A Monte Carlo approach for the bouncer model

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A Monte Carlo approach for the bouncer model

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Departamento de Física, UNESP – Univ. Estadual Paulista, Av. 24A, 1515, Bela Vista, 13506-900, Rio Claro, SP, Brazil

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Article history: Received 18 April 2017 Received in revised form 22 September 2017 Accepted 24 September 2017 Available online xxxx Communicated by C.R. Doering

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Gabriel Díaz, Makoto Yoshida, Edson D. Leonel

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Keywords: Discrete mappings Bouncer model Monte Carlo simulation

A Monte Carlo investigation is made in a dissipative bouncer model to describe some statistical properties for chaotic dynamics as a function of the control parameters. The dynamics of the system is described via a two dimensional mapping for the variables velocity of the particle and phase of the moving wall at the instant of the impact. A small stochastic noise is introduced in the time of flight of the particle as an attempt to investigate the evolution of the system without the need to solve transcendental equations. We show that average values along the chaotic dynamics do not strongly depend on the noise size. It allows us to propose a Monte Carlo like simulation that lead to calculate average values for the observables with great accuracy and fast simulations. © 2017 Published by Elsevier B.V.

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1. Introduction

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The bouncer model [1] consists of a classical particle of mass m colliding with a heavy platform moving periodically in time. The model was proposed as an alternative system to investigate Fermi acceleration [2]. The phenomenon is characterized by the unlimited energy growth due to collisions of the particle with the moving wall. The difference from the Fermi–Ulam model is the mechanism that re-injects the particle for a further collision with the moving wall [3]. In the Fermi–Ulam model, which consists of a particle moving inside of two rigid walls – one fixed and the other moving periodically in time – the returning mechanism is a fixed wall. The particle collides with it and is sent back to the moving wall for a further collision. The interval of time from one collision to the other one in the Fermi–Ulam is inversely proportional to the velocity of the particle, i.e. t ∝ V −1 . It means that for small values of velocities, the interval of time between collisions is large, producing many oscillations of the moving wall during the flight. Consequently, the phase of the moving wall in a given collision is uncorrelated with its further collision. This leads the velocity of the moving wall to exhibit behavior typical of random variables, therefore leading the velocity of the particle to grow along the collisions, therefore leading to diffusion in velocity. The phenomenon is interrupted as soon as the velocity of the particle grows because the interval of time between collisions becomes small, hence producing correlations between the phases before and after the

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E-mail addresses: [email protected] (G. Díaz), [email protected] (M. Yoshida), [email protected] (E.D. Leonel). https://doi.org/10.1016/j.physleta.2017.09.039 0375-9601/© 2017 Published by Elsevier B.V.

collisions. It then brings regularity to the phase space leading to the existence of invariant spanning curves and periodic islands [4]. The invariant spanning curves work as barriers do not letting the flow of particles through them and play important rule on the scaling of the chaotic sea [5]. Such curves block the Fermi acceleration produced by the unlimited diffusion of the velocity. In the bouncer model however, the returning mechanism for a further collision is non longer produced by a fixed wall but rather, it comes from the gravitational field. The interval of time between two collisions scales with the velocity as t ∝ V . Hence, for specific ranges of control parameters, correlations are present in the phases before and after the collisions for small values of V while they are destroyed for large V . This is desired for producing Fermi acceleration. Applications of the models are wide and include research such as vibration waves in a nanometric-sized mechanical contact system [6], granular materials [7–11] dynamic stability in human performance [12], mechanical vibrations [13,14] among many others. The conservative versions of the two models are described via two-dimensional, nonlinear and area preserving mappings for the variables, velocity of the particle and phase of the moving wall right after the collision. Although the physics of the systems are simple, the numerical treatment behind them is very complicated. The difficulty comes from the fact that the instant of the collision of particle and moving wall must be obtained by the solution of a transcendental equation. For few collisions the treatment is feasible, however for statistical properties observed for sufficiently long time and large ensembles, the solution of the transcendental equations could be a difficult task to overcome. Our contribution

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in the present paper is exactly to avoid solving the transcendental equations of the system giving the instant of the collisions without considering the so called static wall approximation [15]. We define a time of flight between collisions using a Monte Carlo approach. To do so, we consider a small stochastic noise in the time of flight of the particle to investigate the evolution of the system without the need to solve transcendental equations. Our results confirm that average values along the chaotic dynamics do not depend strongly on the intensity of the noise. The approach leads us to calculate average values for the observables with great accuracy and fast simulations as compared to the complete solution of the model. The advantage of the model is its easy extension to other models, particularly a class of the so called time dependent billiards [16]. The organization of the paper is as follows. In section 2 we discuss the model, present the equations that describe the dynamics and show typical phase space for the conservative dynamics and snapshots of the phase space for the dissipative case. Section 3 is devoted to discuss a stochastic model and how the random variables are picked-up. A Monte Carlo approach is discussed in section 4 where the three types of possible collisions are discussed namely: tangential, sub-tangential and sup-tangential. Our numerical results are discussed in section 5 while conclusions are presented in section 6.

