Applied Mathematics and Computation 221 (2013) 32–39

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Applied Mathematics and Computation

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A rescaling of the phase space for Hamiltonian map:Applications on the Kepler map and mappings with divergingangles in the limit of vanishing action

0096-3003/$ - see front matter Crown Copyright � 2013 Published by Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.06.045

⇑ Corresponding author at: Departamento de Física, UNESP – Univ Estadual Paulista, Av. 24A, 1515, 13506-900 Rio Claro, SP, Brazil.E-mail addresses: julianoantonio@sjbv.unesp.br (J.A. de Oliveira), edleonel@rc.unesp.br (E.D. Leonel).

Juliano A. de Oliveira a,b,⇑, Edson D. Leonel a,c

a Departamento de Física, UNESP – Univ Estadual Paulista, Av. 24A, 1515, 13506-900 Rio Claro, SP, Brazilb UNESP – Univ Estadual Paulista, Câmpus São João da Boa Vista, São João da Boa Vista, SP, Brazilc Abdus Salam ICTP, 34100 Trieste, Italy

a r t i c l e i n f o

Keywords:Chaotic seaInvariant KAM curveUniversality classesKepler mapWave packet

a b s t r a c t

A rescale of the phase space for a family of two-dimensional, nonlinear Hamiltonian map-pings was made by using the location of the first invariant Kolmogorov-Arnold-Moser(KAM) curve. Average properties of the phase space are shown to be scaling invariantand with different scaling times. Specific values of the control parameters are used torecover the Kepler map and the mapping that describes a particle in a wave packet forthe relativistic motion. The phase space observed shows a large chaotic sea surroundingperiodic islands and limited by a set of invariant KAM curves whose position of the firstof them depends on the control parameters. The transition from local to global chaos isused to estimate the position of the first invariant KAM curve, leading us to confirm thatthe chaotic sea is scaling invariant. The different scaling times are shown to be dependenton the initial conditions. The universality classes for the Kepler map and mappings withdiverging angles in the limit of vanishing action are defined.

Crown Copyright � 2013 Published by Elsevier Inc. All rights reserved.

1. Introduction

Nonlinear systems have requested a lot of dedication of many scientists along the years. The systems are used to describedifferent problems enconutered in the literature including biology [1], chemistry [2,3], condensed matter physics [4,5], non-linear dynamics [6] and also in billiard problems [7]. In statistical mechanics, the phase transitions are related to abruptchanges in the spatial structure of the system [4,5] due to a variation of a control parameter while in dynamical systemsthe phase transitions are linked to modifications in the structure of the phase space of the system [6,7]. Near the phase tran-sition, the dynamics of the system is mostly described by using a scaling function [8,9] where critical exponents characterisethe dynamics nearby the criticality.

Often nonlinear systems may be described in the form of mappings. Possible applications of mappings can be found in thestudy of channel flows [10,11], waveguide [12], transport properties [13], Fermi acceleration [14–17] and also for the studyof magnetic field lines in toroidal plasma devices with reversed shear (like tokamaks) and many other [18,19].

In this paper, we describe a rescale in the phase space for a family of two-dimensional nonlinear mappings using the po-sition of the first invariant tori (sometimes also called as invariant spanning curve or as invariant KAM curve). Given themapping, we focus our attention in a specific control parameter which recovers the Kepler map and the mapping that de-scribes a particle in a wave packet for the relativistic motion. It may show to be relevant in plasma systems [20], while

J.A. de Oliveira, E.D. Leonel / Applied Mathematics and Computation 221 (2013) 32–39 33

applications of the Kepler map may be found in astronomy [21] and to describes the problem of the hydrogen atom in themicrowave [22] among others. We use a connection with the standard mapping [23] near a transition from local to globalchaos. Once the chaotic sea is rescaled, properties for the average observables (like average action) can also be described bythe same kind of scaling. Moreover, a dependence on the initial conditions on the chaotic sea leads the observables to havedifferent time scaling. Thus, the universality classes for the Kepler map and mappings with diverging angles in the limit ofvanishing action are defined.

