Shrimp-shape domains in a dissipative kicked rotator Diego F. M. Oliveira, Marko Robnik, and Edson D. Leonel Citation: Chaos 21, 043122 (2011); doi: 10.1063/1.3657917 View online: http://dx.doi.org/10.1063/1.3657917 View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v21/i4 Published by the AIP Publishing LLC. Additional information on Chaos Journal Homepage: http://chaos.aip.org/ Journal Information: http://chaos.aip.org/about/about_the_journal Top downloads: http://chaos.aip.org/features/most_downloaded Information for Authors: http://chaos.aip.org/authors Downloaded 12 Jul 2013 to 186.217.234.17. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions http://chaos.aip.org/?ver=pdfcov http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/1877665666/x01/AIP-PT/Chaos_PDFCoverPg_0713/FreeContentHand_1640x440.jpg/6c527a6a7131454a5049734141754f37?x http://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Diego F. M. Oliveira&possible1zone=author&alias=&displayid=AIP&ver=pdfcov http://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Marko Robnik&possible1zone=author&alias=&displayid=AIP&ver=pdfcov http://chaos.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Edson D. Leonel&possible1zone=author&alias=&displayid=AIP&ver=pdfcov http://chaos.aip.org/?ver=pdfcov http://link.aip.org/link/doi/10.1063/1.3657917?ver=pdfcov http://chaos.aip.org/resource/1/CHAOEH/v21/i4?ver=pdfcov http://www.aip.org/?ver=pdfcov http://chaos.aip.org/?ver=pdfcov http://chaos.aip.org/about/about_the_journal?ver=pdfcov http://chaos.aip.org/features/most_downloaded?ver=pdfcov http://chaos.aip.org/authors?ver=pdfcov Shrimp-shape domains in a dissipative kicked rotator Diego F. M. Oliveira,1,a) Marko Robnik,1,b) and Edson D. Leonel2,c) 1CAMTP—Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia 2Departamento de Estatı́stica, Matemática Aplicada e Computação, UNESP, Universidade Estadual Paulista, Av. 24A, 1515 Bela Vista, 13506-900 Rio Claro, SP, Brazil (Received 5 June 2011; accepted 12 October 2011; published online 14 November 2011) Some dynamical properties for a dissipative kicked rotator are studied. Our results show that when dissipation is taken into account a drastic change happens in the structure of the phase space in the sense that the mixed structure is modified and attracting fixed points and chaotic attractors are observed. A detailed numerical investigation in a two-dimensional parameter space based on the behavior of the Lyapunov exponent is considered. Our results show the existence of infinite self-similar shrimp-shaped structures corresponding to periodic attractors, embedded in a large region corresponding to the chaotic regime. VC 2011 American Institute of Physics. [doi:10.1063/1.3657917] A strongly dissipative kicked rotator is studied. The investigation is based mainly on the asymptotic behavior of the Lyapunov exponents. Our results show that by reducing the two-dimensional map to a one-dimensional map in the limit of infinite kicks, the Feigenbaum’s d is recovered. An investigation in the parameter space (the dissipation parameter and the kicking parameter) reveals the existence of infinite families of self-similar structures of shrimp-shape embedded in a large region correspond- ing to the chaotic regime. The organization of stability shrimps reported here for the discrete-time kicked rota- tor agrees well with the parameter space organization reported recently in the literature for several flows (con- tinuous-time systems). I. INTRODUCTION Studies in dissipative systems have attracted much atten- tion during the last decades, and it has been used in various fields of science including optics,1,2 turbulence and fluid dy- namics,3,4 nanotechnology,5,6 atomic and molecular physics,7,8 and quantum and relativistic systems.9,10 One sys- tem of special importance for its large applicability, which has been the subject of extensive research, is the standard map. Proposed originally by Chirikov11,12 in 1969, the stand- ard map is a dynamical system which describes the motion of a kicked rotator. Since the pioneering paper of 1969, the standard map has been applied in many different fields of science including solid state physics,13 statistical mechan- ics,14 accelerator physics,15 problems of quantum mechanics and quantum chaos,16,17 plasma physics,18 ratchet trans- port,19 and many others. The dynamics of the kicked rotator is controlled by the kicking parameter K and in the absence of dissipative forces; if K is small enough, the structure of the phase space is mixed20–34 in the sense that Kolmogorov-Arnold-Moser (KAM) invariant tori and regular islands are observed coex- isting with chaotic seas. As the parameter K increases and becomes larger than Kc� 0.971635..., the last invariant spanning curve disappears and the system presents a glob- ally chaotic component in the sense that a chaotic orbit can spread over the phase space. However, the introduction of dissipation in the model changes completely the mixed structure and the system exhibits attractors.35–37 In the regime of strong dissipation, the