Exponential transient rotating waves and their bifurcations in a ring of unidirectionally coupled bistable Lorenz systems

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1 Available olie at Procedia IUTAM 5 (2012 ) IUTAM Symposium o 50 Years of Chaos: Applied ad Theoretical Expoetial trasiet rotatig waves ad their bifurcatios i a rig of uidirectioally coupled bistable Lorez systems Yo Horikawa*, Hiroyuki Kitajima Faculty of Egieerig, Kagawa Uiversity Takamatsu Japa Abstract Rotatig waves i a rig of uidirectioally coupled Lorez systems are studied. Whe a sigle system is bistable ad ochaotic, various rotatig waves are geerated ad bifurcated i the rig of systems. We cosider two kids of rotatig waves: expoetial trasiet rotatig waves ad chaotic rotatig waves. Expoetial trasiet rotatig waves are those the duratio of which icreases expoetially with the umber of systems i the rig. Chaotic rotatig waves are caused through iteractios betwee periodic rotatig waves of multiple wave umbers Published by by Elsevier Elsevier Ltd. Ltd. Selectio Peer-review ad/or Peer-review uder resposibility uder resposibility of Takashi of Takashi Hikihara Hikihara ad Tsutomu ad Tsutomu Kambe Kambe Ope access uder CC BY-NC-ND licese. Keywords: Lorez system; uidirectioal couplig; rotatig wave; expoetial trasiet 1. Itroductio We study rotatig waves i a rig of uidirectioally coupled Lorez systems. A model is give by dx dy dz ( y x 1 z x y x ) x z y (1 N, x 0 x N ) (1) where x i the first term of the equatio for y of the th system is replaced by x -1 of the 1st system. This couplig is equivalet to the liear covectio of x i the equatios for y with stregth equal to. * Correspodig author. Tel.: ; fax: address: horikawa@eg.kagawa-u.ac.jp Published by Elsevier Ltd. Selectio ad/or Peer-review uder resposibility of Takashi Hikihara ad Tsutomu Kambe Ope access uder CC BY-NC-ND licese. doi: /j.piutam

