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

Transcription

1 Iteratioal Joural of Bifurcatio ad Chaos, Vol. 1, No. 5 () c World Scietific Publishig Compay AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC MAPS LI-QUN CHEN ad YAN-ZHU LIU Departmet of Egieerig Mechaics, Shaghai Jiaotog Uiversity, Shaghai 3, P. R. Chia lqche@olie.sh.c liuyzhc@olie.sh.c Received December 3, ; Revised October 1, 1 I this paper, the ope-plus-closed-loop cotrol strategy is adapted to sychroize discrete chaos. Two sychroizatio problems of chaos are studied: oe is to drive a chaotic map with the aim of obtaiig desirable chaotic dyamics; the other is to idetify chaotic behaviors of a oliear map for differet iitial coditios. It is show that i the latter case the eeded cotrol sigal ca be arbitrarily small. Two umerical examples, amely, the Gaussia map ad a hyperchaotic map, are ivestigated i detail for demostratio of the effectiveess of the approach. The results show that sychroizatio of discrete chaos ca be realized if the cotrol is activated i the basi of etraimet. Keywords: Ope-plus-closed cotrol; sychroizatio; discrete system; chaos; hyperchaos. 1. Itroductio Over the past decade, there has bee a great deal of research related to the cotrol ad sychroizatio of chaos [Che & Dog, 1998; Ott et al., 1994]. I particular, Jackso ad Grosu [1995] developed a ew ad powerful method of cotrollig chaos the ope- plus-closed-loop method for cotiuous dyamical systems defied by a set of first-order ordiary differetial equatios. This method may be implemeted i experimets [Jackso, 1995, 1997; Jackso & Grosu, 1995a, 1995b; Weigel & Jackso, 1998]. The idea has bee employed to cotrol chaotic oscillators defied by a set of oautoomous secod- order ordiary differetial equatios [Che & Liu, 1998], as well as discrete dyamical systems defied by a set of differece equatios [Che, 1]. The idea has also bee applied to sychroizatio of two logistic maps with chaotic behaviors [Grosu, 1997]. I this paper, the ope-plus-closed-loop cotrol strategy is adopted to sychroize discrete chaos. Two chaos sychroizatio problems are ivestigated: oe is to drive a chaotic map with the aim of obtaiig desirable chaotic dyamics; the other is to idetify chaotic behaviors of a oliear map for differet iitial coditios. I the latter case, the cotrol sigal ca be arbitrarily small. Two umerical examples are show to demostrate the effectiveess of the approach. More precisely, the first case is cocerig a specific chaotic map the Gaussia map. Its basi of etraimet for a arbitrary orbit is preseted. The Gaussia map is drive to track aother chaotic map, the logistic map. The chaotic behaviors of the Gaussia map for differet iitial coditios ca be sychroized as well. The secod case is a specific hyperchaotic map with two positive Lyapuov expoets. This hyperchaotic map is drive to track aother chaotic map, the Héo map. The chaotic behaviors of the hyperchaotic map for differet iitial coditios ca also be sychroized. 119 转载

2 1 L.-Q. Che & Y.-Z. Liu. The Ope-plus-Closed-Loop Cotrol Law Cosider a cotrollable discrete dyamical system described by the map z +1 = f(z )+u (z, u R m ) (1) where z ad u deote respectively the state of the system ad the cotrol parameters at time step, adf : R R is a smooth map. Prescribig a desired goal g that may be chaotic, ad the liearizig locally Eq. (1) i the small eighborhood of the goal, lead to z +1 = f(g )+Df (z g )+u () where the Jacobi matrix Df = [ f i / z j ] m m is evaluated at g. Itroduce a cotrol law of the form u = g +1 f(g )+(Df A )(g z ) (3) where A is a sequece of m m matrices such that the equatio y +1 = A y (4) has oly a asymptotically stable zero solutio. Oe ca the derive from Eq. (), after some algebraic operatios, the followig: z +1 g +1 = A (z g ). (5) Thus, the dyamical behavior of Eq. (1) is etraied to the goal g, i.e. lim z g = (6) It should be poited out that the cotrol law (3) cosists of two parts: the ope-loop part without feedback is u = g +1 f(g ) (7) ad the closed-loop part with feedback is u c =(Df A )(g z ). (8) Therefore, Eq. (3) presets a ope-plus-closedloop cotrol law. The ope-loop part ad the closed-loop part i cotrol law (3) have differet effects o cotrol: the ope-loop part creates a desired orbit ad the closed-loop part stabilizes it. If the desired orbit g is a ustable periodic or a chaotic orbit of the ucotrolled system, i.e. g +1 = f(g ) (9) the cotrol law (3) oly has a closed-loop part (8). I some practical applicatios, it is coveiet to choose A as a costat matrix A. Ithiscase, the magitudes of all eigevalues of the costat matrix A should be less tha 1. Suppose that the cotrol starts whe =. To force Eq. () to approximate Eq. (1) sufficietly well, oe should apply cotrol i a small eighborhood of g. To esure the desired results, the basis of etraimet of Eq. (1) associated with g ad, amely, { } BE(g, )= z lim z g = (1) should be oempty. It is to be expected that the basi of etraimet will deped o both the particular discrete dyamical system ad some characteristics of the cotrol goal. To avoid a violet reactio of the system i the course of cotrol, the ope-plus-closed-loop cotrol should be activated whe the chaotic orbit is close to the goal orbit. For this reaso the cotrol law (3) takes the followig form: u = S( )S(ε z g )(g +1 f(g ) +(Df A )(g z )) (11) where ε is a small positive umber ad the switchig fuctio is defied by { 1 z S(z) = (1) < 3. Sychroizatio of a Chaotic Map Cosider the Gaussia map with a cotrol parameter i the form x +1 =x 1 x + u (13) Jackso [1991] determied its coverget regios ad basis of etraimet for a give periodic goal. Sychroizatio of chaos for this map is studied i the preset paper. I this special case, the cotrol law (3) leads to u = g +1 g e 1 g ( ) + e 1 g 8g e 1 g A (g x ) (14)