model exhibits a period doubling bifurcation cascade, and the so called Feigen- baum’s d, which is the rate of the bifurcations, can be obtained numerically. On the other hand, when weak dissi- pation is taken into account, a drastic change occurs in the behavior of the average energy. The unlimited energy growth present in the Hamiltonian case38 for the case K�Kc is no longer observed. The average action exhibits a characteristic saturation value which can be described using scaling arguments. Additionally, such a behavior can be described remarkably well by an empirical universal func- tion of the type f(x)¼ xb/(1þ x)b, where b is the acceleration exponent. Such a function can also be applied to many dissi- pative systems.39–42 In this paper, we will explore some properties of a dis- sipative standard map seeking to understand and describe its parameter space. We specifically take into account its two-dimensional parameter space, namely, the dissipation parameter c and the amplitude of the kicks K. We revisit the parameter space of the dissipative standard map where the existence of self-similar structures called shrimps was shown.44 According to Ref. 43, “Shrimps are formed by a regular set of adjacent windows centered around the main pair of intersecting superstable parabolic arcs. A shrimp is a doubly infinite mosaic of stability domains composed by a)Electronic mail: diegofregolente@gmail.com. b)Electronic mail: robnik@uni-mb.si. c)Electronic mail: edleonel@rc.unesp.br. 1054-1500/2011/21(4)/043122/6/$30.00 VC 2011 American Institute of Physics21, 043122-1 CHAOS 21, 043122 (2011) Downloaded 12 Jul 2013 to 186.217.234.17. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions http://dx.doi.org/10.1063/1.3657917 http://dx.doi.org/10.1063/1.3657917 an innermost main domain plus all the adjacent stability domains arising from two period-doubling cascades to- gether with their corresponding domains of chaos. Shrimps should not be confused with their innermost main domain of periodicity.” Results by Gaspard et al.45 in 1984, Rössler et al.46 in 1989, and Komuro et al.47 in 1991 already dem- onstrated the existence of self-similar periodic structures in a 2D-mapping of the Chua’s system, the logistic map, and the mapping of the double scroll circuit, respectively. How- ever, it was after the pioneering paper of Gallas48 in 1993 studying the parameter space of the Hénon map that the pa- rameter space attracted much attention, and since then, it has already been shown that such shrimp-shaped domains can be found in many theoretical models.49–58 Very recently, they were also observed experimentally in a cir- cuit of the Nishio-Inaba family.59 A recent result from one of the pioneers of chaos theory,60 was also devoted to this intriguing and rich parameter space structures. Here, in order to classify regions in the parameter space with regular or chaotic behavior, we use as a tool the Lyapunov expo- nent. We adopt the following procedure: starting with a fixed initial condition, after a long transient, the Lyapunov exponent is obtained and, for each combination of (K, c), a color is attributed. After that we give an increment in the parameters. We use the last value obtained for the dynami- cal variables (I, h) before the increment, as the new initial condition after the increment. This ensures that we are always in the basin of the same attractor. These self-similar very well organized structures of shrimp-shape are shown embedded into a large region corresponding to chaotic attractors. The paper is organized as follows. In Sec. II, we describe all the necessary details to obtain the two-dimensional map that describes the dynamics of the system and also we present and discuss our numerical results. Conclusions are drawn in Sec. III. II. THE MODEL AND THE NUMERICAL RESULTS The Hamiltonian that describes the dynamics of the kicked rotator has the following form:61,62 HðI; h; tÞ ¼ I2 2 þ K cosðhÞ X1 n¼�1 d t� nð Þ; (1) where I and h are the action and angle variables, respec- tively. K is the amplitude of the delta-function pulses (kicks), the kicking parameter. The equations of motion can be easily found and are given by _I ¼ K sinðhÞ X1 n¼�1 d t� nð Þ; _h ¼ I: (2) Assuming that (In, hn) are the values of the variables just before the (nþ 1)th kick, (Inþ1, hnþ1) represent their values just before the (nþ 2)th kick, and introducing a dissipative parameter63 c, the dynamics of a dissipative kicked rotator is described by the following two-dimensional nonlinear map: S : Inþ1 ¼ ð1� cÞIn þ K sinðhnÞ hnþ1 ¼ ½hn þ Inþ1�modð2pÞ ; � (3) where c 2 0; 1½ � is the dissipation parameter. When dissipa- tion is taken into account the structure of the phase space is changed. Then, an elliptical fixed point (generally sur- rounded by KAM islands) turns into a sink. Regions of the chaotic sea might be replaced by chaotic attractors. Figure 1(a) shows the structure of the phase space for the conserva- tive dynamics (c¼ 0) with K¼ 1. As it is well known for such a value of the kicking parameter K, the last invariant torus is