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2. The Chaotic Bouncer Model

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The bouncer model consists of a particle of mass m moving along the vertical under the action of a constant gravitational field g and experiences collisions with a heavy and periodically moving wall. Initial investigations on this problem backs to Pustylnikov [1] and has been studied for many years [17–19] along different approaches. The physics behind the model is reduced to consider two phenomena:

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1. The kinematic equation of motion for the particle and the wall are given with known initial conditions. We are interested to find where and when the particle will meet with the wall in space and time. For this, we need to solve the kinematic equation for t v given by

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x p (t v ) = x f (t v ).

(1)

Such equation means that for a given time t v , the position of the particle, x p and the moving wall, x f , are the same. Both positions depend on the control parameters and initial conditions. In general, the kinematic equation (1) is difficult to solve since it is described by a transcendental equation with possibly more than one mathematical solution for t v . The mathematical solution that has physical interpretation corresponds to the smallest positive t v . Any other solution means the particle has traversed the wall, therefore leading to a nonphysical solution. 2. The velocities of the particle and the wall are given at the instant immediately before the impact. We are interested to know what is the velocity of the particle right after the collision. We consider that the wall is infinitely heavy as compared to the mass of the particle and the collision is inelastic, that means a fractional loss of energy upon collision. From the conservation law we have





V 1 − V f = −γ V 0 − V f ,

(2)

where V 0 and V 1 are the particle’s velocity before and after the collision respectively, V f is the velocity of the wall at the impact and γ is the dissipation parameter that ranges from

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The dynamics of the bouncer model is described by a twodimensional, non-linear and discrete mapping for the two variables velocity of the particle v and time t immediately after the nth collision of the particle with the moving wall. To construct the mapping we consider that the motion of the moving wall is described by x f (t ) = ε cos (ωt ), where ε and ω are parameters that give the amplitude of the wall’s oscillation and its frequency of motion respectively. The particle is uniformly accelerated and its gt 2 , 2

general equation is x p (t ) = x0 + v 0 t − where x0 and v 0 are the particle’s initial position and velocity, g is the acceleration due the gravitational field, which is assumed to be constant. The position and time of the collision is obtained by solving the kinematic equation (1), and the velocity of the particle is calculated using the collision equation (2). Dynamically, the important variables are the particles velocity and the time at the instant of the collision. The mapping gives the evolution of the states from ( v 0 , t 0 ) to ( v 1 , t 1 ), or in general from ( v n , tn ) to ( v n+1 , tn+1 ). Defining dimensionless 2 and hence more convenient variables ωt → φ , ωg v →  V , ωg ε → 

we have the following map

(3)

φn+1 = (φn + t v ) mod (2π ) ,

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with help of the transcendental equation for the dimensionless t v variable

 (cos (φn + t v ) − cos (φn )) − V n t v +

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V n+1 = −γ ( V n − t v ) − (1 + γ )  sin (φn+1 ) ,

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0 (plastic collision) to 1 (elastic collision). Equation (2) comes from the analysis of the collision in the wall’s Galilean referential frame, where the velocity of the particle changes as V 1 = −γ V 0 at the collision.

t 2v 2

= 0.

(4)

We recognize that  is the nonlinear parameter since it causes the dynamics to be complicated when it is different from 0 and trivial otherwise. Indeed if  = 0 and γ = 1 the system is integrable. For γ < 1, the dynamics leads the particle to reach the state of rest for large enough time. For  = 0 and γ = 1 the phase space can be mixed for specific ranges of  where periodic islands coexist with chaotic seas and invariant spanning curves. For the parameter  larger than a critical c , the invariant spanning curves are destroyed and diffusion in velocity is observed leading to Fermi acceleration [2]. The presence of dissipation makes the phase space to shrink in area due to the existence of attractors warranting the suppression of Fermi acceleration. Fig. 1(a) shows the phase space for  = 0.5 and γ = 1. One can observe the mixed dynamics scenario, typical of conservative Hamiltonian systems. It contains stable periodic islands and chaotic seas. In Fig. 1(c) the islands become small and there is almost a single chaotic sea where the velocity diffuses to infinity producing a phenomenon known as Fermi Acceleration [2]. The introduction of dissipation destroys the mixed structure of the phase space. As shown in Fig. 1(b) for  = 0.5 and γ = 0.99, the dense clouds of points suggest the presence of attractors in phase space producing a very complicated dynamics. Finally, Fig. 1(d) shows only one chaotic attractor that does not diverges to infinity along the velocity axis, therefore suppresses the Fermi Acceleration [20,21].

