Using the Poincar\'{e} section technique, we study in detail the dynamical
behaviors of delay differential system and find a new type of solutions $S_i$
in short-time delay feedback. Our numerical results remind us to deny the
opinion that there are no complex phenomena in short-time delay case. Many
similarities between foundamental solution and the new type of solutions are
found. We demonstrate that the scales of $S_i$ increase with exponential growth
via $i$ in the direction of $\mu $, while decrease with exponential decays in
the direction of $x$ or delay time $t_R$.Comment: 4 pages, REVTEX, 4 ps figures, to be published in Phys. Lett. 

Li, H. B.

Liu, Y. W.

Wang, Y. H.

Zhao, H.

赵鸿

English

arXiv

Using the Poincare section technique, we study in detail the dynamical behaviors of delay differential system and find a new type of solutions S-i in Short-time delay feedback. Our numerical results remind us to deny the opinion that: there are no complex phenomena in short-time delay case. Many similarities between the fundamental solution and the new type of solutions are found. We demonstrate that the scales of S-i increase with exponential growth via i in the direction of mu, while decrease with exponential decays in the direction of x or delay time t(R). (C) 1999 Published by Elsevier Science B.V. All rights reserved

Xiamen University Institutional Repository

31 May 1999Ž .Physics Letters A 256 1999 166–172Self-similarity in a system with a short-time delayed feedbackLiu Yaowen 1, Li Haibin, Zhao Hong 2, Wang Yinghai 3Department of Physics, Lanzhou UniÕersity, Lanzhou 730000, ChinaReceived 5 October 1998; received in revised form 23 March 1999; accepted 6 April 1999Communicated by A.R. BishopAbstractUsing the Poincare section technique, we study in detail the dynamical behaviors of delay differential system and find a´new type of solutions S in short-time delay feedback. Our numerical results remind us to deny the opinion that there are noicomplex phenomena in short-time delay case. Many similarities between the fundamental solution and the new type ofsolutions are found. We demonstrate that the scales of S increase with exponential growth via i in the direction of m, whileidecrease with exponential decays in the direction of x or delay time t . q 1999 Published by Elsevier Science B.V. AllRrights reserved.PACS: 05.45.qb; 42.55.Px1. IntroductionOptical feedback systems governed by delay dif-Ž .ferential Eqs. DDEs have attracted much attentionfrom both the applied and the fundamental points ofw xview 1–16 . Generally, the delay-differential systemrelated to optical bistable or hybrid optical bistabledevice is described bytX x t syx t q f x ty t ,m , 1Ž . Ž . Ž . Ž .Ž .˙ RŽ .where x t is the dimensionless output of the systemat time t, t is the time delay of the feedback loop,RtX is the response time of the nonlinear medium, theparameter m is proportional to the intensity of theŽ . Ž .incident light. In Eq. 1 , f x,m is a nonlinearŽ .function of x, characterizing the system, e.g. f x,m1 E-mail: address: chaosun@lzu.edu.cn2 E-mail: address: zhaoh@lzu.edu.cn3 Author to whom correspondence should be addressed. E-mail:wangyh@lzu.edu.cnw Ž .x w xs mp 1 y z cos x y x for Ikeda model 2 ,BŽ . w 2Ž .xf x,m sp Aymsin xyx for Vallee model´0w x Ž . 2Ž .3 , and f x,m smsin xyx for the sine-square0w xmodel 4 .Ž .The understanding of Eq. 1 up to now can besummarized as follows. The first experimental obser-vation of period-doubling bifurcations and chaos in aw xhybrid bistable device was made by Gibbs et al. 1w xfollowing a prediction by Ikeda et al. 2 . The solu-tion of the system, which appears after Hopf bifurca-tion, evolves through a period doubling T ™2TF FPPP ™2 N T , as one parameter is varied. TheseFsolutions are called 2 N periodic and the cascadeaccumulates at the Feigenbaum point. These solu-tions are named fundamental solutions by Ikeda etw xal. 2 , we do so in this paper. Later the two groupsfound that higher- harmonic oscillation states appearsuccessively in the course of transition to developedŽchaos