destroyed and the phase space has one large chaotic sea, the nested structures of thin chaotic layers, and KAM islands. Figure 1(b) shows the basin of attraction where the main fixed points are of period 1 (red and black), 2 (cyan), 3 (maroon), and 4 (green). The dissipation parameter considered is c¼ 10�2. The procedure used to construct the basin of attrac- tion is to divide both I 2 0; 2p½ � and h 2 0; 2p½ � into grids of 1000 parts each, thus leading to a total of 106 different initial conditions. Each initial condition is iterated up to n¼ 5� 105. One can see that many periodic attractors emerge for such a choice of control parameters. It is important to stress that other attractors could in principle exist. If they exist, however, their basins of attraction are too small to be observed. We now consider the regime of strong dissipation. It corresponds to the case where the action I loses more than 70% of its value upon a kick. We considered the case of FIG. 1. (Color online) (a) Phase space for the conservative standard map (c¼ 0) with K¼ 1. (b) Basin of attraction for the attracting fixed points (sinks) of period 1 (red and black), 2 (cyan), 3 (maroon), and 4 (green). The control parameters used to construct the basin of attraction were K¼ 1 and c¼ 10�2. 043122-2 Oliveira, Robnik, and Leonel Chaos 21, 043122 (2011) Downloaded 12 Jul 2013 to 186.217.234.17. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions c¼ 0.80. To explore some typical behavior, we have used the initial conditions (h0, I0)¼ (3, 5.53) and investigated its attraction to periodic orbits, and looked at the bifurcations as K varies in the range where global chaos is observed in the non-dissipative regime, i.e., c¼ 0. Figure 2(a) shows the behavior of the asymptotic action plotted against the control parameter K, where a sequence of period doubling bifurca- tions is evident.65–67 The bifurcations observed in (a) are marked by the vanishing Lyapunov exponent at the same control parameter K as shown in Fig. 2(b). As discussed by Eckmann and Ruelle,64 the Lyapunov exponents are defined as kj ¼ lim n!1 1 n ln jKjj; j ¼ 1; 2; (4) where Kj are the eigenvalues of M ¼ Qn i¼1 Jiðhi; IiÞ and Ji is the Jacobian matrix evaluated over the orbit (hi, Ii). However, a direct implementation of a computational algorithm to evalu- ate Eq. (4) has a severe limitation to obtain M. For the limit of short n, the components of M can assume different orders of magnitude for chaotic orbits and periodic attractors, mak- ing the implementation of the algorithm impracticable. To avoid such a problem, J can be written as J¼HT, where H is an orthogonal matrix and T is a right up triangular matrix. M is rewritten as M ¼ JnJn�1 … J2H1H �1 1 J1, where T1 ¼ H�1 1 J1. A product of J2H1 defines a new J02. In a next step, one can show that M ¼ JnJn�1 … J3H2H �1 2 J02T1. The same procedure can be used to obtain T2 ¼ H�1 2 J02 and so on. Using this procedure, the problem is reduced to evaluate the diagonal elements of Ti : Ti 11; T i 22. Finally, the Lyapunov exponents are given by kj ¼ lim n!1 1 n Xn i¼1 lnjTi jjj; j ¼ 1; 2: (5) If at least one of the kj is positive then the orbit is said to be chaotic. Figure 2(b) shows the behavior of the Lyapunov exponents corresponding to Fig. 2(a). One can see also that FIG. 2. Bifurcation cascade for (a) I�K; (b) the Lyapunov exponent associ- ated to (a). The damping coefficient used was c¼ 0.80. TABLE I. The value of n, the period of the bifurcation, the values of the parameter K where the bifurcation happen, and the convergence of the Feigenbaum’s d considering bifurcations up to the eleventh order. n Period K d 1 2 5.57011554475050 — 2 4 5.75421890196820 4.66723703234012 3 8 5.79366480024635 4.60078306721265 4 16 5.80223853600000 4.65331150449313 5 32 5.80408103800000 4.66432491229193 6 64 5.80447605808000 4.66821746028430 7 128 5.80456067712000 4.66893856195158 8 256 5.80457880094812 4.66910749234449 9 512 5.80458268259534 4.66917075858837 10 1024 5.80458351393080 4.66920050635855 11 2048 5.80458369197744 — FIG. 3. (Color online) (a) Phase diagram of K vs. c where the regular struc- ture of shrimp-shape is shown. The color scale corresponds to the Lyapunov exponent for a given combination of (K, c). Regular regions are shown in a red-yellow scale, while chaotic behavior is shown in a green-blue scale. (b) Magnification of the main structure in (a) where white indicates chaos (posi- tive Lyapunov exponent) and periodic solution (negative Lyapunov expo- nent) is shown in colors, each color indicates a given period, namely, red correspond to period 4, green period 8, and blue period 16 and larger periods are no longer visible. The color coding runs between the maximum and min- imum values of the entire plot and its numerical value is indicated. 043122-3 A dissipative kicked rotator Chaos 21, 043122 (2011) Downloaded 12 Jul 2013 to 186.217.234.17. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://chaos.aip.org/about/rights_and_permissions when the bifurcations occur, the exponent k vanishes. The Lyapunov exponents between 5.66