Scaling investigation of Fermi acceleration on a dissipative bouncer model

Scaling investigation of Fermi acceleration on a dissipative bouncer model

André Luis Prando Livorati, Denis Gouvêa Ladeira, and Edson D. Leonel

Departamento de Estatística, Matemática Aplicada e Computação, IGCE, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, CEP 13506-900, Rio Claro, São Paulo, Brazil

共Received 30 July 2008; published 11 November 2008兲

The phenomenon of Fermi acceleration is addressed for the problem of a classical and dissipative bouncer model, using a scaling description. The dynamics of the model, in both the complete and simplified versions, is obtained by use of a two-dimensional nonlinear mapping. The dissipation is introduced using a restitution coefficient on the periodically moving wall. Using scaling arguments, we describe the behavior of the average chaotic velocities on the model both as a function of the number of collisions with the moving wall and as a function of the time. We consider variations of the two control parameters; therefore critical exponents are obtained. We show that the formalism can be used to describe the occurrence of a transition from limited to unlimited energy growth as the restitution coefficient approaches unity. The formalism can be used to charac-terize the same transition in two-dimensional time-varying billiard problems.

DOI:10.1103/PhysRevE.78.056205 PACS number共s兲: 05.45.Pq, 05.45.Tp

I. INTRODUCTION

Fermi acceleration共FA兲 is a phenomenon in which a clas-sical particle acquires unbounded energy from collisions with a massive moving wall 关1,2兴. Applications of FA have acquired a broad interest in different fields of physics includ-ing plasma physics 关3兴, astrophysics 关4,5兴, atomic physics 关6兴, optics 关7,8兴, and even the well-known time-dependent billiard problems关9,10兴. The interesting question posed that should be answered is whether the FA results from the non-linear dynamics itself, considering the complete absence of any imposed random motion. A one-dimensional model ex-hibiting such a phenomenon, which is modeled by a nonlin-ear mapping, is the bouncer model 关11,12兴. For two-dimensional time-dependent billiards共billiards with moving boundaries兲, the answer to this question is not unique; it depends on the kind of phase space for the corresponding static version of the problem. Therefore, as conjectured by Loskutov, Ryabov, and Akinshin 关13兴 共this conjecture is known in the literature as the LRA conjecture兲, the regular dynamics for a fixed boundary implies a bound to the energy gained by the bouncing particle, but the chaotic dynamics of a billiard with a fixed boundary is a sufficient condition for FA in the system when a boundary perturbation is intro-duced. This conjecture was confirmed in different billiards 关14–16兴. FA was observed recently in a time-varying ellipti-cal billiard 关17兴. The oval billiard, however, seems not to exhibit FA for the breathing case关18兴.

In this paper we reconsider the problem of a classical particle bouncing inelastically from a periodically time-varying wall in the presence of a constant gravitational field 关19–27兴. Our main goal is to understand and describe the phase transition present in this problem. The transition is from limited to unlimited energy growth when the damping coefficient 共restitution coefficient兲 approaches unity. The problem is described using a two-dimensional nonlinear mapping for the variables velocity of the particle and time immediately after a collision with the moving wall. It is known that 关28兴, for the nondissipative version of the prob-lem and depending on both the control parameters and initial

conditions, the particle can experience the phenomenon of FA. This phenomenon is observed in this problem due to the absence of invariant spanning curves in the phase space, which happens for specific conditions of the control param-eters. As we shall show, when the collisions of the particle with the moving wall are innelastic共the particle experiences a fractional loss of energy upon colision with the moving wall兲, the phenomenon of unlimited energy growth is not observed关29兴. This implies that there exists a restitution co-efficient␣苸共0,1兴. Then, for␣= 1 all the collisions are elas-tic. We consider, however, the situation where ␣⬍1. More-over, for ␣⬍1 the mapping possesses the property of shrinking the phase space area; thus, depending on the initial condition, a chaotic attractor can be observed. The principal focus of this paper is to describe the behavior of the average velocities of the chaotic attractor and use scaling arguments to characterize the transition from limited to unlimited en-ergy growth for␣→1, when the dissipation disappears. De-pending on the combination of the initial conditions and con-trol parameters, the model can exhibit FA for␣= 1. Thus the limit ␣→1 characterizes this transition. We describe the model via a two-dimensional nonlinear mapping T that de-scribes the system in terms of the variables velocity of the particle, v, and time t, i.e., T共vn, tn兲=共vn+1, tn+1兲, where n

denotes the index of the nth collision with the wall. We con-sider in this paper both the complete 关30,31兴 and the simpli-fied versions of the model 关32–34兴. The complete model takes into account the full aspects of the dynamics in the sense that the motion of the platform leads to different kinds of collision, which we denote as 共i兲 direct 共or successive— those collisions that the particle has before leaving the region near the moving wall’s allowed positions兲 and 共ii兲 indirect collisions. The simplified model, however, consists of an ap-proximation that disregards the displacement of the oscillat-ing platform. However, it considers that the particle changes energy upon collision with the moving wall as if the wall were periodically moving. This approximation carries the ad-vantage of avoiding the solution of transcendental equations, as they must be solved in the complete model 关31兴. It also retains the nonlinearity of the problem 关35兴. The study of both cases allows us to compare the critical exponents near

the transition and, as far as we know, has not been consid-ered before using such an approach.