2 284 Yo Horikawa ad Hiroyuki Kitajima / Procedia IUTAM 5 ( 2012 ) A total N systems make a closed loop with x 0 = x N. This rig of coupled Lorez systems has bee studied i [1, 2] ad it has the bee show that periodic ad hyperchaotic rotatig waves are geerated from sychroized spatially uiform chaotic states through the Hopf bifurcatios. These rotatig waves have bee observed i a rage of parameters i which a sigle Lorez system is chaotic. I this study, we cosider Eq. (1) cosistig of bistable Lorez systems ad show (i) ustable trasiet rotatig waves the duratio of which icreases expoetially with the umber N of systems (expoetial trasiets); (ii) the stabilizatio of the rotatig waves through pitchfork bifurcatios ad the geeratio of chaotic rotatig waves. I the rest of the paper, bifurcatios of steady states ad rotatig waves i a rig of ie Lorez systems are show i Sect. 2. Expoetial trasiet rotatig waves ad chaotic rotatig waves are show i Sect. 3 ad 4, respectively. Fially coclusio is give i Sect Bifurcatios of a rig of Lorez systems We first cosider bifurcatios of steady states ad rotatig waves i a rig of Lorez systems. The origi (x = y = z = 0 (1 N)) is a steady state of Eq. (1). The eigevalues of the Jacobia matrix of Eq. (1) evaluated at the origi are give by [1 2 4exp( i2 / N) ] 1/ / 2 (0 N), (2) We set = 10, = 8/3 ad use for a bifurcatio parameter. Figure 1 shows bifurcatio diagrams of a rig of ie Lorez systems (N = 9), i which the sum S x = of x 1 (1 N) is plotted. Figure 1(a) shows the bifurcatio diagram of the steady states. The origi (S x = 0) is stable whe 0 < 1 ad is destabilized through the Hopf bifurcatio at = 1. A pair of stable spatially uiform steady states (x = y 1/ 2 = [ ( 1)], z = 1 (1 N)) is geerated at the same time so that Eq. (1) becomes bistable. A ustable symmetric rotatig wave (S x = 0) (RW1) is the geerated through the Hopf bifurcatio from the origi as icreases at 1.24 (H1). I the geerated symmetric rotatig wave (RW1), each system oscillates with phase differece 2/N betwee the adjacet systems. Figure 2(a) shows a time course of x 1 of the RW1 at = 1.5 (a upper pael) ad a spatiotemporal patter of it (a lower pael), i which black ad white regios correspod to the states x (t) of positive ad egative sigs, respectively. (It should be oted that a rotatig wave for N = 10 is plotted sice it is observed with computer simulatio i the ivariat space: x = x + N/2, y = y + N/2, z = z + N/2, (1 N/2) whe N is eve.) Further, a ustable symmetric rotatig wave of secod harmoics (RW2), which has two spatial periods, is geerated through the secod Hopf bifurcatio from the origi at 2.62 (H2) successively. O the other had, a pair of the ozero spatially uiform steady states causes the Hopf bifurcatios at 6.2 ad 7.5 (H) ad is destabilized. Ustable asymmetric rotatig waves are the geerated though they are ot plotted i Fig. 1(a). Figure 1(b) shows the bifurcatio diagram of the first symmetric rotatig wave (RW1) geerated at the origi. The stability of the RW1 chages alterately through successive pitchfork bifurcatios at 1.92, 2.53 ad 3.88 (PFs). Three pairs of asymmetric rotatig waves (S x 0) (RW11 13) are geerated at the same time. Figure 2(b) shows a spatiotemporal patter of the secod oe (RW12) at = 7.0, which is stable. Such pitchfork bifurcatios ad stabilizatio of rotatig waves have bee show i a rig eural etwork with iertia [3]. More complicated bifurcatios of the rotatig waves occur i Eq. (1). First, a stable quasiperiodic rotatig wave, i which S x chages periodically, is geerated from the RW1 through the Neimark-Sacker bifurcatio at (NS), which is destabilized at A spatiotemporal N x

Yo Horikawa ad Hiroyuki Kitajima / Procedia IUTAM 5 ( 2012 ) 283 287 285 patter of the quasiperiodic rotatig wave at = 10.6 is show i Fig. 2(c).

3 Yo Horikawa ad Hiroyuki Kitajima / Procedia IUTAM 5 ( 2012 ) patter of the quasiperiodic rotatig wave at = 10.6 is show i Fig. 2(c). Figure 3(c) the shows the bifurcatio diagram of the secod symmetric rotatig wave (RW2) geerated at the origi. It causes a pitchfork bifurcatio at 5.78 (PF) ad a pair of asymmetric rotatig waves (RW21) is geerated. The geerated RW21 causes the Neimark-Sacker bifurcatio at 5.93 (NS) ad a period doublig bifurcatio at 6.07 (PD) successively. The rotatig wave (RW22) of period two geerated at PD disappears through the saddle-ode bifurcatio with the RW12 at 8.62 (SN1). The symmetric RW2 is stabilized through the Neimark-Sacker bifurcatio at (NS). A chaotic rotatig wave, i which S x chages itermittetly, coexists with the stable RW1 i betwee SN1 ad NS (8.62 < < 10.34). Figure 2(d) shows a spatiotemporal patter of the chaotic rotatig wave at = Fig. 1. Bifurcatio diagrams of Eq. (1) with N = 9. (a) steady states; (b) first rotatig wave; (c) secod rotatig wave. Fig. 2. Spatiotemporal patters of rotatig waves i Eq. (1) with N = 9. (a) ustable symmetric rotatig wave (N = 10); (b) stable asymmetric rotatig wave; (c) stable quasiperiodic rotatig wave; (d) chaotic rotatig wave