The work reported here is organised as follows. In the Section 2 we present the mapping, discuss the variables, the controlparameters used and show applicability to other models known in the literature. The position of the first invariant KAMcurve is estimated by using a connection with the standard mapping near a transition from local to global chaos. The stabilityof the fixed points is studied and we show a rescaling of the phase space. Section 3 is devoted to show a merger of all curvesof the average action onto a single plot after an appropriate rescaling of the axis. Our conclusions and final remarks aredrawn in Section 4.

2. The model

In this paper, we consider a family of two-dimensional Hamiltonian mappings given by [24]

T :

Jnþ1 ¼j Jn � � sinðhnÞ j

hnþ1 ¼ hn þ 1Jcnþ1

� �modð2pÞ

8<: ð1Þ

where � and c are the control parameters. Note that the control parameter � controls a phase transition from integrability(� ¼ 0) to non-integrability (�– 0). Connections with several mappings known in the literature are easily made. To illustratejust a few of them, for the case of c ¼ 1 and considering the transformations J ! V and h! /, one can recover the Fermi-Ulam accelerator model [25]. For the case where J ! c and 2ph! X where c in this transformation represents angular coor-dinate instead of control parameter (as is the case for this letter), one can have the periodically corrugate waveguide [12]. Onthe other hand, for the case of c ¼ 1=2, one can describe the dynamics of a classical particle confined inside an infinitely deepbox of potential containing a periodically moving square well [26,27] or time varying barrier [28]. This value of c describesthe motion of a non-relativistic particle in the field of a wave packet [20], and describes also a map for the classical Morseoscillator driven by time-periodic force [29]. For the case of c ¼ 3=2 and considering the transformations J ! P and h! g,where P corresponds the energy and g is the angle of Jupiter, when the comet passes the perihelion, the map describesthe motion of comets in the Solar system [21]. This value of c also may be used to describe the problem of the highly excitedhydrogen atom in the microwave [22]. For c ¼ 2 and considering the transformation J ! I, where I corresponds the momen-tum and h denotes variations of phase, the map describes a particle in a wave packet for the relativistic motion, it is relevantin plasma systems [20]. For c < 0, depending on the initial conditions and control parameters, unlimited growth for the var-iable J may be observed. Such growth is observed since large values of J imply in a large number of oscillations for the sinefunction. Then, in the regime of very large oscillations, the sine function works more likely a random function yielding in anunlimited growth for J, which is unwanted in this description.

The stability of the fixed points of the mapping (1) for the control parameter c 2 ½2=3;1� was studied in Ref. [24]. The crit-ical exponents also were found using extensive numerical simulations in the study of the deviation of average J for chaoticorbits. Recently an analytical approach for the critical exponents obtained numerically in Ref. [24] was presented using aconnection with standard mapping to estimate the position of the first invariant KAM curve [30]. In this paper, given thenew physical applications of the mapping (1) in astronomy and plasma systems, we concentrate our attention in somedynamical properties for the values of c ¼ 3=2 and c ¼ 2, results so far not published yet.

The phase space generated from iteration of the mapping (1) is shown in Fig. 1.One sees that the phase space is mixed thus containing a set of periodic islands that are surrounded by a large cha-

otic sea that is limited by a set of invariant KAM curves. We used in the construction of Fig. 1 a grid of 9� 14 differentinitial conditions iterated over 103 iterations for the range h 2 ½0;2p� and: (a) J 2 ½10�2;0:21� and (b) J 2 ½10�2;0:33�. Thecontrol parameters used were � ¼ 10�2 and: (a) c ¼ 3=2 and (b) c ¼ 2. It is expected that the size of the chaotic seavaries as the control parameters vary. As for example, when the parameter c rises, one may expect that the positionof the lowest invariant KAM curve in Fig. 1(a) and (b) (green line) rises too; it indeed happens. Similar discussion alsoholds in general for the other parameters too. The connection with the standard mapping is quite obvious. For the stan-dard mapping (see Ref. [23] for specific details), when the control parameter K < 0:9716 . . ., the phase space is mixedand invariant KAM curves confine the chaotic sea into closed regions. On the other hand, when K P 0:9716 . . ., allthe invariant KAM curves are destroyed. Consequently, the chaotic sea spreads over the phase space. In our case, asshown in Fig. 1(a), for J < 0:1935 and Fig. 1(b), for J < 0:2896, the chaotic sea can spread over the phase space (withexception of the periodic islands) from zero up to the first invariant KAM curve. Such region is then correspondentto the globally chaotic region in the formalism of standard mapping. For values J > 0:1935 in Fig. 1(b), andJ > 0:2896 in Fig. 1(b) the periodic and quasi-periodic behaviour dominate the dynamics. One can observe chaos ofcourse, but only for small and confined regions. The region above the first invariant KAM curve corresponds to thelocally chaotic region in the terminology of the standard mapping.

Fig. 1. Phase space generated by the mapping (1) for the control parameters � ¼ 10�2 and (a) c ¼ 3=2 and (b) c ¼ 2. (c) and (d) rescaling phase space by J�

given by Eq. (3) using the same control parameters used in (a) and (b), respectively. We see that after rescaling the J axis, the position of the first invariantKAM curve is pushed to be close to 0:9716 ffi 1.

Table 1Fixed points and their classification for c ¼ 3=2 and c ¼ 2.

c Fixed point 1 Fixed point 2 Elliptic Hyperbolic

32 0; 1

ð2pmÞ

� �2=3� �

p; 1ð2pmÞ

� �2=3� �

0 < m < 24=3

3�p5=3

� �3=5m > 24=3

3�p5=3

� �3=5

20; 1

ð2pmÞ

� �1=2� �

p; 1ð2pmÞ

� �1=2� �

0 < m < 21=2

2�p3=2

� �2=3m > 21=2

2�p3=2

� �2=3

34 J.A. de Oliveira, E.D. Leonel / Applied Mathematics and Computation 221 (2013) 32–39

The fixed points of the mapping (1) can be obtained by matching the following conditions: Jnþ1 ¼ Jn ¼ J andhnþ1 ¼ hn ¼ hþm, where m ¼ 1;2;3; . . .. Given the physical applications of the mapping (1) for c ¼ 3=2 and c ¼ 2, we ob-tained the respective fixed points in Table 1. We stress that fixed point 1, as labelled in Table 1, is always hyperbolic, whilefixed point 2 can be elliptic or hyperbolic, according to the combination of control parameters. Some of the elliptic fixedpoints are shown in Fig. 1 as (red) circles while some of the hyperbolic are identified as (blue) squares.

In order to estimate analytically the position of the first invariant KAM curve we suppose that around it J can be written as

Jnþ1 ffi J� þ DJnþ1; ð2Þ

where J� is a typical value along of the invariant tori and DJnþ1 is a small perturbation of Jnþ1. Using Eq. (2) and making similaralgebraic procedure in Eq. (1) as shown in Refs. [30,31], we obtain the localisation of the invariant KAM curve given by

J� ffi �c0:9716 . . .

� �1=ð1þcÞ

: ð3Þ

From Eq. (3) we conclude that the size of the chaotic sea is proportional to �cð Þ1=ð1þcÞ. This result recover the localisation ofthe invariant KAM curve for the problems discussed in Refs. [21,23,32].

A plot of Eq. (3) together with the numerical value obtained for J� is shown in Fig. 2 as a function of c. The (red) bulletsfurnish the numerical data of the average position of the first invariant KAM curve while the (black) line corresponds to theanalytical result given by Eq. (3). It is clear that the position of the first invariant KAM curve exhibits a growing behaviour asa function of c. Moreover, the numerical data gratefully approaches the theoretical result.

Fig. 2. Plot of J� vs. c for the parameter � ¼ 10�3. The (red) bullets furnish the numerical data of the average position of the first invariant KAM curve, whilethe (black) line indicates the analytical data given by Eq. (3).

Fig. 3. Plot of different curves x as a function of n for three different values of � and control parameters: (a) c ¼ 3=2 and (b) c ¼ 2. (c) and (d) show theoverlaps of the curves shown in (a) and (b) onto a single and universal plot.

J.A. de Oliveira, E.D. Leonel / Applied Mathematics and Computation 221 (2013) 32–39 35

Fig. 1 shows a plot of the phase space obtained from iteration of mapping (1) for the axis J rescaled by J� given by Eq. (3).The control parameters used were � ¼ 10�2 and: (c) c ¼ 3=2 and (d) c ¼ 2. We see that after rescaling the J axis, the positionof the first invariant KAM curve is pushed to be close to 0:9716 ffi 1. The procedure was also used to other values of c, leadingto similar rescaling, therefore confirming that the procedure is valid for other values of c.

3. Scaling properties

In this section we concentrate to discuss some scaling properties present in the chaotic sea. The average quantity we areexploring is the behaviour of the deviation of the average J for chaotic orbits, which we denote it as x. In fairness, the behav-iour of x shows the same properties of the average J. It is defined as

Fig. 4.respect

36 J.A. de Oliveira, E.D. Leonel / Applied Mathematics and Computation 221 (2013) 32–39

xðn; �Þ ¼ 1M

XM

i¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiJ2

i ðn; �Þ � Ji2ðn; �Þ

q; ð4Þ

where M corresponds to an ensemble of different initial conditions hi 2 ð0;2pÞ randomly chosen for a fixed J0 ¼ 10�3� and �Ji

is given by

Jiðn; �Þ ¼1n

Xn

j¼1

Jj;i: ð5Þ

Fig. 3 shows the behaviour of x vs. n for different values of � (as labelled in the figure) using very long simulations of5� 108 iterations. The ensemble average used was M ¼ 5� 103. Fig. 3(a) was obtained for c ¼ 3=2, while Fig. 3(b) we con-sidered c ¼ 2.

Let us now discuss the behaviour observed in Fig. 3(a) and (b). We see that the curves start growing for small n and then,after reaching a critical crossover iteration number, nx, they bend towards a regime of convergence. Based on the behaviourseen in Fig. 3(a) and (b) we can suppose that:

(i) For n� nx;x grows according to a power law of the type

x / n�2 b

; ð6Þ

where b is a critical exponent;(ii) For large n, say n� nx, the behaviour of x is

x / �a; ð7Þ

where a is a critical exponent;(iii) The crossover nx, that characterises the transition of the growing regime for the saturation is

nx�2 / �z; ð8Þ

where z is a critical exponent.

Considering these three scaling hypotheses, we propose that x might be described in terms of a scaling function of thetype

Plots of: (a) and (c) nx vs. � for the control parameter c ¼ 3=2 and c ¼ 2, respectively, while (b) and (d) show xsat vs. � using c ¼ 3=2 and c ¼ 2,ively.

Table 2Compar

c

3=22

J.A. de Oliveira, E.D. Leonel / Applied Mathematics and Computation 221 (2013) 32–39 37

xðn�2; �Þ ¼ lx la1 n�2; lb1�� �

; ð9Þ

where l is a scaling factor and a1 and b1 are scaling exponents. They have to be related with the exponents b;a and z. Usingalgebraic manipulation we obtain the following relation to the exponents: (i) b ¼ �1=a1, and (ii) a ¼ �1=b1, and (iii)z ¼ a1=b1. The critical exponents a and z can be obtained from extensive numerical simulation. Firstly, fitting the initial re-gime of growth, we obtain that the critical b ffi 0:5. The other critical exponents are obtained from specific plots. So it isshown in Fig. 4(a) and (c) the behaviour of nx vs. �, while Fig. 4(b) and (d) plot the behaviour of xsat vs. � for the controlparameters c ¼ 3=2 and c ¼ 2, respectively. We show the values of critical exponents in Table 2 as labelled in Fig. 4.