The paper is organized as follows. In Sec. II, we present all the details needed to construct a complete version of the model. We show some chaotic orbits and their positive Lyapunov exponent. We also discuss the behavior of the positive Lyapunov exponent as a function of the control pa-rameters. Our numerical results obtained for the average ve-locity and deviation around the average veve-locity are pre-sented in this section. We also present a simplified version of the problem and compare the results and critical exponents with those from the full model. In Sec. III the scaling de-scription is made by using the averages obtained as a func-tion of time. The critical exponents are obtained and a spe-cial discussion is made connecting the results found as a function of the collision number and time. Finally, in Sec. IV we summarize our results and present our concluding re-marks.

II. THE MODEL, THE COMPLETE VERSION OF THE MAP, AND RESULTS FOR QUANTITIES AVERAGED

OVER THE NUMBER OF COLLISIONS

The model under consideration consists of a classical par-ticle of mass m which is suffering inelastic collisions with a periodically moving wall given by y共t兲=␧ cos共wt兲, where ␧ is the amplitude of oscillation and w is the angular frequency. Moreover, the particle is under the action of a constant gravi-tational field g. Thus the return mechanism of the particle for the next collision with the wall is due only to the gravita-tional field. We have assumed that the dissipation is intro-duced via a coefficient of restitution ␣ such that ␣苸关0,1兴. The limit of␣= 1 recovers the results for the nondissipative case while for␣= 0 it is possible to observe the phenomenon of locking 关22,23兴. For the case where ␣⬍1, the particle experiences a fractional loss of energy upon collisions with the moving wall. The description of the dynamics of the problem is made by use of a two-dimensional nonlinear map-ping for the variables velocity of the particle vn and time tn

immediately after the nth collision of the particle with the moving wall. The so-called complete version takes into ac-count the movement of the moving wall. There are then two distinct kinds of collision that may arise from such a version, namely,共i兲 multiple collisions of the particle with the moving wall, which can happens before the particle leaves the colli-sion zone 共the collision zone is defined as the region y 苸关−␧,␧兴兲; or 共ii兲 a single collision of the particle with the moving wall. In the case共ii兲 the particle leaves the collision zone after suffering a single collision with the moving wall. Before we write the equations of the mapping, we should point out that there are an excessive number of control pa-rameters, four in total, namely,␧, g, w, and␣. The dynamics, however, does not depend on all of them, and we can define the dimensionless variables Vn=vnw/g,⑀=␧w2/g, and

mea-sure the time as ␸n= wtn. Considering this set of new

vari-ables and assuming that the initial conditions for the dynam-ics are Vn⬎0,␸n苸关0,2␲兴, and the position of the particle at

the instant ␸n is yp共␸n兲=⑀cos共␸n兲, we obtain the mapping

written as T:

再

Vn+1= −␣共Vn *_−␾ c兲 − 共1 +␣兲⑀sin共␾n+1兲, ␾n+1=共␾n+⌬Tn兲 mod共2␲兲,

冎

共1兲 where the expressions for V_n*and⌬Tn depend on what kind

of collision happens. For the case共i兲, i.e., multiple collisions, the corresponding expressions are V_n*= Vnand⌬Tn=␸c. The

term ␸cis obtained from the condition that matches the

po-sitions for the particle and the moving wall. This condition gives rise to the transcendental equation

G共␾c兲 =⑀cos共␾n+␾c兲 −⑀cos共␾n兲 − Vn␾c+

1 2␾c

2_{. 共2兲}

If the particle leaves the collision zone, then case 共ii兲 applies, and the corresponding expressions are V_n* = −

冑

V_n2+ 2⑀关cos共␸n兲−1兴 and ⌬Tn=␸u+␸d+␸c, with ␸u= Vn

denoting the time spent by the particle moving in the upward direction until it reaches the null velocity, and ␸d

=

冑

V_n2+ 2⑀关cos共␸n兲−1兴 corresponds to the time that the

par-ticle takes to move from the place where it had zero velocity to the entrance of the collision zone. Finally, the term␸chas

to be found numerically from the equation F共␸c兲=0, where

F共␾c兲 =⑀cos共␾n+␾u+␾d+␾c兲 −⑀− V_n*␾c+

1 2␾c

2

. 共3兲 After some straightforward algebra, it is easy to show that the mapping 共1兲 shrinks the phase space measure, since the determinant of the Jacobian matrix is given by

DetJ =␣2

冉

Vn+⑀sin共␾n兲 Vn+1+⑀sin共␾n+1兲

冊

. 共4兲

In order to see the behavior that can emanate from itera-tion of the mapping共1兲, Fig.1shows the behavior of chaotic orbits for four different sets of control parameters. In Fig. 1共a兲 ␣= 0.99 and ⑀= 10, while in Fig. 1共b兲 ␣= 0.999 and

⑀= 10. For Fig.1共c兲␣= 0.99 and⑀= 100 and finally for Fig. 1共d兲 ␣= 0.999 and ⑀= 1000. Each initial condition was iter-ated up to 107 _{collisions with the moving wall. We can see}

that, even for a long evolution of 107_{collisions, the orbits do}

not seem to show unlimited energy growth. To check whether the orbits shown in Fig.1are chaotic, the Lyapunov exponent must be evaluated. It is known that the Lyapunov exponent is a common tool used to characterize the sensitiv-ity to the initial conditions. The procedure consisits in evolv-ing the system over a long time from two slightly different initial conditions. If the two trajectories diverge exponen-tially in time the orbit is called chaotic and the Lyapunov exponent obtained is positive. Let us describe briefly the pro-cedure used to obtain the Lyapunov exponents numerically. They are defined关36兴 as

␭j= lim

n→⬁ln兩⌳j兩, j = 1,2,

where⌳jare the eigenvalues of M =兿k=1 n

Jk共Vk,␸k兲 and Jkis

the Jacobian matrix evaluated along the orbit共Vk,␸k兲.

Figure2shows the convergence of the positive Lyapunov exponents as a function of the number of collisions with the moving wall for the complete version of the dissipative bouncer model. We see that, when five different initial

ditions are evolved in time, all of them converge to a con-stant value for large n. This convergence allows us to obtain the average value for the positive Lyapunov exponent over the chaotic attractor. The control parameters used in the con-struction of the Fig. 2 were 共a兲 ␣= 0.99 and ⑀= 10; 共b兲 ␣ = 0.999 and⑀= 10;共c兲␣= 0.999 and⑀= 100;共d兲␣= 0.999 and

⑀= 1000. The corresponding medium values are shown in the figure.

Let us now discuss the behavior of the positive Lyapunov exponent as a function of the control parameters ⑀ and ␣. Figure3shows the behavior of共a兲 ␭¯ vs 共1−␣兲 and 共b兲 ␭¯ vs

⑀. We can see in Fig.3共a兲that the positive Lyapunov expo-nent is almost constant over the two initial decades in the log-linear graph and then decreases for 共1−␣兲⬇10−3.

Thus an exponential fit give us that ␭¯

= 2.302 50共2兲exp关−0.861共5兲共1−␣兲兴. On the other hand, the positive Lyapunov exponent is more sensitive to the depen-dence on ⑀, as shown in Fig. 3共b兲. The log-linear graph shows an increase in␭¯ as⑀increases. Such behavior is better fitted by a logarithmic function of the type␭¯=−0.001 71共6兲 + 0.999 76共1兲ln共⑀兲.

We now concentrate on obtaining some statistical proper-ties of the chaotic time series. We shall investigate the be-havior of the deviation of the average velocity for chaotic orbits. The deviation of the average velocity therefore exhib-its similar properties as the average velocity关29,37兴. Before we define the deviation of the average velocity properly, it is

important first to know the average velocity, which is given by V ¯ 共n,⑀,␣兲 =1 n

兺

_i=0 n Vi. 共5兲

Then, the deviation around the average velocity is defined as

␻共n,⑀,␣兲 = 1 M

兺

i=1 M

冑

V2i共n,⑀,␣兲 − V¯i 2_共n,⑀ ,␣兲. 共6兲 The behavior of␻ as a function of n is shown in Fig.4. To construct the curves shown in Fig. 4 we have evolved an ensemble of 103 different initial phases for the same initial velocity V0=⑀. The situation where different initial velocities

are considered can be found, for a version of the Fermi-Ulam model关38兴, in 关30,32兴. Each initial condition was iterated up to 107collisions with the moving wall. One can see that the curve starts growing for a small number of collisions and then suddenly it bends toward a regime of convergence marked by a constant plateau. The changeover from growth to saturation is characterized by a typical number of colli-sions, indicated in the figure as nx. This kind of behavior is

observed for different combinations of control parameters. FIG. 1. Chaotic orbits for a full version of the dissipative

bouncer model. The control parameters used in the construction of the figures were共a兲␣=0.99 and ⑀=10; 共b兲 ␣=0.999 and ⑀=10; 共c兲 ␣=0.99 and ⑀=100; 共d兲 ␣=0.999 and ⑀=1000. 2.28 2.285 V₀= 7.52 φ0= 1.56 V₀= 8.20 φ₀= 0.79 V₀= 7.03 φ₀= 0.20 V₀= 3.17 φ₀= 2.62 V₀= 5.67 φ₀= 1.08 2.295 2.3 V₀= 7.52 φ₀= 1.56 V₀= 8.20 φ₀= 0.79 V₀= 7.03 φ₀= 0.20 V₀= 3.17 φ₀= 2.62 V₀= 5.67 φ₀= 1.08 4.6 4.61 V₀= 97.42 φ₀= 6.04 V₀= 11.40 φ₀= 2.48 V₀= -24.37 φ₀= 1.48 V₀= 80.41 φ₀= 2.09 V₀= 84.92 φ₀= 1.45 6.905 6.91 V₀= -394.03 φ₀= 1.79 V₀= 912.20 φ₀= 0.55 V₀= -70.64 φ0= 1.58 V₀= 86.93 φ₀= 1.01 V₀= -414.79 φ₀= 1.27 λ λ = 2.28296(8) λ λ = 2.3003(1) λ λ = 4.6021(1) λ λ = 6.9044(8) (a) (b) (c) (d) 107 _n 108 107 _{n 10}8 107 _n 108 107 n 108

FIG. 2. 共Color online兲 Convergence for the positive Lyapunov exponent. The control parameters used were 共a兲 ␣=0.99 and ⑀=10; 共b兲 ␣=0.999 and ⑀=10; 共c兲 ␣=0.999 and ⑀=100; 共d兲 ␣=0.999 and ⑀=1000.