in the long-time delayed case i.e. delay time.is longer than the response time . These solutions0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9601 99 00237-6( )L. Yaowen et al.rPhysics Letters A 256 1999 166–172 167coexist and each follows a period-doubling cascade.The oscillation period of these harmonic states aregiven by T rn, where n is the odd integer and T isF Fthe period of the fundamental solution. As the studyof the dimension of the chaotic attractor, the behav-w xior of the DDE exhibits high-dimensional chaos 5 .Ikeda and Matsumoto have given an estimate of theLyapunov dimension of the attractor for the Ikedamodel, and it ranges approximately from 2 to 13when some bifurcation parameter is varied. Recently,some researchers demonstrated that the behavior ofquasiperiodicity followed the hierarchy of the Fareyw xtree 6–10 and the chaotic itinerancy phenomenonswitches among some different unstable local chaoticw xorbits 11–13 . We reported two new types of solu-tions found in moderate-time and short-time delayw xregimes 15 , which are different from the fundamen-tal solution and the odd harmonic solution. In thispaper, we study in detail the dynamical behaviors ofthe new type of solution found in short-time delayregimes.This paper is organized as follows: In Section 2,the numerical methods used in this paper are intro-duced. By using the Poincare section technique, we´can easily observe the course of bifurcation of DDE,and easily distinguish the new type of solutions andthe fundamental solution. In Section 3, we demon-strate our numerical results. In the short-time delayŽ .case, there is a new type of solutions S is1,2,3 . . . ,iwhich has many similarities compared to the funda-mental solution. Moreover, these new solutions arealike to each other. In Section 4 we summarize ourresults and conclude.2. Numerical methodsMeasuring the delay time in units of t , one canRŽ .rewrite Eq. 1 ast x t syx t q f x ty1 ,m , 2Ž . Ž . Ž . Ž .Ž .˙where tst Xrt characterizes the effect of the timeRdelay when t X is fixed. In this paper, we study Eq.Ž .2 with the special feedback functionf x ,m s1ym x 2 . 3Ž . Ž .This feedback function can be considered as the firstnonlinear term of the Taylor expansion of the gen-Ž .eral nonlinear function f x,m in the vicinity of asteady state. It should keep the general nonlinearw xproperties of DDE, as shown in Refs. 14,15 .Ž .Eq. 2 can be solved numerically and a fourth-order Adam’s interpolation is suitable for that. Inorder to trace the evolution of a DDE, one mightŽ .investigate the evolution curve of the variable x tvs the time t. However, it is difficult to distinguishŽ .different solutions if one only observes the x t y tŽ .relation. Some of us Zhao et al. have offered aw xmethod in Ref. 10 to represent the solutions of aone-variable DDE by using the Poincare section´technique. This method has been proved to be apowerful tool in exploring the evolution of bifurca-tion of DDE. Let us review this method briefly. LetŽ . Ž . Ž .x u 'x tqu ,y1FuF0, then x u is deter-t t2Ž . Ž .mined by x u uniquely according to Eq. 2 , wheret1t - t . We approach the section mapping as follows:1 2choose an appropriate constant x gR; integrate Eq.cŽ . Ž . Ž .2 numerically till x t )x and x tqh -x ,c cwhere h is the length of the integrating step; thenproceed a simulation procedure to get t as well asiŽ . Ž .x u such that x 0 sx . To be simple, we denotet t ci iŽ . Ž .x u as x u in the following discussions. In thist iiŽ .way we convert the flow of Eq. 2 into a mappingŽ . Ž .which maps the curve x u onto the curve x u .i iq1We regard this curve-to-curve mapping as thePoincare map of a DDE. A periodic solution of Eq.´Ž . Ž . Ž .2 with period T , x t sx tqT , corresponds to aperiodic solution of the Poincare map with period N,´Ž . Ž .x u sx u , where N is an integer. For practi-i iqNŽ .cal applications, we can take n discrete points x ui jŽ .on the curve x u to represent the solution, whereiŽ .u g y1,0 and js1,2, . . . ,n. Then the curve-to-jcurve mapping appears as a point-to-point mappingin Rn. In order to exhibit the bifurcation process,here we usually need a one-dimensional mappingŽ .representation x u with the bifurcation parameter.i 13. ResultsAs usually considered, there is no complex phe-Ž .nomenon in the short-time delay region since Eq. 2will approach a normal one-dimensional ordinarydifferential equation. Our results remind us that thisis not the truth.