Now we suppose that J is dependent on the position of the lowest invariant KAM curve, as foreseen by Eq. (3), which leadsus to conclude that

a ffi 1ð1þ cÞ : ð10Þ

Using scaling arguments (similar as done in Ref. [24]), one can show that z ¼ a=b� 2, therefore leading to

z ffi 1bð1þ cÞ � 2� �

: ð11Þ

The analytical procedure to obtain the critical exponents shown in Eqs. (10) and (11) was developed in Ref. [30] for the rangeof c 2 ½0;1�. In the present manuscript given the new physical applications for the mapping (1) using the control parametersc ¼ 3=2 and c ¼ 2, we confirm the analytical procedure is valid to obtain the critical exponents. It is shown in Table 2.

Using a proper rescaling of the axis obtained from Eq. (9) with the critical exponents shown in Table 2, all the curves of xgenerated for control parameters c ¼ 3=2 and c ¼ 2 overlap each other onto a single and universal plot, as shown in Fig. 3(c)and (d).

Now we suppose that J is dependent of the initial conditions. Fig. 5 shows the behaviour of J vs. n for different values of �(as labelled in the figure) using 5� 108 iterations of the mapping. The ensemble average used was M ¼ 5� 103. Fig. 5(a) wasobtained for c ¼ 3=2, while Fig. 5(b) we considered c ¼ 2.

We see from Fig. 5(a) and (b) that the curves of J depend sensible on the initial J0. For J0 ffi 0 the curves start growing untilbend towards a regime of saturation. The changing from growth to the saturation is marked by a crossover n0x. On the otherhand, when J0 � 0, the curves present a second plateau therefore having a second crossover, which we call as n00x < n0x. WhenJ0 ¼ 10�6, we have n00x 0, and we see in Fig. 5(a) and (b) that the curves for � ¼ 10�3 and � ¼ 10�4 show only two regimes: (i)a growth in power law for n00x � n0x and (ii) the saturation regime for n� n0x. For this case we suppose that for n00x � n0x; J growsaccording to a power law of the type

J / ðn�2Þb; ð12Þ

(iii) for n� n0x, the J reaches a saturation regime that is described as

Jsat / �a; ð13Þ

and (iv) that the crossover iteration number n0x marking the approach to saturation is

n0x / �z: ð14Þ

b, a and z are critical exponents. With these initial suppositions, we can now describe J in terms of a scaling function of thetype

Jðn�2; �; J0Þ ¼ lJ lan�2; lb�; lcJ0

� �; ð15Þ

where l is a scaling factor and a; b; c are scaling exponents must be related to the critical exponents b;a and z. Using algebraicmanipulation similar those shown in Ref. [25], we obtain the following relation to the exponents: (i) b ¼ �1=a, (ii) l ¼ ��1=b,

where �1=b ¼ a and (iii) c ¼ �1=ð2aÞ. When J0 ¼ 10�3 and � ¼ 10�4 in Fig. 5 and J0 ¼ 31=2 � 10�3 and � ¼ 3� 10�4 in Fig. 5(a)and (b), we have n00x < n0x, and we see three regimes for such a curve. For n00x � n0x, J is basically constant. When n00x < n < n0x, the

curve grows and begins to follow the curve of J0 ¼ 10�6 and the same �. In this interval of n, we have a growth with a smallereffective exponent b. Finally, for n� n0x, we have the saturation regime. Since we have obtained the critical exponents asshown in Table 2 we show in Fig. 5(c) and (d) a merger of three different curves J generated for different values of controlparameters � and J0 into a single and universal plot. If Keff is written in terms of scaled variables, we have

ison of the critical exponent 1=ð1þ cÞ and a;1=½bð1þ cÞ� � 2 and z. The range considered was � 2 10�4;10�2h i

for c ¼ 3=2 and c ¼ 2.

1ð1þcÞ a b 1=½bð1þ cÞ� � 2 z

0:4 2=5 0:419ð4Þ 0:495ð4Þ �1:191ð3Þ �1:156ð8Þ0:333 . . . 1=3 0:348ð2Þ 0:495ð5Þ �1:326ð6Þ �1:297ð6Þ

Fig. 5. (a) Plot of different curves of J as a function of n for three different values of � and control parameters: (a) c ¼ 3=2 and (b) c ¼ 2. (c) and (d) show theoverlap of the curves plotted in (a) and (b) onto a single and universal plot.