However, those curves constructed for different values of⑀ start growing at different positions, thus indicating that n is not a good variable. However, if we apply the transformation n→n⑀2 _{all curves start growing together, as shown in Fig.}

4共b兲.

The behavior of the curves shown in Fig.4共b兲motivates us to suppose the following scaling hypotheses.

共1兲 For a small number of collisions with the moving wall, the behavior of␻can be described by a power law of the type

␻⬀ 共n⑀2_兲␤_{, 共7兲}

where␤ is the critical exponent. Applying a power law fit-ting in Fig.4共b兲, we obtain␤= 0.479共8兲⬵0.5.

共2兲 For a sufficient long n, the saturation of␻共we denote it as␻sat兲 is given by

␻sat⬀ 共1 −␣兲␥1⑀␥2, 共8兲

where both␥1and␥2are also critical exponents.

共3兲 Finally, the number of collisions that characterizes the changeover from growth to the saturation regime is given by

nx⑀2⬀ 共1 −␣兲 z_1⑀z₂

, 共9兲

where z₁and z₂ are called dynamical exponents.

Instead of considering the variable ␣, we do better to choose to describe the average properties as functions of 1 −␣. This transformation brings the criticality to 共1−␣兲→0.

To obtain the critical exponents ␥₁ and ␥₂, we must evolve a simulation for a very long n. Thus, using extrapo-lation we can obtain the values of ␻sat as functions of

共1−␣兲 and ⑀. The exponents z1 and z2 are obtained via a

crossing of the values obtained for the saturation and the initial growth regime. The behavior of ␻_satand nx as

func-tions of both 共1−␣兲 and ⑀ is shown in Fig. 5. One of the most important results of this paper is evident from a careful investigation of Eqs. 共8兲 and 共9兲 for the critical exponents

␥1⬵−0.5 and z1⬵−1. Clearly, both Eqs. 共8兲 and 共9兲 diverge

for the limit of ␣→1. In particular, Eq. 共8兲 recovers the phenomenon of Fermi acceleration for the limit of ␣→1.

Considering these scaling hypotheses, we can now de-scribe␻in terms of a scaling function of the type

␻„共1 −␣兲⑀y

,n⑀2… = l␻„la_{共1 −␣兲⑀}y

,lb共n⑀2兲…, 共10兲 where l is the scaling factor and a and b are scaling expo-nents. Choosing l =共n⑀2兲−1/b, we have,

10

-5

10

-4

10

-3

10

-2

(1 - α)

2.29

2.3

10

-5

10

-4

10

-3

10

-2 Numerical Data Best Fit

10

1

10

2

10

3

10

4

ε

3

6

9

10

1

10

2

10

3

10

4 Numerical Data Best Fit

λ = 2.30250(2)e

- 0.861(5)(1−α)

λ = − 0.00171(6) + 0.99976(1)ln(ε)

(b)

(a)

λ

ε = 10

α = 0.999

λ

FIG. 3. 共Color online兲 Plot of the averaged positive Lyapunov exponent as a function of共a兲 共1−␣兲 and 共b兲 ⑀.

FIG. 4. 共Color online兲 共a兲 Plot of␻ as a function of n. 共b兲 The same plot shown in共a兲 after the transformation n→n⑀2_{. The control}

parameters used in the construction of these curves are shown in the figure.

␻„⑀y_{共1 −␣兲,n⑀}2_{… = 共n⑀}2_兲−1/b_␻

1„共n⑀2兲−a/b共1 −␣兲⑀y…,

共11兲 where we assume that ␻1 is constant for nxⰇn. From Eq.

共11兲 we obtain the following expression for the crossover number:

nx⑀2⬀ 关共1 −␣兲⑀y兴b/a. 共12兲

Using the above relation and Eq. 共9兲, we obtain b

a= z1, yb

a = z2. 共13兲

Thus the numerical value for y is

y =z2

z₁= − 2.07⫾ 0.01. 共14兲

Now choosing, l =关共1−␣兲⑀y_兴−1/a_{, we have}

␻„⑀y_{共1 −␣兲,n⑀}2_{… = 关共1 −␣兲⑀}y_兴−1/a_␻

2„共1 −␣兲⑀y−b/a共n⑀2兲…,

共15兲 where we assume␻2 as constant for nxⰆn. From Eq. 共15兲

we obtain the expression for␻sat,

␻sat⬀ 关共1 −␣兲⑀y兴−1/a. 共16兲

From the above equation and Eq.共8兲, we have

−1

a =␥1, − y

a=␥2. 共17兲

These expressions give us the numerical values of the expo-nents a and y as a = − 1 ␥1 = 2.00⫾ 0.03, 共18兲 y =␥2 ␥1 = − 2.00⫾ 0.03. 共19兲

After this discussion, we can see that both Eq. 共14兲 and Eq. 共19兲 give y⬵−2. Considering the limit n⑀2_Ⰶn

x⑀2, we

can rewrite Eq. 共15兲 as

␻„⑀y_{共1 −␣兲,n⑀}2_{… = 关共1 −␣兲⑀}y_兴−1/a_{兵关共1 −␣兲⑀}y−b/a_共n⑀2_兲兴其␤_.

共20兲 Thus, for the regime of growth we have

−y a− ␤by a = − 1 a− ␤b a = 0. 共21兲

Considering Eq.共21兲 and Eqs. 共13兲 and 共20兲, we obtain b = −1

␤= − 2.08⫾ 0.03, 共22兲

b = − z2

␥2

= − 2.007⫾ 0.005. 共23兲

We therefore obtain that b⬇−2. Considering Eqs. 共13兲, 共20兲, and共21兲, we have the numerical value for a as

a = y

␤z2

= 1

␤z1

⬇ 2. 共24兲

To check whether these initial suppositions are correct, let us now obtain a universal plot of different curves generated from different control parameters. The behavior of ␻ as a function of n for different values of control parameters is shown in Fig.6. One can see that, after reescaling the coor-dinate axes according to the critical exponents obtained above, all curves generate a single and universal plot.

A. A simplified model and numerical results for averages over the number of collisions

Let us discuss in this section a simplified version of the problem. The simplification consists basically in considering that, when the particle hits the wall, it suffers an exchange of energy and momentum as if the wall were moving 关33,35兴. This is a common approximation used to speed up the nu-merical simulation, since no transcendental equations must be solved as they must be in the complete version. The sim-plified version also retains the nonlinearity of the problem. The mapping is then written as

10-5 10-4 10-3 10-2 (1 − α) 102 103 ω sat 10-5 10-4 10-3 10-2 102 103 Numerical Data Power Law Fit

10-5 10-4 10-3 10-2 (1 − α) 104 105 106 107 n x ε 2 10-5 10-4 10-3 10-2 104 105 106 107 Numerical Data Power Law Fit

101 102 103 104 ε 103 104 105 ω sat 101 102 103 104 103 104 105 Numerical Data Power Law Fit

101 102 103 104 ε 106 109 n x ε 2 101 102 103 104 106 109 Numerical Data Power Law Fit

γ₁= − 0.498(8) _z

1= - 0.969(9)

γ₂= 0.9985(9) z₂= 2.004(7)

(a) _(b)

(c) (d)

FIG. 5. 共Color online兲 Behavior of 共a兲␻_satvs共1−␣兲, 共b兲 n_xvs 共1−␣兲, 共c兲 ␻satvs⑀, and 共d兲 nx⑀2vs⑀.

T:

再

Vn+1=兩␣Vn−共1 +␣兲⑀sin共␾n+1兲兩,

␾n+1=关␾n+ 2Vn兴 mod共2␲兲.

冎

共25兲 This mapping also shrinks the area on the phase space for

␣⬍1 since Det J=␣sgn关␣Vn−共1+␣兲⑀sin共␸n+1兲兴, where the

function sgn共u兲=1 if u⬎0 and sign共u兲=−1 if u⬍0, and J is the Jacobian matrix. It is important to say that in the simpli-fied model negative velocities of the particle after a collision with the wall are forbidden, because such velocities would be equivalent to the particle moving beyond the wall. In order to avoid such problems, if after the collision with the wall the particle acquires a negative velocity, we inject it back with the same modulus of velocity. This procedure is effected by use of a modulus function. Note that the velocity of the particle is reversed by the modulus function only if, after the collision, the particle remains traveling in the nega-tive direction. The modulus function has no effect on the motion of the particle if the particle moves in the positive direction after the collision.

Chaotic orbits are also present in this version of the model. The convergence of the Lyapunov exponents is ob-served by using the same algorithm as used for the complete version. Thus, we can obtain the behavior of the positive Lyapunov exponent as a function of ⑀ and ␣. In Fig. 7 is shown the behavior of␭¯ vs⑀and␭¯ vs 共1−␣兲. The functions that fit the numerical data are the same as those obtained for the complete model. However, the coefficients are a little different. Their corresponding fits are shown in the figure.

Let us now discuss the differences obtained for the posi-tive Lyapunov exponents based on the behavior of the

histo-grams of frequency concerning the chaotic attractors for both the complete and simplified versions of the model. Such an analysis is important since we have differences in the asymptotic values for the positive Lyapunov exponents. It is shown in Fig. 8the behavior of the histogram of frequency for the共a兲 simplified and 共b兲 complete versions of the model. We can clearly see that the two are rather different from each other in the low- and average-energy regimes and quite simi-lar for the high-energy regime, where they experience a de-cay. The difference in the low-energy regime is basically due to the reinjection of the particle that occurs in the simplified version of the model. This reinjection is caused by the modu-lus function. In the simplified model, the frequency is slightly sparse in the regime of low energy, mainly because of the action of the modulus function, and then it converges to a thin curve which experiences a decay for the regime of high energy. The behavior of the histogram of frequency for the complete model is different in the regime of low energy. We can see that the frequency at low energy is low but very well defined. Then it rises rapidly, passing by a maximum value, and experiences a decay for the regime of high energy. Thus the complete model has a preference for average ve-locities. This kind of behavior is the main explanation for the differences in values obtained for the positive Lyapunov ex-ponents when we compare the simplified and complete ver-sions of the model.

The behavior of the histogram of frequency for the com-plete model seems to be universal since it has the same gen-eral form for different control parameters. Figure9共a兲 illus-trates the behavior of the histogram of frequency for different control parameters, as labeled in the figure. We can FIG. 6. 共Color online兲 Universal plot for different curves of

␻. 10-5 10-4 10-3 10-2 (1 − α) 2.992 2.993 2.994 2.995 2.996 2.997 Numerical Data Best Fit 101 102 103 104 ε 2 4 6 8 10 λ Numerical Data Best Fit

(a)

(b)

λ = 2.996(5)e-0.121(2)(1−α) λ = 0.6929(1)+0.9999(6)ln(ε)

ε = 10

α = 0.999

λ

see that the peaks of the histograms depend on the control parameters. We suppose that the velocity of the peaks can be described as

Vp⬀⑀␮共1 −␣兲␦, 共26兲

where ␮ and ␦ are exponents to be obtained. Considering different combinations of control parameters, we obtain the values of the velocity in the peak, which we denote as Vp, by

fitting a polynomial of second order around the region of peak. Thus, by finding the value where this polynomial as-sumes a maximum, we easily obtain the peak value. This value is then plotted against the control parameters, as shown in Fig. 10. Thus, after fitting power laws, we obtain that

␮= −0.487共5兲⬵−0.5 and ␦= 1.00共6兲⬵1. With these values, we can then apply a reescaling in the velocity axis and col-lapse all the curves onto a single and universal curve, as shown in Fig. 9共b兲.

B. Scaling as function of the number of collisions As discussed in the previous section, the simplified ver-sion of the model also allows us to describe their statistical properties by using scaling arguments. The results as a func-tion of the damping coefficient are already known in the literature共see Ref. 关37兴兲, but not the description as a function of the parameter ⑀, to our knowledge. After doing extensive simulation to obtain the critical exponents, we found that their values are rather similar, considering the margin of un-certainty. The critical exponents we found for the simplified

version are ␥1= −0.4983共9兲, z1= −0.9818共1兲, ␥2= 0.9983共1兲,

and z2= 1.9940共3兲. Therefore, the collapse of all the curves

of the deviation of the average velocity is pretty similar to the one shown in Fig. 6.

III. ANALYSIS OF THE SIMPLIFIED VERSION ON VARIABLE TIME

In this section we describe the scaling properties of the bouncer model when average properties of the system are evaluated over time. The main interest in such an analysis lies in the fact that the collision number is not necessarily FIG. 8. Frequency histograms, for the variable velocity, of

cha-otic attractors for 共a兲 simplified and 共b兲 complete versions of the model. The control parameters used were␣=0.999 and ⑀=10.

FIG. 9.共Color online兲 共a兲 Frequency histograms, for the variable velocity, regarding chaotic attractors for the complete version of the model.共b兲 Their collapse onto a single and universal curve.

FIG. 10. 共Color online兲 Behavior of Vp as a function of the

proportional to time, as discussed by Lichtenberg and Lie-berman关39兴.

A. Scaling of the deviation of the velocity

Before we present the scaling description let us introduce the average quantities that we are going to deal with. Regard-ing at first the orbit of an initial condition i, we evaluate the average velocity using the expression

Vi共␧,␣,t兲 =

1 t

冕

0

t

Vi共␧,␣,␶兲d␶. 共27兲

Now we regard an ensemble of M chaotic trajectories and we obtain the average value of the deviation of the velocity,

␻共␧,␣,t兲 = 1 M

兺

_i=1

M

冑

Vi2共␧,␣,t兲 − Vi2共␧,␣,t兲. 共28兲

The scaling description is more straightforward when we replace t by the new variable t␧2_{. Figure 11 shows the ␻}

curves for different values of the control parameters

␣r= 1 −␣and␧, as labeled in the figure.

We observe in Fig. 11 that the initial growth of the ␻ curves depends only on t␧2 _{according to the expression}

␻⬀ 共t␧2_兲␤_{, 共29兲}

where␤ is the growth exponent. Performing power law fit-tings to the numerical data in the growth regime, we obtain the average value␤= 0.334⫾0.008共⬇1/3兲.

In the limit of large values of t␧2_{, when the deviation of}

the velocity reaches constant values,␻depends on␧ and␣r.

Thus we describe the saturation regime of␻as

␻sat⬀ ␧␥1␣r␥2, 共30兲

where␥₁and␥₂are the saturation exponents. We obtain the exponent␥1by plotting␻satversus␧ with a fixed value of␣r

and, similarly, we obtain␥2by fitting the numerical data in a

plot of␻_satas a function of␣rfor a fixed value of␧. Figures

12共a兲 and 12共b兲 illustrate the procedure, which furnishes

␥1= 0.99⫾0.01 and␥2= −0.510⫾0.001. Because␥2is

nega-tive we observe that ␻ diverges in the limit ␣r→0, an

ex-pected result.

