12. (a) Different curves of the V for five different control parameters. (b) Their collapse onto a single and universal plot. A power law fitting gives us that δ =.48(3) 1/2. Such value was obtained from the range of γ [.999,.99999]. A power law fitting on the plot Vsat (1 γ ) furnishes α =.37(5) [see Fig. 11(a)]. Finally, using a similar procedure as that one used for the non-dissipative case, we show that z = α δ =.77(1), (4) which is quite close to the value obtained numerically, as shown in Fig. 11(b). A confirmation of the initial hypotheses is made by a collapse all the curves of V n onto a single and universal plot, as shown in Fig. 12(b). Besides showing that the system is scaling invariant, we also shown that dissipation causes a drastic change in the behaviour of V. The unlimited energy growth, observed for the non-dissipative case for long time, is not observed anymore for the dissipative version since V shows a regime of saturation when n n x (see Fig. 1). These results are confirmed analytically looking at the Eqs. (38) and (39) and the previous values for the critical exponents α and z. Observe that both α and z are negative, which lead to V sat 1/(1 γ ) α and n x 1/(1 γ ) z. Note that when γ 1, implies that V sat and n x, too, thus recovering the results for the conservative case, i.e., Fermi acceleration. However, when γ is slightly less than 1, it implies that the systems possess a characteristic saturation value V sat and a crossover iteration number n x. Our results for a two-dimensional time-dependent billiards confirm that when dissipation, via inelastic collision, is introduced into the model, it is assumed as a sufficient condition to suppress the unlimited energy growth. 4. Conclusion We have studied a classical version of an oval-shaped billiard considering both the static and time-varying boundary. For the static version we obtain the condition where the invariant spanning curves, present in the mixed phase space are destroyed. For the time-dependent boundary, we consider two situations: (i) non-dissipative and (ii) dissipative version. For the non-dissipative version, we show that Fermi acceleration is observed and that the behaviour of the average velocity is described using a scaling approach. For the dissipative version, the average velocity achieves a constant value for large enough collisions of the particle with the boundary. Then Fermi acceleration is suppressed. The average velocity was also described using a scaling approach. Acknowledgements D.F.M.O gratefully acknowledges CAPES and FAPESP. E.D.L. is grateful to FAPESP, CNPq and FUNDUNESP, Brazilian agencies and comissão mista CAPES/FULBRIGHT. The authors also thank Prof. Alexander Loskutov for the fruitful discussions. Appendix In this Appendix we present the procedure used to obtain the expression of the critical control parameter ɛ c. Increasing the control parameter ɛ, the shape of the boundary changes (see Fig. 1). The expression for the critical value ɛ c, which marks a change where curvature of the boundary varies locally from a positive (κ > ) to a negative (κ < ). Hence, using polar coordinates the expression for κ(θ) is given by κ(θ) = X (θ)y (θ) X (θ)y (θ). [X 2 (θ) + Y 2 (θ)] 3 2 (41)

13 12 D.F.M. Oliveira, E.D. Leonel / Physica A 389 (21) For the general case, the expressions for X (θ), Y (θ), X (θ) and Y (θ) are where dr(θ) dθ X (θ) = dr(θ) dθ Y (θ) = dr(θ) dθ X (θ) = d2 R(θ) dθ 2 Y (θ) = d2 R(θ) dθ 2 and d2 R(θ) dθ 2 cos(θ) R(θ) sin(θ), sin(θ) + R(θ) cos(θ), cos(θ) 2 dr(θ) dθ sin(θ) + 2 dr(θ) dθ are given by dr(θ) = ɛp sin(pθ), dθ d 2 R(θ) = ɛp 2 cos(pθ). dθ 2 sin(θ) R(θ) cos(θ), cos(θ) R(θ) sin(θ), The expression for ɛ c is obtained when κ =. The expression for ɛ c as function of p is (42) (43) ɛ c = p 2. Thus, when ɛ < ɛ c the boundary is strictly convex. However, if ɛ > ɛ c one can observe non-convex pieces on the boundary. References [1] E. Fermi, Phys. Rev. 75 (1949) [2] L.A. Bunimovich, Commun. Math. Phys. 65 (1979) 295. [3] Y.G. Sinai, Russian Math. Surveys 25 (197) 137. [4] M.V. Berry, European J. Phys. 2 (1981) 91. [5] M. Robnik, J. Phys. A: Math. Gen. 16 (1983) [6] M. Robnik, M.V. Berry, J. Phys. A: Math. Gen. 18 (1985) [7] S.O. Kamphorst, S.P. Carvalho, Nonlinearity 12 (1999) [8] R. Markarian, S.O. Kamphorst, S.P. de Carvalho, Commun. Math. Phys. 174 (1996) 661. [9] V. Lopac, I. Mrkonjić, D. Radić, Phys. Rev. E 66 (22) [1] V. Lopac, I. Mrkonjić, N. Pavin, D. Radić, Physica D 217 (26) 88. [11] A.J. Lichtenberg, M.A. Lieberman, Appl. Math. Sci., vol. 38, Springer Verlag, New York, [12] A.M. Ozório de Almeida, Hamiltonian Systems: Chaos and Quantization, Cambridge Univ. Press, Cambridge, [13] S.O. Kamphorst, E.D. Leonel, J.K.L. Silva, J. Phys. A: Math. Gen. 4 (27) F887. [14] E.D. Leonel, D.F.M. Oliveira, A. Loskutov, Chaos 19 (29) [15] A. Loskutov, A.B. Ryabov, L.G. Akinshin, J. Phys. A: Math. Gen. 33 (2) [16] E.D. Leonel, J. Phys. A: Math. Theor. 4 (27) F177. [17] D.G. Ladeira, E.D. Leonel, Chaos 17 (27) [18] A.L.P. Livorati, D.G. Ladeira, E.D. Leonel, Phys. Rev. E 78 (28) [19] E.D. Leonel, P.V.E. McClintock, J.K.L. Silva, Phys. Rev. Lett. 93 (24) [2] D.F.M. Oliveira, R.A. Bizão, E.D. Leonel, Math. Probl. Eng. (29) ID , 13 pages, doi:1.1155/29/ [21] J.P. Eckmann, D. Ruelle, Rev. Modern Phys. 57 (1985) 617. [22] D.F.M. Oliveira, E.D. Leonel, Brazilian J. Phys. 38 (28) 62. (44)