Boundary crisis and transient in a dissipative relativistic standard map

Physics Letters A 375 (2011) 3365–3369

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Physics Letters A www.elsevier.com/locate/pla

Boundary crisis and transient in a dissipative relativistic standard map Diego F.M. Oliveira a,∗ , Edson D. Leonel b , Marko Robnik a a b

CAMTP, Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000, Maribor, Slovenia Departamento de Estatística, Matemática Aplicada e Computação, UNESP, Univ. Estadual Paulista, Av. 24A, 1515, Bela Vista, 13506-900, Rio Claro, SP, Brazil

a r t i c l e

i n f o

Article history: Received 21 June 2011 Accepted 22 July 2011 Available online 29 July 2011 Communicated by A.R. Bishop Keywords: Chaos Standard map Crisis

a b s t r a c t Some dynamical properties for a problem concerning the acceleration of particles in a wave packet are studied. The model is described in terms of a two-dimensional nonlinear map obtained from a Hamiltonian which describes the motion of a relativistic standard map. The phase space is mixed in the sense that there are regular and chaotic regions coexisting. When dissipation is introduced, the property of area preservation is broken and attractors emerge. We have shown that a tiny increase of the dissipation causes a change in the phase space. A chaotic attractor as well as its basin of attraction are destroyed thereby leading the system to experience a boundary crisis. We have characterized such a boundary crisis via a collision of the chaotic attractor with the stable manifold of a saddle fixed point. Once the chaotic attractor is destroyed, a chaotic transient described by a power law with exponent −1 is observed. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The theory of Hamiltonian area preserving maps developed very fast during the last decades including many important works [1,2]. One of the most studied models is the standard map proposed by Chirikov [3,4] in 1969 which describes the motion of the kicked rotator. Since the pioneering paper of 1969 it has been shown that the standard map can be applied in different fields of science [5–10]. Its is well known that the structure of the phase space depends on the individual characteristics of each system. Basically they are classified in three different groups, namely, (i) integrable, (ii) ergodic and (iii) mixed. In case (i) the phase space consists of a set of invariant tori and an applied example is the circular billiard [11], whose integrability comes from the conservation of the angular momentum. In case (ii), the ergodic systems have only one chaotic component filling the entire phase space. Such a structure can be observed, for example in dispersing twodimensional billiards such as the stadium billiard [12] and in the Lorentz gas [13]. Finally, case (iii) corresponds to the majority and is most common between the three. An important characteristic in mixed type systems is that chaotic seas are generally surrounding Kolmogorov–Arnold–Moser (KAM) islands which are confined by a set of invariant spanning curves. Intensive research has been done in such a kind of systems considering both classical as well as the quantum approach [14–24]. However, the introduction of

*

Corresponding author. E-mail addresses: [email protected] (D.F.M. Oliveira), [email protected] (E.D. Leonel), [email protected] (M. Robnik). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.07.045

dissipation into the model changes completely the structure of the phase space in the sense that elliptical fixed points may turn into sinks (attracting fixed points) and chaotic seas may be replaced by chaotic attractors. It is important to mention that much attention has been devoted to dissipative systems and an extensive research has been done to explain phenomena present in different fields of science including optics [25], fluid dynamics [26] and nanotechnology [27]. Recently, dissipation, namely, collisional dissipation [28] and in-flight dissipation [29], has been introduced in time dependent billiards and it has been shown that it works well as a mechanism of suppression of the unlimited energy growth also called Fermi acceleration [30]. Considering the problem of a particle in the electric field of wave packet [31], it has been shown that if the motion of the particle is such that the Newton equations can be applied, the energy of the particle cannot grow unlimited due to the existence of invariant spanning curves limiting the size of the chaotic sea. However, the position of the first invariant spanning curve is very sensitive and it changes as a power of the control parameter with exponent 2/3, as discussed in Ref. [32]. Additionally, it has been shown that the standard deviation of the kinetic energy for an ensemble of particles grows as a power law for short time and after a crossover it saturates for long enough time [32]. Such a behavior, close to the transition from integrability to non-integrability, can be described using scaling arguments with very well defined exponents [33]. If the acceleration of particles is sufficiently large the Newton equations cannot be applied and relativistic equations must be taken into account. In this Letter we revisit the problem concerning the acceleration of particles considering the relativistic motion, also seeking to understand and describe the consequences of dissipation in the