The change from growth to the saturation regime defines the variable tz␧2. This crossover is a function of ␧ and ␣r

according to

tz␧2⬀ ␧z1␣r z₂

, 共31兲

where z1 and z2 are the dynamical exponents. We obtain

these exponents by fitting the numerical data tz␧2 versus ␧

and tz␧2versus␣r, as we present, respectively, in Figs.12共c兲

and 12共d兲. The procedure furnishes z1= 2.98⫾0.02 and

z2= −1.49⫾0.01. Because z2is negative we observe that the

time until the saturation regime diverges for ␣r→0. This

result indicates that␻diverges in the elastic collision limit. In Fig. 11 we observe that␻ depends on␧,␣r, and t␧2.

However, the appropriate scaling variables of the average quantity␻are␧y_␣

rand t␧2, where y is an exponent related to

the critical exponents. Near the transition from limited to unlimited energy growth, ␣r⬇0, we suppose that ␻ obeys

the homogeneous equation

␻共␧y_␣

r,t␧2兲 = l␻共la␧y␣r,lbt␧2兲. 共32兲

Choosing l =共␧y_␣

r兲−1/a, we have that the crossover is

de-scribed by tz␧2⬀共␧y␣r兲b/a. This relation and Eq.共31兲 furnish

z1= yb/a and z2= b/a. These relations give that the exponent

y of Eq.共32兲 obeys the expression y=z1/z2= −2.00⫾0.03.

In the limit of large values of t␧2_{we have that␻does not}

depend on time. Thus Eqs.共32兲 and 共30兲 furnish the relations

␥1= −y/a and ␥2= −1/a. These expressions give us

a = −1/␥2= 1.961⫾0.004 and y =␥1/␥2= −1.94⫾0.02.

Regarding the uncertainties, we observe that y = z1/z2

=␥1/␥2⬇−2.

Regarding the limit t␧2_Ⰶt

z␧2, we have for the growth

regime that the exponents satisfy the relation −y/a−␤by/a = −1/a−␤b/a=0. We combine this result with the relations obtained above for the exponents z1, z2, ␥1, and ␥2

and we obtain that␥₁−␤z₁= 0 and␥₂−␤z₂= 0. Moreover, we have that the scaling exponent b satisfies the relation b = −1/␤= −z1/␥1⬇−3. Furthermore, we find that a=−1/␥2

= −1/共␤z2兲=−y/␥1⬇2.

With the scaling exponents a and b above we obtain the universal behavior of the curves of ␻. Figure 13shows the collapse of the ␻ curves onto a single and universal curve. This result confirms that the transition from unlimited to lim-ited growth of ␻ presents scaling behavior, as initially pro-posed in Eq.共32兲.

As showed in Ref.关34兴, the average energy of the simpli-fied version of the Fermi-Ulam model decays in the limit of large values of time. This result contrasts with the saturation of ␻ in the limit of large times, as we see in Fig. 11. Al-though the Fermi-Ulam and bouncer models are similar, the mathematical description of these systems is different. In the Fermi-Ulam model the time between two collisions is ⌬t = 2/V, while in the bouncer model ⌬t=2V. Thus these sys-tems present different behaviors for small and large values of the velocity and different behaviors for large values of time. FIG. 11.共Color online兲 Log-log plot of␻ as a function of t␧2_for

B. Scaling of the average collision number

We define Ni共t兲 as the collision number between the

par-ticle and the moving wall until the time t. Thus, regarding an

ensemble of M different initial conditions, we define the av-erage collision number as

N共t兲 = 1 M

兺

_i=1

M

Ni共t兲. 共33兲

Figure14illustrates some curves of the average collision number as a function of t␧2_{. These figures illustrate the}

be-havior of N for different values of ␣rand␧.

We observe in Fig. 14 that the N curves present two growth regimes. The initial growth regime does not depend on␣r. Thus we describe this initial regime as

N⬀ 共t␧2兲␰1␧␩1_. 共34兲 We obtained the growth exponent␰1 from power law

fit-tings to the numerical data for different collision number curves. Thus we obtained the average value␰₁= 0.69⫾0.02. Figure 15共a兲shows the numerical data in an N/共t␧2_兲␰1 _vs ␧ plot. The best fit to these data furnishes␩1= −2.12⫾0.01.

For large values of t␧2 _{we describe the average collision}

number as

(b) (a)

(c) (d)

FIG. 12. 共Color online兲 共a兲 Saturation values of deviation of the velocity as function of ␧. A power law fit gives␥₁= 0.99共1兲. 共b兲 shows the least-squares fit to the numerical values of␻satas a function of the parameter␣r. The procedure furnishes␥2= −0.510共1兲. 共c兲 and 共d兲

show, respectively, the crossover value t_z␧2_{as a function of␧ and␣}

r. The corresponding fits furnish z1= 2.98共2兲 and z2= −1.49共1兲.

FIG. 13. 共Color online兲 Collapse of the␻ curves into a single and universal curve.