3 where g is a desired chaotic goal ad A < 1 To ivestigate the etraimet capability, oe sets x = g + ε (15) Isertig Eqs. (14) ad (15) ito Eq. (13) ad usig Taylor s expasio, oe obtais ε +1 = ε (A c ε )+o(ε 3 ) (16) A Ope-plus-Closed-Loop Approach to Sychroizatio 11 x where c =4 eg (3 4g )e g (17) Carryig a liearlized stability aalysis, oe fids out that Eq. (16) has a stable fixed poit for A < 1. Defie a Lyapuov fuctio for a geeral discrete dyamical system by Lasalle [1976] V (ε )=ε (18) The, V >, ad V = V (ε +1 ) V (ε ) =((A c ε ) 1)ε + o(ε5 ) (19) Thus, V < for x -g A 1 c <ε < 1+A c () Hece, the cotrol law (14) for the Gaussia map (13) has a basi of etraimet give by { BE G = x g + A 1 <x <g + 1+A } c c (1) for ay desired chaotic goal g. However, eve though the iitial states are iside the basi, the chaotic behavior may escape from the basi abruptly, because the basi of etraimet has ot bee prove to be ivariat. The problem of trackig a give chaotic orbit is cosidered here. As is well kow, chaos appears i the logistic map u g +1 =4.g (1 g ) () The cotrolled dyamical behavior, the differece betwee the cotrolled dyamical behavior ad the goal, ad the cotrol iput give by Eq. (14) are respectively depicted i Figs I this case, = ad A =.5. Fig. 1. Drivig the Gaussia map to track the logistic map. The cotrolled dyamical behavior. Differece betwee the output ad the goal. The cotrol iput.

4 1 L.-Q. Che & Y.-Z. Liu The chaotic behaviors of the Gaussia map for differet iitial coditios are sychroized. The cotrolled Gaussia map starts at x 1 =.5, ad the goal Gaussia map starts at x 1 =.1. The cotrolled map, the differece betwee the cotrolled map ad the goal, ad the cotrol iput give by Eq. (14) are respectively depicted i Figs.. I this case, = ad A =.5. x 4. Sychroizatio of a Hyperchaotic Map The two-dimesioal map x +1 =1 a(x + y ) y +1 = abx y (3) ca describe the Zagzag patter i coupled map lattices. Numerical results [Liu et al., 1999] idicate that for suitable parameters (for istace, a = 1.95 ad b =1.6), the map has a spreadig hyperchaotic attractor with two positive Lyapuov expoets. Here, cosider the case with cotrol iputs: x +1 =1 a(x + y )+u y +1 = abx y + v (4) The cotrol law (3) for the desired goal (g x,g y )is x -g - u = g x +1 1+a(gx + g y ) (A 1 ag x )(x g x ) +ag y (y g y ) v = g y +1 +abgx g y +abg y (x g x ) (A abg x )(y g y ) (5) where oe chooses the matrix A =diag[a 1,A ], ad A 1 < 1, A < 1. The cotrol law (14) for the hyperchaotic map (4) has a basi of etraimet [Che, 1] u BE H = {(x,y ) A (x g x ) ab(x g x ), A 1 (x g x ) a((x g x ) +(y g y ) ) } (6) Now, the hyperchaotic map is drive to track aother chaotic map, the Héo map g x +1 =1+g y 1.4g x, g y +1 =.3gx (7) - Fig.. Sychroizig the Gaussia map for differet iitial coditios. The cotrolled dyamical behavior. Differece betwee the cotrolled map ad the goal. The cotrol iput.

5 A Ope-plus-Closed-Loop Approach to Sychroizatio x y x -g x y -g y u v - Fig. 3. Drivig the hyperchaotic map to track the Héo map. The cotrolled dyamical behavior. Differece betwee the output ad the goal. The cotrol iput.

6 14 L.-Q. Che & Y.-Z. Liu x y x -g x y -g y u v Fig. 4. Sychroizig the hyperchaotic map for differet iitial coditios. The cotrolled dyamical behavior. Differece betwee the output ad the goal. The cotrol iput.

7 A Ope-plus-Closed-Loop Approach to Sychroizatio 15 I the case that a =1.95, b =1.6, A 1 = A =.9, =, the results obtaied by applyig the cotrol law (5) are show i Fig. 3. The chaotic behaviors of the hyperchaotic map for differet iitial coditios are sychroized. The cotrolled hyperchaotic map, i which a = 1.95, b =1.6, starts at (.8,.1), ad the goal is the same map startig at (.5,.). The cotrolled map, the differece betwee the cotrolled map ad the goal, ad the cotrol iput give by Eq. (14) are respectively depicted i Figs I this case, = ad A 1 = A = Coclusios A ew ad powerful cotrol method, the opeplus-closed-loop approach, has bee employed to sychroize chaotic ad hyperchaotic maps i this paper. Two umerical examples, the Gaussia map ad a hyperchaotic map, are ivestigated to demostrate the effectiveess of the strategy. The results show that sychroizatio of discrete chaos ca be realized if the cotrol is activated i the basi of etraimet. Two chaos sychroizatio problems, amely, to drive a chaotic map to desired chaotic dyamics ad to idetify a chaotic map startig from differet iitial coditios, have bee studied. I the first case, the cotrol iput may be large if the goal dyamics is differet from the origial system without cotrol. I the secod case, the goal dyamics is the dyamics of the same system, so the cotrol ca be arbitrarily small, if a log-time trasitio to achieve the goal is acceptable. I fact, it is the problem of stabilizig ustable chaos. It has bee proved i [Che, 1] that the ope-plus-closed-loop cotrol law is robust. That is, oe ca desig the ope-plus-closed-loop cotrol law based o the model system, to achieve good cotrol results for the real system, if the model errors are small. Ackowledgmets The research is supported by the Natioal Natural Sciece Foudatio of Chia (Project No. 183) ad Shaghai Sciece ad Techology Developmet Foudatio of Higher Educatio Istitutes. Refereces Che, G. & Dog, X. [1998] From Chaos to Order: Methodologies, Perspectives ad Applicatios (World Scietific, Sigapore). Che, L.-Q. & Liu, Y.-Z. [1998] The modified ope-plusclosed-loop cotrol of chaos i oliear oscillatios, Phys. Lett. A45, Che, L.-Q. [1] A ope-plus-closed-loop cotrol for discrete chaos ad hyperchaos, Phys. Lett. A81, Grosu, I. [1997] Robust sychroizatio, Phys. Rev. E56, Jackso, E. A. [1991] O the cotrol of complex dyamic systems, Physica D5, Jackso, E. A. [1995] OPCL migratio cotrols betwee five attractors of the Chua systems, It. J. Bifurcatio ad Chaos 5, Jackso, E. A. & Grosu, I. [1995a] A ope-plus-closedloop (OPCL) cotrol of complex dyamic systems, Physica D85, 1 9. Jackso, E. A. & Grosu, I. [1995b] Toward experimetal implemetatios of migratio cotrol, It. J. Bifurcatio ad Chaos 5, Jackso, E. A. [1997] The OPCL cotrol method for etraimet, model-resoace, ad migratio actios o multiple-attractor systems, Chaos 7, Lasalle, J. P. [1976] The Stability of Dyamical Systems (SIAM, Philadelphia). Liu, Z., Che, L. & Yag, L. [1999] O properties of hyperchaos: Case study, Acta Mech. Siica 15, Ott, E., Sauer, T. & Yorke, J. A. [1994] Copig with Chaos (Joh Wiley, NY). Weigel, R. & Jackso, E. A. [1998] Experimetal implemetatio of migratios i multiple-attractor systems, It. J. Bifurcatio ad Chaos 8,