( )L. Yaowen et al.rPhysics Letters A 256 1999 166–1721683.1. The bifurcation of fundamental solutionBefore discussing the new type of solutions, let usfirst review the bifurcation process of the fundamen-Žtal solution. In the long-time delay case i.e. t is. w xvery small , Ikeda et al. 2 have shown theoreticallythat instabilities and chaotic behavior can occur inthe system. As t is fixed and m is increased, asquare-wave solution appears after the Hopf bifurca-tion of a steady state. With a further increase of m,this square wave solution undergoes a square ofbifurcation with its period doubling itself succes-sively and then becomes chaotic. We define thissolution as the fundamental solution of the systemand marked it as S .0Ž .When t increases i.e. delay time decreases , thefundamental solution exhibits mirror-similar bifurca-Ž .tion behavior as shown in Fig. 1 a . With the contin-uous increase of t , the period-doubling bifurcationwith less and less order takes place in the course ofbifurcation. At ts1.13, S undergoes only period-0Ž .two bifurcation via m. Fig. 1 b shows a bifurcationdiagram just below this value. With a further in-crease t , we can no longer observe the period-dou-Ž . Ž .bling bifurcation of S , see Fig. 1 c – f . We regard0the regime of t)1.13 as the short-time delay case.Fig. 1. The bifurcation diagrams of S and S with the increase of t .0 i( )L. Yaowen et al.rPhysics Letters A 256 1999 166–172 169In this regime, the delay time t is smaller than theRresponse time t X because tst Xrt , and the funda-Rmental solution exhibits only a one-period limit cy-cle state.3.2. New type of solutionsIn the short-time delay case, fundamental solutionshows no bifurcation and chaos. This is not to saythat there is no chaos in this system with the varyingof m at a fixed parameter t . In fact, there still existsanother chaotic attractor, which locates behind S in0Ž . Ž .the direction of m, as shown in Fig. 1 c – f . WeŽ .marked them as S is1,2,3, . . . . In the direction ofim, every S evolves a period-doubling bifurcation.iFig. 2 exhibits the evolution courses of fundamentalsolution and S , which is located on period-one limiticycle state of themselves, respectively.From Fig. 1 and Fig. 2, we can easily find thatthere are not only similarities but also differencesamong S and S . Firstly, with the increase of t ,0 ieach S appears the same bifurcation process as Si 0Ž . Ž .does. Comparing Fig. 1 c with Fig. 1 a , we canfind that the diagram of S at ts3.0 displays the1same shape as that of S at ts0.80. In fact, at0certain parameter, S , S and S et al. also show the2 3 4similar shape. Secondly, S has more and moreioscillation with the increase of the subscript i. Si-multaneously, i increases with the increase of t . S0has only one peak within one period. In contrast toŽ . Ž . Ž . Ž .Fig. 2. The evolution of S and S versus time. a S ,ts0.3,ms1.50; b S ,ts1.5,ms8.50; c S ,ts1.5,ms9.42; d S ,ts3.5,m0 i 0 1 2 3Ž .s31.45; b S ,ts40,ms2585.4( )L. Yaowen et al.rPhysics Letters A 256 1999 166–172170S , S has not only a peak but also another small0 1peak; S ,S and S have more and more peaks2 3 4within one period respectively, see Fig. 2. Thirdly,with the increase of t , S appears one by one and Si 1follows S , S follows S in the direction of m.0 2 13.3. The scales of S Õia iiFrom Fig. 1 one can find that S appears continu-iously with the increase of t . In order to find the lawwhich exhibits the appearance order of S and theiscales of S , we should choose a standard to compareiS with each other. In this paper, we choose theicritical values of t as the standard, t is the value ati iwhich the second period-doubling bifurcation of theperiod-1 solution of S takes place with the decreaseiof t . In fact one could also choose another standard.For this specific choice, the bifurcation diagram of SiŽ . Ž .appears as the patterns in Fig. 3 a – e respectively.Our numerical solutions show that t increases withiexponential growth against the increase of i. Fig.