N⬀ 共t␧2兲␰2␧␩2␣ r

␨_{. 共35兲}

We obtained the growth exponent␰₂for different curves and the average value is␰2= 0.9999⫾0.0002. With a fixed value

of ␣rwe performed a power law fit to the numerical data in

a plot of N/共t␧2_兲␰2 _vs ␧. The procedure furnishes

␩2= −2.97⫾0.01, as presented in Fig.15共b兲. In the large-t␧2

limit we obtain the exponent␨from an N/共t␧2_兲␰2 _vs␣ rplot

with a fixed value of␧. Figure15共c兲illustrates the procedure that furnishes ␨= 0.492⫾0.005.

The change between the growth regimes defines the quan-tity tz␧2. We describe this crossover as

tz␧2⬀ ␧z1␣r z₂

. 共36兲

The values of the dynamical exponents are z1⬇3 and

z2⬇−3/2, essentially the same as those that we obtained in

the previous section, when we studied the scaling properties of ␻as a function of time.

The above results describe the dependence of the average collision number on t␧2_,␣

r, and ␧. The appropriate scaling

variables of the average collision number are␣rand␧x共t␧2兲.

We will show that x is related to the critical exponents. Near the transition from unlimited to limited energy growth we suppose that the average collision number obeys the relation N„␣r,␧x共t␧2兲… = lN„ld␣r,le␧x共t␧2兲…. 共37兲

Choosing l =␣r

−1/d_{, the above expression becomes}

N„␣r,␧x共t␧2兲… =␣r−1/dN1„␣r−e/d␧

x_共t␧2_{兲…. 共38兲}

Therefore, we rewrite the above expression as N„␣r,␧x共t␧2兲… ⬀␣r

−1/d_„␣ r

−e/d_␧x_共t␧2_兲…␰_{. 共39兲}

In the initial growth regime of N we have that ␰=␰1. In

this limit the quantity N does not depend on ␣r. From Eq.

共34兲 we find −1/d−e␰₁/d=0 and x␰₁=␩₁. In the second growth regime we have that ␰=␰2 and from Eq. 共35兲 we

obtain −1/d−e␰2/d=␨and x␰2=␩2.

The expression共38兲 furnishes the crossover tz␧2⬀␣r e/d_␧−x_.

Thus we have from Eq.共36兲 that z₁= −x and z₂= e/d. Regard-ing both the t␧2_Ⰶt

z␧2and t␧2Ⰷtz␧2regimes, we have from

the above relations that x =␩1/␰1=␩2/␰2= −z1⬇−3,

d = −1/共␰1z2兲=−1/共␨+␰2z2兲⬇1, and e = −1/␰1

=共␨−␰₁z₂兲/共␰₁␰₂z₂兲⬇−3/2.

Finally, Fig. 16 shows that by performing the variable transformations N→N/l and t␧2_→le_t␧2_␧x_{, l =␣}

r

−1/d_{, we}

ob-tain the collapse of the average collision number into a single and universal curve. This result confirms that the scaling hypotheses are correct.

FIG. 14. 共Color online兲 Average collision number curves as functions of t␧2_{for different values of␧ and␣}

r.

(b) (a)

(c)

FIG. 15. 共Color online兲 共a兲 A least-squares fit to the numerical data in an N/共t␧2兲␰1 _vs ␧ plot furnishes␩₁= −2.12⫾0.01. 共b兲 The best fit to the data in an N/共t␧2兲␰2 _vs ␧ plot furnishes ␩2= −2.97⫾0.01. 共c兲 The best fit to the numerical

data in an N/共t␧2兲␰2 _vs ␣_{r plot furnishes} ␨=0.492⫾0.005.

IV. DISCUSSIONS AND CONCLUSIONS

In Sec. II we showed that␻grows with the variable n␧2 according to the relation ␻共n兲⬀共n␧2_兲␤_{, with␤⬇1/2. In the}

analysis of the variable t␧2 _{共Sec. III A兲 we obtained that␻}

grows according to␻共t兲⬀共t␧2_兲␤_{, with␤⬇1/3.}

As we discussed in Sec. III B the average collision num-ber curves present two growth regimes, as we see in Fig.14.

The initial growth regime of N depends on time according to N⬀共t␧2兲␰1. Thus we rewrite this expression as t␧2⬀N1/␰1_{. We} replace this result in Eq. 共29兲, writing ␻⬀N␤/␰1_{. Because}

␰1⬇2/3 and␤⬇1/3 we find, therefore,␻⬀N1/2.

As a short summary, we have studied the bouncer model, considering both the complete and simplified versions. We have described the behavior of the deviation around the av-erage velocity for chaotic orbits using the description in terms of n 共number of collisions with the moving wall兲 and time t. Despite the difference of having multiple collisions with the moving wall before leaving the collision zone in the complete version, a situation that is forbidden to happen in the simplified model, the critical exponents obtained are the same for both complete and simplified versions. The descrip-tion in terms of time also allows a characterizadescrip-tion in terms of a scaling analysis. However, since a particle with low energy can suffer many more collisions with the moving wall as compared to a high-energy particle in the same interval of time, the critical exponents obtained are different from those obtained for the description in terms of n. Therefore, both descriptions allow the characterization of the phase transition from limited to unlimited energy growth when the control parameter␣→1.

ACKNOWLEDGMENTS

The authors acknowledge Dr. Amjed A. H. Mohammed for fruitful discussions. We also acknowledge the support from CNPq, FAPESP, and FUNDUNESP, Brazilian agencies.

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