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model. We show that the phase space is mixed and the dynamics of the system can be described in terms of a two-dimensional nonlinear map. However the introduction of dissipation changes the structure of the phase space and attractors are observed, namely, attracting fixed points and chaotic attractor each of them with their own basin of attraction. Then, increasing the strength of the dissipation the edge of the basin of attraction of the attracting fixed points and of the chaotic attractor (which corresponds to the stable and unstable manifolds of a saddle fixed point) touch each other (they could even cross infinitely many times) and as a result the chaotic attractor as well as its basin of attraction is destroyed. This destruction is called boundary crisis [34,35]. The Letter is organized as follows. In Section 2 we describe the two-dimensional map that describes the dynamics of the system and discuss our numerical results. Conclusions are drawn in Section 3. 2. The model and numerical results

∞ 

E n sin(kn x − ωn t ),

(1)

n=−∞

where E n is the amplitude of the nth Fourier component of the electric field wave. Considering that the wave packet has a broad spectrum such that one can assume E n = E 0 , kn = k0 and ωn = nω , then we get

E (x, t ) = E 0 sin(k0 x)

∞ 

cos(nωt ),

(2)

n=−∞

and therefore using the Fourier decomposition of the periodic Dirac delta function we have

E (x, t ) = E 0 T sin(k0 x)

∞ 

δ(t − nT ),

(3)

where T = 2π /ω . Assuming that the motion of an electron with rest mass m0 and charge −e in an electric field given by Eq. (3) can be described by the relativistic Hamiltonian



p 2 c 2 + m20 c 4 −

e E0 T k0

cos(k0 x)

∞ 

δ(t − nT ), (4)

n=−∞

where c is the speed  of light and the relativistic momentum is given by p = m0 v / 1 − ( v /c )2 . Thus between the delta kicks we have free motion of the particle and therefore the system of the two first order ordinary differential equations for x˙ and p˙ emerging from Eq. (4) are given by

x˙ = 

pc 2

(5)

,

p 2 c 2 + m20 c 4

p˙ = −e E 0 T sin(k0 x)

∞ 

S:

I n+1 = (1 − δ) I n + K sin(θn ), θn+1 = θn + √ I n+1 2 ,

(7)

1+(β I n+1 )

where K = 2π e E 0 k0 /m0 ω2 is the control parameter which controls the transition from integrability (K = 0) to non-integrability (K = 0). In the limit of β → 0 the relativistic standard map is reduced to the Newtonian standard map and δ ∈ [0, 1] is the dissipation parameter. In the ultra-relativistic limit β → ∞ or I n → ∞ we observe trend towards integrability, which is easily seen in Eq. (7). If δ = 0 all the results for the Hamiltonian area-preserving relativistic standard map are recovered. From the map S (see Eq. (7)), the Jacobian matrix, J , is defined as

J=

 ∂ I n +1

∂ In ∂θn+1 ∂ In

∂ I n +1 ∂θn ∂θn+1 ∂θn



(8)

,

∂ I n +1 = 1 − δ, ∂ In ∂ I n +1 = K cos(θn ), ∂θn   ∂ I n +1 β 2 I n +1 ∂θn+1 1  , =1+ − 3 ∂ In ∂ In 1 + (β I n+1 )2 [1 + (β I n+1 )2 ] 2   ∂θn+1 ∂ I n +1 β 2 I n +1 1  . = − 3 ∂θn ∂θn 1 + (β I n+1 )2 [1 + (β I n+1 )2 ] 2

δ(t − nT ).