Ž .4 a demonstrates the result, where the scatters arethe values of t and the dotted line is the fittingicurve which is an exponential growth function withFig. 3. Bifurcation diagrams of S and S at the critical values of t .0 i i( )L. Yaowen et al.rPhysics Letters A 256 1999 166–172 171Ž . Ž . Ž . Ž . XFig. 4. a t , b t , c dm, and d d x of S versus i. Here t is as same as Fig. 3 and t st rt . dm is the length and d x is thei R i i Rmaximum height of period-2 solution in Fig. 3 respectively. The square scatter is the numerical result and the dotted line is the fittingŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .curves. a t s2.026exp ir0.817 ; b t i s1.135exp yir0.747 ; c dm i s1.072exp ir0.616 ; d d x i s0.125exp yir0.458 .i RŽ .the increase of i as follows: t sAexp irB , whereiA and B are fitting coefficients.From Fig. 1 one can find that the scale of thebifurcation diagrams of S along m increases, whileialong x it decreases with the increase of i. Using thestandard chosen above, we can measure the lengthdm of the period-2 solution in the direction of m att and use it to characterize the scale of S along m.i kŽ .Fig. 4 c shows that dm also increases with exponen-tial growth via i, and the exponential function isŽ .dm i sAexp irB . On the other hand, by measur-Ž .ing the maximum height d x of the period-2 solutionat t , we find that d x exhibits the exponential decayiŽ . Ž .with the increase of i: d x i sAexp yirB , seeŽ .Fig. 4 d .4. ConclusionIn this paper we have studied in detail the dynam-ics of a short-time delay differential system. Ournumerical result remind us that it is not correct toassume that there is no complex behavior in short-time delay feedback. By using the Poincare section´w xtechnique in DDE 10 , a new type of solutions Siwas found in this case and it has many similaritieswith the fundamental solutions in the bifurcationdiagrams. We found the law of S and showed theiscales of S with the increase of i. The scales of Si iincreases along m and t , while it decreases along xand delay time.AcknowledgementsThis work is supported in part by the NationalNatural Science Foundation of China, and in part byDoctoral Education Foundation of National Educa-tion Committee.Referencesw x1 H.M. Gibbs, F.A. Hopf, D.L. Kaplan, R.L. Shoemaker, Phys.Ž . Ž .Rev. Lett. 46 1981 474; Phys. Rev. A 25 1982 2172.w x Ž .2 K. Ikeda, H. Daido, O. Akimoto, Phys. Rev. Lett. 45 1980Ž .709; 49 1982 1467.( )L. Yaowen et al.rPhysics Letters A 256 1999 166–172172w x Ž .3 R. Vallee, C. Delisle, Phys. Rev. A 34 1986 309.´w x Ž .4 J.P. Goedgebuer, L. Larger, H. Porte, Phys. Rev. E 57 19982795.w x5 B. Dorizzi, B. Grammaticos, M. Le Berre, Y. Pomeau, E.Ž .Ressayre, A. Tallet, Phys. Rev. A 35 1987 328; K. Ikeda,Ž .K. Matsumoto, Physica 29D 1987 223.w x6 D. Baums, W. Elsasser, E.O. Gobel, Phys. Rev. Lett. 63¨ ¨Ž .1989 155.w x Ž .7 J. Mørk, J. Mark, B. Tromborg, Phys. Rev. Lett. 65 19901999.w x8 J. Sacher, D. Baums, P. Panknin, E.O. Gobel, Phys. Rev. A¨Ž .45 1992 1893.w x Ž .9 J. Ye, H. Li, J.G. McInerney, Phys. Rev. A 47 1993 2249.w x10 H. Zhao, F.Z. Zhang, J. Yan, Y.H. Wang, Phys. Rev. E 54Ž .1996 6925.w x Ž .11 K. Otsuka, Phys. Rev. Lett. 65 1990 329.w x12 I. Fischer, O. Hess, W. Elsaßer, E. Gobel, Phys. Rev. Lett.¨ ¨Ž .73 1994 2188.w x Ž .13 C. Masoller, Phys. Rev. A 50 1994 2569.w x Ž .14 J.N. Li, B.L. Hao, Commun. Theor. Phys. 11 1989 265.w x15 H. Zhao, Y.W. Liu, Y.H. Wang, B. B Hu, Phys. Rev. E 58Ž .1998 4383.w x Ž .16 A.R. Volkovskii, N.F. Rulkov, Tech. Phys. Lett. 19 199397.

Physics Letters A

Self-similarity in a system with a short-time delayed feedback

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Self-similarity in a system with short-time delayed feedback

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