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3. A Stochastic-Chaotic Bouncer Model

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The bouncer model described by Eq. (3) presents the behavior of an ideal particle colliding with a moving platform. However in real experiments, the deterministic equations are not enough to describe the dynamics since there are more variables and physical effects to take into account that influences the behavior of the

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Fig. 3. Evolution of V , mean velocity over an ensemble of orbits, as a function of n with σ ranging from 1 × 10−10 to 1 × 10−1 . All curves show a convergence to a stationary state. We notice that for σ > 1 × 10−5 the stationary state is almost independent of σ . The control parameters are  = 100 and γ = 0.95.

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Fig. 1. Representation of the phase space for the non dissipative bouncer model shown in (a) and (c). The control parameters used were: (a)  = 0.5 and γ = 1; (c)  = 10 and γ = 1. In (b) and (d) are a snapshot of the orbits in the phase plane for the dissipative dynamics. The parameters used were: (b)  = 0.5 and γ = 0.99; (d)  = 10 and γ = 0.99.

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Fig. 4. Plot of the final value of V and V rms as function of σ for the control parameters  = 100 and γ = 0.95. We notice for σ > 1 × 10−5 there is a constant plateau.

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Fig. 2. The value of time of flight in the Stochastic-Chaotic model, t v ( S −C ) , is a random number of a Gaussian distribution with mean t v (C ) , solution of equation (4), and standard deviation σ .

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particle. To mimic such extra phenomena we consider that the particle suffers a stochastic deviation from its deterministic trajectory. In the present investigation we added a small stochastic noise in the particle’s time of flight. It is written to be of Gaussian type  N μ, σ 2 . We considered in every successive collision the time of flight obtained through Eq. (4) as a mean time of flight, μ = t v (C ) , and selected an actual t v ( S −C ) , for the Stochastic-Chaotic model, as a random number taken from a Normal distribution along that  mean with a specific standard deviation σ , t v ( S −C ) ∼ N t v (C ) , σ 2 . See Fig. 2, in the case of t v ( S −C ) ≤ 0, another value was drawn from the same distribution until t v ( S −C ) > 0. Fig. 3 shows different values of σ , the average velocity over an ensemble of orbits as a function of the collision number n. We can see that all ensembles converge to a stationary state with different rates. However the values for the average velocity and crossover

time depend on the value of σ . For larger σ the convergence to stationary state is achieved faster. The convergence value for the velocity appears to grow with σ until it reaches a final value that no longer depends on σ . The dependence of both the final value of the average velocity and the root mean square velocity as a function of the parameter σ is better shown in Fig. 4. At first the value stays almost constant until it reaches a crossover σ , where the average grows rapidly until it reaches a plateau of convergence where the dependence with σ is not observed anymore. We point out that the Fig. 4 is in logarithmic scale.

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4. A Monte-Carlo approach

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Due the stochastic nature of particle-wall collisions and the large ensemble of elapsed particle time of flight, a Monte Carlo procedure can be proposed to pick up time intervals at random, this made to estimate the corresponding points along the phase space and to extend the movement from this point as if the system has been evolved from the past exactly as described by the equations of motion. Taking this into consideration we developed a type of Monte Carlo simulation to search for averages in the dynamics that are less time-consuming as compared to the traditional study of solving transcendental equations. To do this we proceeded in

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Fig. 6. Acceptance/rejection probability distribution as function of G (t v ) for the Chaotic simulation. In an algorithm in double precision machine usually β = 1 × 10−14 .

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In a numerical simulation the probability to have a tangential collision is almost zero. So we needed to worry about the other kinds of collisions, if it was known that the collision would be sub-tangential, with the interpolation analysis described before, we picked up a random number for the time of flight between zero and the time of flight for a tangential collision. If the collision was known to be sup-tangential then we calculated the number of oscillations n that the wall has had until the next collision. To do this we used the fact that the trajectory of the particle was described by

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Fig. 5. Classification of the trajectories for the bouncer model. Each kind of trajectory has its proper domain for piking randomly the time of flight shown in (a). Zoom of the sub-tangential domain in (b).