38 J.A. de Oliveira, E.D. Leonel / Applied Mathematics and Computation 221 (2013) 32–39

Keff ffi2pcðlb�ÞðlcJ0Þ

ð1þcÞ ¼2pc�lb

Jð1þcÞ0 lcð1þcÞ : ð16Þ

From Eq. (16) we observe that the maximum initial J0 inside the chaotic sea is J0;max 2p�cð Þ1=ð1þcÞ implying that the secondtime scale has a maximum value of n00x 2pcn0x. This result confirms the scale invariance of the phase space presented in pre-vious section. Additionally, an initial J� > J0 � 0 produces an addition time scaling which, when taken into account, can beused to overlap all curves onto a single plot.

Now we discuss on the universality classes. The critical exponents of the mapping (1) for the control parameterc 2 ½2=3;1� were obtained in Ref. [24] using extensive numerical simulations in the study of the deviation of average J forchaotic orbits. Recently an analytical approach to obtain the critical exponents was developed using a connection withstandard mapping to estimate the position of the first invariant KAM curve [30]. It was observed that the critical exponentsa ¼ 0:518ð4Þ, b ¼ 0:495ð6Þ and z ¼ �1 obtained for c ¼ 1 recovered the same critical exponents found for the systems: (i)Fermi-Ulam model (a classical particle confined to bounce between two walls; one of them is fixed and the other one is peri-odically time varying – collisions are elastic and no damping forces are present) [25] and (ii) corrugate waveguide modelconsiders the description of a light-beam moving inside two mirrors where one of them is flat and the other one is period-ically corrugated [12]. The models are totally different from each other, but near such a transition, the chaotic sea has qual-itatively the same general behaviour. Thus the two models belong to the same class of universality, near this transition. Thecritical exponents: a ¼ 0:659ð1Þ, b ¼ 0:492ð5Þ and z ¼ 1:33ð1Þ using the control parameter c ¼ 1=2 recover the critical expo-nents for the model that consists of a classical particle confined to move inside an infinitely deep potential well which con-tains: (i) oscillating square well [27] or (ii) oscillating potential barrier [28]. Therefore, the critical exponents are the sameand the models belong to the same class of universality near this phase transition from integrability to non-integrability.

4. Conclusion

To summarise, we have studied in this work the rescale of the phase space for a family of two-dimensional, area-preserv-ing Hamiltonian map using the position of the first invariant KAM curve. The procedure used was to describe locally (nearthe first invariant KAM curve) the dynamics by using the standard mapping. A rescaling in the phase space was made andshown that the size of the chaotic sea and consequently the position of the first invariant KAM curve is scaling invariant. Therescale of the phase space was confirmed by the overlap of curves describing the average action onto a single plot. The

J.A. de Oliveira, E.D. Leonel / Applied Mathematics and Computation 221 (2013) 32–39 39

different scaling times showed to be dependent on the initial conditions for deviation around the average quantities for thechaotic sea. We have concentrated our attention to discuss the scaling properties using the exponents c ¼ 3=2 and c ¼ 2 torecover the Kepler map and the mapping that describes a particle in a wave packet for the relativistic motion. Thus, the uni-versality classes for the Kepler map and mappings with diverging angles in the limit of vanishing action were defined. Theprocedure used can be applicable to many other systems.

Acknowledgement

JAO thanks CNPq, PROPe and CAPES. EDL kindly acknowledges the financial support from CNPq, FAPESP andFUNDUNESP, Brazilian agencies. This research was supported by resources supplied by the Center for Scientific Computing(NCC/GridUNESP) of the São Paulo State University (UNESP). The authors kindly acknowledge Prof. Rui M. G. Vasconcellosand Prof. Gerson Santarine for a careful reading on the manuscript.

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