4 286 Yo Horikawa ad Hiroyuki Kitajima / Procedia IUTAM 5 ( 2012 ) Expoetial trasiet rotatig waves We cosider trasiet states whe a pair of the stable spatially uiform steady states ad the ustable symmetric rotatig wave (RW1) coexist ad Eq. (1) is bistable. I trasiet states, asymmetric rotatig waves are geerated, i which the umbers of positive (x > 0) ad egative (x < 0) states are ot equal. It is show that the largest eigevalue of the Poicare map of the RW1 decreases to zero double expoetially with the umber N of systems. It meas that the relaxatio time of the RW1 to coverge to oe of the steady states icreases expoetially with N. As a result, the mea duratio m(t) of trasiet rotatig waves geerated from radom iitial states icreases expoetially with N. Figure 3 shows the results of computer simulatio of 10 4 rus uder Gaussia radom iitial coditios: x (0), y (0), z (0) ~ N(0, 1) for each N whe = 1.5. Such expoetial trasiet states are commo i static kik ad pulse patters i symmetric bistable reactio-diffusio systems [4] ad have bee recetly foud i dyamic rotatig waves i a spatially discrete coupled system (a rig eural etwork) [5]. Fig. 3. Mea duratio m(t) of radomly geerated trasiet rotatig waves of vs the umber N of systems 4. Chaotic rotatig waves A chaotic rotatig wave exists i 8.62 < < whe N = 9 (Fig. 1(c) ad Fig. 2(d)). Figure 4 shows a sequece {y 1 (t k )} at t k whe the sig of x 1 chages from a egative to a positive at kth time whe = Further, Fig. 5 shows a retur map of the successive maxima y 1 mi m i {y 1 (t k )}. We ca see type-ii itermittecy i these figures. The largest Liapuov expoet is estimated to be It is worth otig that this chaotic rotatig wave appears i a rage of parameters i which a sigle Lorez system is bistable ad ochaotic ( < 24.06). This is i cotrast to chaotic rotatig waves show i [1, 2]. Fig. 4. Sequece {y 1 (t k )} at which the sig of x 1 chages i chaotic rotatig wave

5 Yo Horikawa ad Hiroyuki Kitajima / Procedia IUTAM 5 ( 2012 ) Fig. 5. Retur map of the successive maxima y 1 mi m i {y 1 (t k )} 5. Coclusio A rig of uidirectioally coupled Lorez systems i which each system is bistable ad ochaotic was studied. It was show that ustable rotatig waves are geerated from the origi ad they are bifurcated ito asymmetric rotatig waves i a rig of ie systems. I the bistable regime, trasiet rotatig waves the duratio of which icreases expoetially with the umber of systems were show. Chaotic rotatig waves with type-ii itermittecy were also show i a rig of ie oscillators. Refereces [1] Matías MA, Pérez-Muñuzuri V, Lorezo MN, Mariño IP, Pérez-Villar V. Observatio of a fast rotatig wave i rigs of coupled chaotic oscillators. Phys. Rev. E 1997; [2] Sáchez E, Pazó D, Matías MA. Experimetal study of the trasitios betwee sychroous chaos ad a periodic rotatig wave. Chaos 2006; /1-10. [3] Horikawa Y, Kitajima H. Bifurcatio ad stabilizatio of oscillatios i rig eural etworks with iertia. Physica D 2009; [4] Ward MJ. Metastable dyamics ad expoetial asymptotics i multi-dimesioal domais. I: Joes CKRT, Khibik AI, editors. Multiple-Time-Scale Dyamical Systems, IMA Volumes i Mathematics ad its Applicatios, vol. 122, New York: Spriger; 2001, p ; ad refereces therei. [5] Horikawa Y, Kitajima H. Duratio of trasiet oscillatios i rig etworks of uidirectioally coupled euros. Physica D 2009;

AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC MAPS

http://www.paper.edu.c Iteratioal Joural of Bifurcatio ad Chaos, Vol. 1, No. 5 () 119 15 c World Scietific Publishig Compay AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC