Bicritical Behavior of Period Doublings in Unidirectionally-Coupled Maps ∗ Sang-Yoon Kim Department of Physics Kangwon National University Chunchon, Kangwon-Do 200-701, Korea We study the scaling behavior of period doublings in two unidirectionally-coupled one- dimensional maps near a bicritical point where two critical lines of period-doubling transition to chaos in both subsystems meet. Note that the bicritical point corresponds to a border of chaos in both subsystems. For this bicritical case, the second response subsystem exhibits a new type of non-Feigenbaum critical behavior, while the first drive subsystem is in the Feigenbaum critical state. Using two different methods, we make the renormalization group analysis of the bicritical behavior and find the corresponding fixed point of the renormalization transformation with two 9 relevant eigenvalues. The scaling factors obtained by the renormalization group analysis agree well 9 with those obtained by a direct numerical method. 9 1 PACS numbers: 05.45.+b, 03.20.+i, 05.70.Jk n a J I. INTRODUCTION [4]. In this paper, using two different methods, we also 6 make the RG analysis of the bicriticality, the results of which agree well with those of previous works. Period-doubling transition to chaos has been ex- 1 This paper is organizedas follows. In Sec. II we study tensively studied in a one-parameter family of one- v the scaling behavior near a bicritical point (A ,B ), cor- 3 dimensional (1D) unimodal maps, c c responding to a border of chaos in both subsystems, by 0 0 x =1 Ax2, (1) directly following a period-doubling sequence converging 1 t+1 − t to the point (Ac,Bc) for a fixed value of C. For this 0 where x is a state variable at a discrete time t. As the bicriticalcase,anewtypeofnon-Feigenbaumcriticalbe- t 9 controlparameterA is increased,the 1D map undergoes havior appears in the second subsystem, while the first 9 aninfinitesequenceofperiod-doublingbifurcationsaccu- subsystemisintheFeigenbaumcriticalstate. Employing / n mulating at a critical point A , beyond which chaos sets twodifferentmethods,wemaketheRGanalysisofthebi- c y in. Usingarenormalizationgroup(RG)method, Feigen- critical behavior in Sec. III. To solve the RG fixed-point d baum [1] has discovered universal scaling behavior near equation,wefirstuseanapproximatetruncationmethod - o the critical point Ac. [5],correspondingtothelowest-orderpolynomialapprox- a Hereweareinterestedintheperioddoublingsinasys- imation. Thus we analytically obtain the fixed point, h tem consisting of two 1Dmaps with a one-waycoupling, associated with the bicritical behavior, and its relevant c eigenvalues. Compared with the previous numerical re- : v x =1 Ax2, y =1 By2 Cx2, (2) sults [4],these analytic resultsarenotbadasthe lowest- i t+1 − t t+1 − t − t order approximation. To improve accuracy, we also em- X where x and y are state variables of the first and sec- ploythe“eigenvalue-matching”RGmethod[6],equating ar ond subsystems, A and B are control parameters of the thestabilitymultipliersoftheorbitofleveln(period2n) subsystems, and C is a coupling parameter. Note that to those of the orbitof the next leveln+1. Thus we nu- the first (drive)subsystem acts on the second(response) merically obtain the bicritical point, the parameter and subsystem, while the second subsystem does not influ- orbitalscaling factors, and the critical stability multipli- ence the first subsystem. This kind of unidirectionally- ers. We note that the accuracy is improved remarkably coupled1Dmapsareusedasamodelforopenflow[2]. A with increasing the level n. Finally, a summary is given new kind of non-Feigenbaumscaling behavior was found in Sec. IV. in a numerical and empirical way near a bicritical point (A ,B ) wheretwocriticallines ofperiod-doublingtran- c c sition to chaos in both subsystems meet [3]. For this II. SCALING BEHAVIOR NEAR THE bicriticalcase,a RGanalysiswasalsodevelopedandthe BICRITICAL POINT correspondingfixedpoint,governingthebicriticalbehav- ior,wasnumerically obtainedby directly solvingthe RG In this section we fix the value of the coupling pa- fixed-point equation using a polynomial approximation rameter by setting C = 0.45 and directly follow a ∗ Electronic address: [email protected] 1 period-doubling sequence converging to the bicritical firstsubsystemexhibitsasequenceofperiod-doublingbi- point (A ,B ), which corresponds to a border of chaos furcations at the vertical straight lines, where λ = 1. c c 1 − in both subsystems. For this bicritical case, the second For small values of the parameter B, the period of os- subsystemexhibitsanewtypeofnon-Feigenbaumcritical cillation in the second subsystem is the same as that in behavior, while the first subsystem is in the Feigenbaum the first subsystem, as in the case of forced oscillation. critical state. As B is increased for a fixed value of A, a sequence of The unidirectionally-coupled 1D maps (2) has many period-doublingbifurcationsoccursinthesecondsubsys- attractors for fixed values of the parameters [7]. For the tem when crossingthe non-verticallines where λ2 = 1. − caseC =0,itbreaksupintothetwouncoupled1Dmaps. The numbers inside the different regions denote the pe- Iftheybothhavestableorbitsofperiod2k,thenthecom- riod of the oscillation in the second subsystem. posite systemhas 2k different stable states distinguished We consider a pair of the parameters (An,Bn), at by the phase shift between the subsystems. This mul- which the periodic orbit of level n (period 2n) has the tistability is preserved when the coupling is introduced, stability multipliers λ1,n = λ2,n = 1. Hence, the point − at least while its value is small enough. Here we study (An,Bn)correspondstoathresholdofinstabilityinboth onlytheattractorswhosebasinsincludetheorigin(0,0). subsystems. Someofsuchpointsaredenotedbytheopen Suchattractorsbecomein-phasewhenA=BandC =0. circles in Fig. 1. Then such a sequence of (An,Bn) con- Stability ofanorbitwith periodq isdetermined byits verges to the bicritical point (Ac,Bc), corresponding to stability multipliers, a border of chaos in both systems, with increasing the leveln. Thebicriticalpointisdenotedbythe solidcircle q q in Fig. 1. To locate the bicritical point with a satisfac- λ1 = 2Axt, λ2 = 2Byt. (3) toryprecision,we numericallyfollowthe orbitsofperiod − − tY=1 tY=1 q = 2n up to level n = 21 in a quadruple precision, and obtain the sequences of both the parameters (A ,B ) n n Here λ1 and λ2 determine the stability of the first and andthe orbitpoints (xn,yn) approachingthe origin. We secondsubsystems,respectively. Anorbitbecomesstable firstnotethatthesequencesofA andx inthefirstsub- n n when the moduli of both multipliers are less than unity, systemare the same asthose inthe 1D maps [1]. Hence, i.e., −1<λi <1 for i=1,2. onlythesequencesofBn andyn inthesecondsubsystem are given in Table I. TABLEI. Sequencesoftheparameterandtheorbitpoint, {Bn} and {yn}, in thesecond subsystem. 1.15 16 n Bn yn 10 1.090088955364 5.019189×10−3 11 1.090092109910 -3.333775×10−3 12 1.090093416851 2.214467×10−3 16 13 1.090093959979 -1.471024×10−3 1.10 8 14 1.090094186392 9.771970×10−4 B 15 1.090094280906 -6.491561×10−4 16 1.090094320376 4.312391×10−4 17 1.090094336865 -2.864762×10−4 8 16 18 1.090094343755 1.903092×10−4 1.05 4 19 1.090094346634 -1.264245×10−4 20 1.090094347837 8.398518×10−5 21 1.090094348340 -5.579230×10−5 1.36 1.38 1.40 A We now study the asymptotic scaling behavior of the FIG. 1. Stability diagram of the periodic orbits born via period-doubling sequences in both subsystems near the period-doubling bifurcations for C = 0.45. The numbers in bicritical point. The scaling behavior in the first sub- thedifferentregionsrepresenttheperiodofmotioninthesec- system is obviously the same as that in the 1D maps ond subsystem. The open circle also denotes the point, cor- [1]. Thatis,thesequences A and x accumulateto responding to a threshold of instability in both subsystems, { n} { n} their limit values, A = A (= 1.401155189092 ) and where λ1 = −1 and λ2 = −1. Such open circles accumulate c ··· x=0, geometrically as follows: tothebicritical point,denoted bythesolid circle, which cor- responds to a border of chaos in both subsystems. For other A A δ−n, x α−n forlargen. (4) details, see thetext. n− c ∼ 1 n ∼ 1 The scaling factors δ and α are just the Feigenbaum 1 1 Figure 1 shows the stability diagram of periodic or- constantsδ (=4.669 )andα(= 2.502 )forthe1D ··· − ··· bits for C = 0.45. As the parameter A is increased, the maps, respectively. However, the second subsystem ex- 2 hibitsanon-Feigenbaumcriticalbehavior,unlikethecase the region in the small box (containing the origin) by ofthefirstsubsystem. Thetwosequences B and y the scaling factor α for the x axis and α for the y n n 1 2 { } { } also converge geometrically to their limit values B =B axis, and then we get the picture in Fig. 2(c). Note that c (= 1.090094348701)and y = 0, respectively, where the the picture in Fig. 2(c) reproduces the previous one in valueofB isobtainedusingthesuperconvergingmethod Fig. 2(a) approximately. Repeating the above procedure c [8]. Toobtaintheconvergenceratesofthetwosequences, once more, we obtain the two pictures in Figs. 2(d) and we define the scaling factors of level n: 2(e). Thatis,Fig.2(d) showsthe hyperchaoticattractor with σ 0.061 and σ 0.01 for A = A +∆A/δ2 and δ2,n ≡ BBnn−−1B−nB+n1, α2,n ≡ yynn−−1y−n+yn1. (5) Bwit=hBthc≃e+s∆caBli/nδg22.faMctaogr2sn≃iαfy21infogrththeerexg-iaoxnisicnanthdeαs22mfao1lrlbthoxe y-axis, we also obtain the picture in Fig. 2(e), which re- Thesetwosequences δ2,n and α2,n arelistedinTable produces the previous one in Fig. 2(c) with an increased { } { } II, and they converge to their limit values, accuracy. δ 2.3928, α 1.5053, (6) 2 2 ≃ ≃− respectively. Note that these scaling factors are com- 1.0 pletely different from those in the first subsystem (i.e., the Feigenbaum constants for the 1D maps). y 0.0 TABLEII. Sequencesoftheparameterandorbital scaling factors, {δ2,n} and {α2,n}, in thesecond subsystem. -1.0 (a) n δ2,n α2,n -1.0 0.0 1.0 x 10 2.4298 -1.5057331 11 2.4137 -1.5055154 12 2.4063 -1.5054281 1.0 1.0 13 2.3988 -1.5053753 14 2.3956 -1.5053440 y 0.0 y2 0.0 15 2.3946 -1.5053316 a 16 2.3937 -1.5053256 (b) (c) -1.0 -1.0 17 2.3931 -1.5053215 -1.0 0.0 1.0 -1.0 0.0 1.0 18 2.3930 -1.5053198 x a x 1 19 2.3929 -1.5053191 20 2.3928 -1.5053186 1.0 1.0 y 0.0 2y2 0.0 Forevidenceofscaling,wecomparethechaoticattrac- a tors, shown in Fig. 2, for the three values of (A,B) near (d) (e) -1.0 -1.0 the bicritical point (A ,B ). All these attractors are the c c -1.0 0.0 1.0 -1.0 0.0 1.0 hyperchaoticoneswithtwopositiveLyapunovexponents x a 2x [9], 1 FIG. 2. Hyperchaotic attractors for the three val- 1 m 1 m ues of (A,B) near the bicritical point (Ac.Bc); in (a) σ1 =ml→im∞m ln|2Axt|, σ2 =ml→im∞m ln|2Byt|. (A,B) = (Ac +∆A,Bc +∆B) (∆A = ∆B = 0.1), in (b) Xt=1 Xt=1 and (c) (A,B)=(Ac+∆A/δ1,Bc+∆B/δ2), and in (d)and (7) (e) (A,B)=(Ac+∆A/δ12,Bc+∆B/δ22). The picture in (c) is obtained by magnifying the region in the small box in (b) Here the first and second Lyapunov exponents σ1 and withthescalingfactorsα1forthex-axisandα2forthey-axis. σ denote the average exponential divergence rates of Similarly,wealsoobtainthepicutre(e)bymagnifyingthere- ne2arby orbits in the the first and second subsystems, re- gioninsidethesmallboxin(d)withthescalingfactorsα21 for spectively. Figure 2(a) shows the hyperchaotic attractor the x-axis and α22 for the y-axis. Comparing the pictures in (a), (c), and (e), one can see that each successive magnified with σ 0.242 and σ 0.04 for A = A +∆A and 1 2 c ≃ ≃ picture reproduces the previous one with an accuracy with B = B +∆B, where ∆A = ∆B = 0.1. This attrac- c thedepth of resolution. tor consists of two pieces. To see scaling, we first rescale ∆A and ∆B with the parameter scaling factors δ and 1 δ , respectively. The attractor for the rescaled parame- Sofarwehaveseenthescalingnearthebicriticalpoint, 2 ter values of A = A +∆A/δ and B = B +∆B/δ is and now turn to a discussion of the behavior exactly at c 1 c 2 shown in Fig. 2(b). It is also the hyperchaotic attrac- the bicritical point (A ,B ). There exist an infinity of c c tor with σ 0.121 and σ 0.02. We next magnify unstable periodic orbits with period 2n at the bicritical 1 2 ≃ ≃ 3 point. The orbit points x and y , approachingthe zero wherex andy arethe state variablesata discrete time n n t t in the first and second subsystems, vary asymptotically t in the first and second subsystems, respectively. Trun- in proportion to α−n and α−n, respectively. The stabil- cating the map (9) at its quadratic terms, we have 1 2 ity multipliers λ and λ of the orbits with period 2n 1,n 2,n a c e al∗so converge to the crit∗ical stability multipliers λ∗1 and TP : xt+1 = b +bx2t, yt+1 = d +dyt2+ dx2t, (10) λ , respectively. Here λ (= 1.601191 ) in the first 2 1 − ··· subsystem is just the critical stability multiplier for the which is a five-parameter family of unidirecionally- case of the 1D maps [1]. However, as listed in Table III, coupled maps. P represents the five parameters, i.e., the second subsystem has the different critical stability P = (a,b,c,d,e). The construction of Eq. (10) corre- multiplier, sponds to a truncation of the infinite dimensional space of unidirectionally-coupled maps to a five-dimensional ∗ λ2 =−1.17885···. (8) space. Theparametersa,b,c,d,andecanberegardedas thecoordinatesofthetruncatedspace. Wealsonotethat Consequently, the periodic orbits at the bicritical point this truncation method corresponds to the lowest-order ∗ ∗ have the same stability multipliers λ and λ for suffi- 1 2 polynomial approximation. ciently large n. We look for fixed points of the renormalization op- erator in the truncated five-dimensional space of TABLE III. Sequences of the second stability multipliers, R unidirectionally-coupled maps, {λ2,n} of the orbits with period 2n at the bicritical point. (T)=ΛT2Λ−1. (11) n λ2,n R 10 -1.178829 Here the rescaling operator Λ is given by 11 -1.178842 12 -1.178839 α 0 13 -1.178850 Λ= 1 , (12) 14 -1.178855 (cid:18) 0 α2 (cid:19) 15 -1.178854 whereα andα arethe rescalingfactorsinthe firstand 1 2 16 -1.178854 second subsystems, respectively. 17 -1.178855 The operation in the truncated space can be rep- 18 -1.178855 R resented by a transformation of parameters, i.e., a map 19 -1.178854 from P (a,b,c,d,e) to P′ (a′,b′,c′,d′,e′), ≡ ≡ a′ =2a2(1+a), (13a) 2 ′ b = ab, (13b) III. RENORMALIZATION GROUP ANALYSIS α1 OF THE BICRITICAL BEHAVIOR a2 c′ =2c(c+c2+e ), (13c) b2 Employing two different methods, we make the RG ′ 2 d = cd, (13d) analysis of the bicritical behavior. We first use the trun- α 2 cation method, and analytically obtain the correspond- 4 ′ e = ce(a+c). (13e) ing fixed point and its relevant eigenvalues. These ana- α2 1 lytic results are not bad as the lowest-order approxima- tion. To improve the accuracy, we also use the numeri- The fixed point P∗ = (a∗,b∗,c∗,d∗,e∗) of this map can caleigenvalue-matchingmethod,andobtainthebicritical be determinedbysolvingP′ =P. The parametersband point, the parameterand orbitalscaling factors, and the dsetonlythescalesinthexandy,respectively,andthus critical stability multipliers. Note that the accuracy in they are arbitrary. We now fix the scales in x and y by the numerical RG results is improved remarkably with setting b =d=1. Then, we have, from Eqs.(13a)-(13e), ∗ ∗ ∗ increasing the level n. fiveequationsforthefiveunknownsα ,a ,α ,c ,ande . 1 2 We thus find one solution, associated with the bicritical behavior,aswillbe seenbelow. The map(10)withaso- A. Truncation Method lution P∗ (TP∗) is the fixed map of the renormalization ∗ transformation ; forbrevityTP∗ willbe denotedasT . R We first note that Eqs. (13a)-(13b) are only for the In this subsection, employing the truncation method ∗ unknowns α and a . We find one solution for α and [5], we analytically make the RG analysis of the bicriti- 1 1 ∗ a ,associatedwiththeperiod-doublingbifurcationinthe calbehaviorintheunidirectionally-coupledmapTofthe first subsystem, form, α T : xt+1 =f(xt), yt+1 =g(xt,yt), (9) α1 =−1−√3=−2.732···, a∗ = 21. (14) 4 ∗ Substituting the values for α and a into Eqs. (13c)- The 5 5 Jacobian matrix J has a semi-block form, 1 ∗ ∗ × (13e), we obtain one solution for α , c , and e , associ- because we are considering the unidirectionally-coupled 2 ated with the bicriticality, case. Therefore, one can easily obtain its eigenvalues. The first two eigenvalues, associated with the first sub- 1 α = (1+√3 √5 √15)= 1.688 , (15a) system, are those of the following 2 2 matrix, 2 2 − − − ··· × c∗ = α22, e∗ =1+ 12(√15−3√3)=0.338···. (15b) M1 = ∂∂((aa′,,bb′))(cid:12)(cid:12)P∗ =(cid:18)32−/αα11 10(cid:19). (17) (cid:12) Compared with the values, α1 = 2.502 and α2 = (cid:12) ′ − ··· Hence the twoeigenvaluesofM , δ andδ ,aregivenby 1.505 , obtained by a direct numerical method, the 1 1 1 − ··· analyitc results for α and α , given in Eqs. (14) and 1 2 δ =4+√3=5.732 , δ′ =1. (18) (15a), are not bad as the lowest-order approximation. 