(6)

n=−∞

Defining θ = k0 x and introducing the auxiliary variables β = ω/kc, I = k0 p /m0 ω and assuming that ( I n , θn ) are the values of the variable immediately after the nth kick, ( I n+1 , θn+1 ) represent their values after the (n + 1)th kick, and introducing the dissipation

(9)

After some algebra one can show that the area preservation is broken for δ = 0 since the determinant of the Jacobian matrix is det( J ) = 1 − δ . It is known that the Lyapunov exponents are an important tool to identify whether the model has chaotic regions or not. As discussed in [40], the Lyapunov exponents are defined as

λ j = lim

n=−∞

H (x, p , t ) =



with coefficients given by

Let us consider the model of a particle in a wave packet, as introduced by Zaslavsky et al. [31,36,37], where the electric field E (x, t ) written as

E (x, t ) =

parameter δ , the dissipative relativistic standard map1 [38,39] is written as

n→∞

1 n

ln |Λ j |,

j = 1, 2,

(10)

n

where Λ j are the eigenvalues of M = i =1 J i ( I i , θi ) and J i is the Jacobian matrix evaluated over the orbit ( I i , θi ). If at least one of the λ j is positive then the orbit is classified as chaotic. Additionally, in conservative systems λ1 + λ2 = 0 while in dissipative systems λ1 + λ2 < 0, which is a consequence of the Liouville’s theorem. The phase space for the mapping (7) is shown in Fig. 1(a). For the same control parameter used in Fig. 1(a) K = 3.5 and β = 0.2, we have also evaluated numerically the positive Lyapunov exponent as shown in Fig. 1(b). The average of the positive Lyapunov exponent for the ensemble of the 10 time series gives λ¯ = 0.577 ± 0.001 where the value 0.001 corresponds to the standard deviation of the ten samples. As can be seen in Fig. 1(b) the Lyapunov exponents fluctuate a little at the regime of short n and experience a small decay. However after few iterations the positive Lyapunov exponents tend towards a regime of convergence marked by a constant plateau. The short deviation is due to the fact that the particle has been confined into a sticky region for about 3.4 × 105 iterations [green/lowest curve in Fig. 1(b)]. When dissipation is taken into account the mixed structure of the phase space is changed and an elliptical fixed point, generally surrounded by KAM islands, turns into a sink and regions of the chaotic sea may be replaced by chaotic attractors. A phenomenon that may occur in dissipative systems is called a boundary crisis [41–43]. The main characteristic is the sudden destruction of the

1

For aesthetic reasons we have shifted θ → θ + π .

D.F.M. Oliveira et al. / Physics Letters A 375 (2011) 3365–3369

Fig. 1. (Color online.) (a) Phase space generated from Eq. (7) for the control parameters K = 3.5 and β = 0.2; (b) Convergence of the positive Lyapunov exponent for the same parameters used in (a).

chaotic attractor and its basin of attraction via a crossing of a stable and an unstable manifolds of the same saddle point. In order to characterize such an event one needs to find the expressions for the fixed points. They are obtained by solving I n+1 = I n and θn+1 = θn + 2mπ . The solutions of these equations give us the saddle fixed points as

I=



2mπ 1 − (2mπ β)2

,

θ (1) = arcsin

Iδ K

 ,

(11)

where m = 0, ±1, ±2, ±3, . . . , while the solution for the attracting fixed points are: (i) if m > 0, θ (2) = π − θ (1) , and (ii) if m < 0, θ (2) = π + θ (1) . Observe that the fixed points are real only if β < (2mπ )−1 . If β = (2mπ )−1 the fixed points become infinity and this case has been considered by Chernikov et al. [36]. A saddle point has the corresponding stable and unstable manifolds. The unstable manifolds are obtained by orbits heading directly to the attracting fixed point or the chaotic attractor and they are obtained just via the iteration of the mapping S with the appropriated initial conditions. On the other hand, the stable manifolds consist basically in orbits heading to the saddle point and the construction of the stable manifolds can be, depending on the model, slightly more complicated than the unstable manifolds. The complication comes from the fact that the inverse of the mapping S has to be obtained and, sometimes, it cannot be obtained analytically/explicitly. Fortunately, the mapping describing the dynamics of a dissipative relativistic standard map is simple enough that allows us to obtain analytically the corresponding inverse of S which we call S −1 . The basic procedure is that S −1 ( I n+1 , θn+1 ) = ( I n , θn ), thus the map S −1 is given by

S

−1

:

⎧ ⎨ In =

⎩ θn =

I n+1 − K sin(θn ) , (1−δ) I n +1 θn+1 − √ . 1+(β I n+1 )2

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Fig. 2(a) that the two branches of the unstable manifolds evolve as follows: the upward branch generates the attracting fixed point (red) while the downward branch creates the chaotic attractor (green).2 On the other hand, the two branches of the stable manifold (obtained via the iteration of the inverse mapping S −1 ) generate the basin boundaries for both the chaotic and fixed point attractor and they are shown in Fig. 2(a), (b) in blue and brown. After a tiny increase of the dissipation parameter, reducing the value of δ , the stable manifold touches3 the unstable manifolds implying the sudden destruction of the chaotic attractor as well as of its basin of attraction. We also emphasize that before the crisis and for the combination of control parameters used in Fig. 2(a) (see details in the caption of that figure), there are three different attractors, namely: (i) two of them are attracting fixed points related to m = 1 and m = −1 and (ii) a chaotic attractor [see Fig. 3(a)]. Thus we have three different basins of attraction, as shown in Fig. 3(b). The procedure used to obtain the basin of attraction for both the chaotic and attracting fixed points consist in iterating a grid of initial conditions in the plane I × θ and observing their asymptotic behavior. We have used a range for the initial I as I ∈ [−500, 500] and θ ∈ [0, 2π ]. Both ranges of I and θ were divided in 1500 parts each, leading to a total of 2.25 × 106 different initial conditions and for the values of the control parameters considered, each combination of ( I , θ) was iterated up to 105 times. It is important to stress that, once the chaotic attractor is destroyed due to the crossing of the two manifolds (unstable and stable), a chaotic transient is observed. This transient is a characteristic time that the particle spends until it finds a way following to one of the two attracting fixed points that survive the boundary crisis. Such a transient can described by a power law of the type n = μρ , (13) t

where μ = δ − δc , with δ < δc . For the combination of control parameters considered, namely, K = 8 and β = 0.15 we found the critical value of the dissipation parameter δ is δc = 0.27901052 . . . . The average transient is the time that an ensemble of initial conditions spends until being attracted by one of the two attracting fixed points, and it is defined by

n¯t =

B 1

B

nti ,

(14)

i =1

where the index i denotes a member of an ensemble of initial conditions, and B is the number of different initial conditions. We considered B = 2 × 104 in our simulations and n¯t was obtained from two different procedures namely: (i) the variable I 0 was fixed as I 0 = 10−2 and θ0 was randomly distributed on the interval [0, 2π ], which produced an exponent ρ = −1.091(9) as can be seen in Fig. 4; (ii) all the initial conditions were randomly chosen along the chaotic attractor just before the crisis (δ = 10−8 ), leading to an exponent ρ = −1.092(8). It is important to say that a boundary crisis was also observed in the one-dimensional Fermi–Ulam model, which consists of a classical particle confined to bounce between two rigid walls. One of them is assumed to be fixed and the other one is periodically moving in time. The dissipation is introduced upon collisions of the particle with the two walls and are controlled by two damping coefficients. The relevant control parameter is the relative amplitude of oscillation of the moving wall, namely . In the absence of dissipation, the phase space is mixed containing either KAM islands, chaotic seas and invariant spanning curves. The position of

(12)

Let us now discuss the behavior of the stable and unstable manifolds of the saddle point for m = ±1. It is easy to see in

2 For the fixed point m = −1 the opposite happen, namely, the two branches of the unstable manifolds evolves as follow: the upward branch generate the chaotic attractor (green) while the downward branch creates the attracting fixed point (red). 3 In fairness it can be proved that there happen infinitely many crossings.

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Fig. 2. (Color online.) Characterization of a boundary crisis. The control parameters used were K = 8 and β = 0.15 and: (a) δ = 0.3 (just before the crisis); (b) δ = 0.278 (immediately after the crisis). The unstable manifolds of the saddle fixed points are shown in red and green, while the stable ones are shown in blue and gray.