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the following way. After experiencing a collision with the moving wall, the particle’s trajectory can have a further collision with the moving platform in one of the following three different forms: (i) a tangential way, the particle velocity was the same as the wall velocity when colliding; (ii) a sub-tangential way, the velocity of the particle after the first collision was smaller than the velocity necessary for second collision to be a tangential one; and (iii) a sup-tangential way, the velocity of the particle was larger than the one needed for a tangential collision, see Fig. 5. After the definition of the type of collision, we found the domain of time where a further collision could happen. Then a random number is drawn from such a domain of time to obtain the time of flight for the collision. Since it was not an actual calculation, Eq. (4) was not accomplished. However we were able to find how far from zero was the value, or how far were the particle and the wall for the proposed time of flight. The idea was to accept the collision with a probability that grows as a function of the distance from the particle to the wall goes to zero. If the time of flight is rejected we proposed another time of flight and proceeded with the draw again. The condition for the tangential velocity for any phase was calculated considering that, before the tangential collision, both particle and wall had the same velocity. Then we reversed the time in the dynamics and found which initial conditions led to such a tangential collision. Knowing the tangential velocity condition for some phases, then for any phase, the velocity that lead to a tangential collision, could be calculated via interpolation. With that in mind we had to do a few of actual simulations and store the data. It will be later used to know, via interpolation, if the particle’s velocity for a given phase would left to a sub-tangential, tangential or sup-tangential collision.

x f = x0 + V t −

t

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(5)

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where x0 =  cos (φ), the final position was x f =  , see Fig. 5 where the specific sub-domain was between two maximum of position and the time elapsed to reach that position was t = 2nπ − φ . Recalling that φ and V gave the phase and velocity of the current collision. So the final inequality that related the number of oscillations, n, in terms of φ and V was

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2nπ − φ 2

 (cos (φ) − 1) 2 (n + 1) π − φ  (cos (φ) − 1) − −
(6)

The idea was to find an integer n in such way that the given velocity V fulfills the last inequality. Given a value of n, it is possible to define a specific finite domain for the time of flight [2nπ , 2 (n + 1) π ] from where a random number is to be drawn. Once we drew a random time of flight, t v , we used equation (4) to calculate G (t v ). With this value we either accept or reject the proposed time t v . The acceptance/rejection decision was done in a − (G (t v ))

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2β 2 Monte Carlo style. We calculated P = e were β was a parameter introduced to control the acceptance/rejection ratio. Then drew a random number, from a uniform distribution between [0, 1] and asked whether the random number was larger (or smaller) than P . If it was larger, we drew another random time of fight t v , otherwise we accepted the time of flight and the respective collision and proceed once again. The method proposed for the Monte Carlo simulation has some resemblances with the actual method for the Chaotic model. While in the later we assume a new time of flights via particle dynamics and use the acceptance/rejection probability distribution shown in Fig. 6, in the former we consider a new time of flight in the described manner, and use the acceptance/rejection probability dis-

tribution

2 − (G (t v2)) 2β e

, see Fig. 7.

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Model can be understood by means of 1(b) where we can see the existence of various attractors in plane space of C model, this phenomenon is not captured nor by S-C neither M-C.

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Fig. 7. Acceptance/rejection probability distribution as function of G (t v ) for the Monte-Carlo simulation. In the numerical simulation we used β ∝  .

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Table 1 Comparison for the V variable regarding the Chaotic model (C), the Stochastic-Chaotic model (S-C) and the Monte Carlo simulation (M-C). The bracketed quantity is the standard deviation.

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γ

V C

V S −C

V M −C

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0.999 0.999 0.999 0.95 0.95 0.95

407.50(8) 3959.7(1) 39608(9) 11.621(6) 69.156(4) 640.97(3)

413.25(7) 4021.0(3) 39492(5) 54.754(6) 545.15(2) 5445.2(1)

405.26(9) 3946.4(5) 39352(3) 52.890(9) 529.22(7) 5295.7(2)

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was independent of the size of the noise perturbation. This phenomenon allowed us to propose a Monte Carlo scheme to simulate a stochastic bouncer model in which simulation was faster and achieved similar results for the averaged V as the ones obtained in the Stochastic-Chaotic model. Moreover for certain parameters both Monte Carlo approach reached almost the same result as the original Chaotic Bouncer Model in a tenth of the simulation time. The approach proposed in this paper can be generalized to many other different dynamical systems, including time dependent billiards.

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Acknowledgements

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References

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5. Numerical results In this section we perform the numerical simulation of the different Bouncer Models, i.e. the Chaotic, the Stochastic-Chaotic and the Monte Carlo approach, described in last sections. We focus in the statistical investigations using averages in the post-transient regime. An observable which is immediate is the average velocity over an ensemble of different initial conditions, hence leading to

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GDI thanks to the Brazilian agency CAPES. EDL thanks FAPESP (2017/14414-2) and CNPq (303707/2015-1) for financial support.

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Based in the well known Bouncer Model we proposed a new Stochastic-Chaotic Bouncer Model that tries to take into account possible experimental unwanted physical effects. This new model has a Stochastic Gaussian noise, with a standard deviation σ in the time of flight. We found that the system achieves a thermal equilibrium for every σ , and for large enough noise, this equilibrium

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6. Conclusions

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V i =

M 1 

M

Vi,

(7)

i =1

where M represents the number of initial conditions of the ensemble. We are interested in the limit i → ∞, the post-transient average value V . Table 1 shows a comparison of the V variable for the Chaotic Bouncer Model (C), the Stochastic-Chaotic Model (S-C), and the Monte Carlo approach (M-C), where the M-C simulation is ten times faster than C simulation. We see that, for certain parameters values, all simulations achieve almost the same numerical values, while for other parameters values just the Stochastic-Chaotic model and the Monte Carlo approach reaches similar results. The deviation between this values and those of the Chaotic Bouncer

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

[19] [20] [21]

Lev Davidovich Pustyl’nikov, Theor. Math. Phys. 57 (1983) 1035. Enrico Fermi, Phys. Rev. 75 (1949) 1169. M.A. Lieberman, A.J. Lichtenberg, Phys. Rev. A 5 (1852) (1972). A.J. Lichtenberg, M.A. Lieberman, Regular and Chaotic Dynamics, Appl. Math. Sci., vol. 38, Springer-Verlag, New York, 1992. E.D. Leonel, P.V.E. McClintock, J.K.L. da Silva, Phys. Rev. Lett. 93 (2004) 014101. N.A. Burnham, A.J. Kulik, G. Gremaud, G.A.D. Briggs, Phys. Rev. Lett. 74 (1995) 5092. P. Dainese, P.St.J. Russell, N. Joly, J.C. Knight, G.S. Wiederhecker, H.L. Fragnito, V. Laude, A. Khelif, Nat. Phys. 2 (2006) 388. M. Scheel, R. Seemann, M. Brinkmann, M. Di Michiel, A. Sheppard, B. Breidenbach, S. Herminghaus, Nat. Mater. 7 (2008) 189. M.K. Müller, S. Ludinga, Thorsten Pöschel, Chem. Phys. 375 (2010) 600. P. Müller, M. Heckel, A. Sack, T. Pöschel, Phys. Rev. Lett. 110 (2013) 254301. F. Pacheco-Vázquez, F. Ludewig, S. Dorbolo, Phys. Rev. Lett. 113 (2014) 118001. D. Sternad, M. Duarte, H. Katsumata, S. Schaal, Phys. Rev. E 63 (2000) 011902. A.C.J. Luo, R.P.S. Han, Nonlinear Dyn. 10 (1) (1996). J.J. Barroso, M.V. Carneiro, E.E.N. Macau, Phys. Rev. E 79 (2009) 026206. A.K. Karlis, P.K. Papachristou, F.K. Diakonos, V. Constantoudis, P. Schmelcher, Phys. Rev. Lett. 97 (2006) 194102. E.D. Leonel, M.V.C. Gallia, L.A. Barreiro, D.M.O. Fregolente, Phys. Rev. E 94 (2016) 062211. Phillip J. Holmes, J. Sound Vib. 84 (173) (1982). John Guckenheimer, Philip J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42, Springer Science & Business Media, 2013. R.M. Everson, Phys. D: Nonlinear Phenom. 19 (1986) 355. Edson D. Leonel, André L.P. Livorati, Phys. A, Stat. Mech. Appl. 387 (2008) 1155. André Luis Prando Livorati, Denis Gouvêa Ladeira, Edson D. Leonel, Phys. Rev. E 78 (2008) 056205.

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