Fig. 3. (Color online.) (a) Attracting fixed points for m = 1 red and m = −1 in green and a chaotic attractor. (b) Their corresponding basin of attraction. The control parameters used in (a) and (b) were K = 8, β = 0.15 and δ = 0.3. The region in black is the basin of attraction of the chaotic attractor; the red color is the basin for the attracting fixed point for m = 1; in green color is the basin of attraction for the m = −1 attracting fixed point.

two models behaves quite differently being characterized by the exponent −2 for the Fermi–Ulam and −1 for the relativistic standard map.

3. Conclusion

Fig. 4. (Color online.) Plot of nt vs. μ for the parameters K = 8 and β = 0.15. A power law fitting gives us ρ = −1.091(9).

the first invariant spanning curves, as discussed in Ref. [33] scales with 1/2 . For the Fermi–Ulam model and immediately after the crisis, the transient was characterized as a power law with exponent ∼ = −2 (see Ref. [42]). In spite of the fact that the two models have the same nonlinearity, namely a sine function, and both exhibit a boundary crisis, the chaotic transient after the crisis for the

We have studied the problem of a relativistic charged particle in the electric field of a wave packet. The model is described by a two-dimensional nonlinear map, which is a relativistic generalization of the standard map. For the conservative dynamics, we have shown that the phase space is mixed in the sense that KAM islands and a set of invariant spanning curves are observed coexisting with ¯ = 0.577(1). When dissipaa chaotic sea with Lyapunov exponent λ tion is taken into account, the property of area preservation is no longer observed and the system exhibits attractors including periodic and/or chaotic attractors. We have found the expressions for the saddle fixed points and constructed their corresponding unstable manifold (obtained by iterating the map S) and the stable manifolds (obtained by iterating the inverse map S −1 ). We have shown that increasing the dissipation, the unstable manifold touches the stable manifolds and a drastic change in the dynamics happens. It corresponds to the abrupt destruction of the chaotic attractor and its basin of attraction and this destruction is caused by a boundary

D.F.M. Oliveira et al. / Physics Letters A 375 (2011) 3365–3369

crisis. Moreover, we have shown that after the boundary crisis the average transient follows a power law with the exponent ρ ∼ = −1. Acknowledgements D.F.M.O. acknowledges the financial support by the Slovenian Human Resources Development and Scholarship Fund (Ad futura Foundation). E.D.L. is grateful to FAPESP, CNPq and FUNDUNESP, Brazilian agencies. M.R. acknowledges the financial support by The Slovenian Research Agency (ARRS). References [1] A.J. Lichtenberg, M.A. Lieberman, Regular and Chaotic Dynamics, Appl. Math. Sci., vol. 38, Springer Verlag, New York, 1992. [2] G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford, 2006. [3] B.V. Chirikov, Research concerning the theory of nonlinear resonance and stochasticity, Preprint No. 267, Institute of Nuclear Physics, Novosibirsk, 1969. [4] B.V. Chirikov, Physics Reports 52 (1979) 263. [5] F.M. Izraelev, Physica D 1 (1980) 243. [6] T.H. Stix, Phys. Rev. Lett. 36 (1976) 10. [7] H.L. Cycon, R. Froese, W. Kirsch, B. Simon, Schrödinger Operators, Springer, Berlin, 1987. [8] H.-J. Stöckmann, Quantum Chaos: An Introduction, Cambridge University Press, Cambridge, England, 1999. [9] F. Haake, Quantum Signatures of Chaos, Springer-Verlag, New York, 2001. [10] G. Casati, I. Guarneri, J. Ford, F. Vivaldi, Phys. Rev. A 34 (1986) 1413. [11] S.O. Kamphorst, S.P. de Carvalho, Nonlinearity 12 (1999) 1363. [12] L.A. Bunimovich, Commun. Math. Phys. 65 (1979) 295. [13] D.F.M. Oliveira, J. Vollmer, E.D. Leonel, Physica D 240 (2011) 389. [14] M.V. Berry, Eur. J. Phys. 2 (1981) 91.

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