1 ··· 1 Consider an infinitesimal perturbation ǫδP to a fixed pointP∗ofthetransformationofparameters(13a)-(13e). Here the relevant eigenvalue δ1 is associated with the Linearizing the transformation at P∗, we obtain the scaling of the control parameter in the first subsystem, ′ equation for the evolution of δP, while the marginal eigenvalue δ1 is associated with the scale change in x. When compared with the numerical δP′ =JδP, (16) value,δ1 (=4.669 ),the analyticresultforδ1,givenin ··· Eq. (18), is not bad as the lowest-orderapproximation. where J is the Jacobian matrix of the transformation at The remaining three eigenvalues, associated with the P∗. secondsubsystem,arethoseofthefollowing3 3matrix, × ′ ′ ′ 4c∗+6c∗2+2a∗2e∗ 0 2a∗2c∗ ∂(c,d,e) ∗ M2 = ∂(c,d,e) (cid:12)(cid:12)P∗ = 4e∗(a∗2+/α22c∗)/α2 2c 0/α2 4c∗(a∗+0 c∗)/α2 . (19) (cid:12)  1 1  (cid:12) The three eigenvalues of M , δ , δ′, and δ′′, are given the orbits with period 2n (n=0,1,2,...) have the same 2 2 2 2 by stability multipliers, which are just the critical stability ∗ ∗ multipliers λ and λ . Thatis, the criticalstability mul- 1 2 δ2 =(u+√v)/2=3.0246 , (20a) tipliers have the values of the stability multipliers of the δ′ =(u √v)/2=0.1379···, δ′′ =1, (20b) fixed point (x∗,y∗) of the fixed map T∗, 2 − ··· 2 ∗ ∗ ∗ ∗ λ =2x = 1.5424 , λ =2y = 0.8899 , (22) where 1 − ··· 2 − ··· where u=(17+7√3 5√5 3√15)/2, (21a) − − v =104+53√3 44√5 23√15. (21b) x∗ =(1 3+2√3)/2, y∗ =(1 √w)/2, (23a) − − −q − Thefirsteigenvalueδ isarelevanteigenvalue,associated w =5+3√3 2√5 √15 2 − − with the scaling of the control parameter in the second + 3+2√3(2 3√3+√15). (23b) ′ subsystem, the second eigenvalue δ is an irrelavantone, q − 2 and the third eigenvalue δ′′ is a marginal eigenvalue, as- We also note that the analytic values for λ∗ and λ∗ are 2 1 2 sociated with the scale change in y. We also compare not bad, when compared with their numerical values. the analytic result for δ , given in Eq. (20a), with the 2 numerical value (δ 2.3928) in Eq. (6), and find that 2 ≃ the analytic one is not bad as the lowest-order approxi- B. Eigenvalue-Matching Method mation. As shown in Sec. II, stability multipliers of an orbit Inthis subsection,we employthe eigenvalue-matching with period 2n at the bicritical point converge to the method [6]and numericallymake the RG analysisof the ∗ ∗ critical stability multipliers, λ1 (= −1.601···) and λ2 bicriticalbehaviorintheunidirectionally-coupledmapT (= 1.178 ) as n . We now obtain these critical of Eq. (2). As the level n increases, the accuracy in the − ··· → ∞ stability analytically. The invariance of the fixed map numerical RG results are remarkably improved. T∗ under the renormalization transformation implies The basic idea is to associate a value (A′,B′) for that, if T∗ has a periodic point (x,y) with pReriod 2n, each value (A,B) such that T(n+1) locally resembles thenΛ−1(x,y)isaperiodicpointofT∗ withperiod2n+1. (A′,B′) Sincerescalingdoesnotaffectthestabilitymultipliers,all T(n) , where T(n) is the 2nth-iterated map of T (i.e., (A,B) 5 T(n) = T2n). A simple way to implement this idea is to the analytic formulas for the eigenvalues δ and δ of 1,n 2,n linearizethemapsintheneighborhoodoftheirrespective the matrix ∆ , n fixed points and equate the corresponding eigenvalues. anLde2tn{+z1t,}raenspde{cztit′v}eblye,tiw.eo.,successivecyclesofperiod2n δ = dλd1A,n′+1(cid:12)∗, (31a) 1,n dλ1,n (cid:12)(cid:12) zt =T((An,)B)(zt), zt′ =T((An′+,B1)′)(zt′); zt =(xt,yt). (24) dA (cid:12)(cid:12)∗ ∂λ2,n+1(cid:12) Here xt depends only on A, but yt is dependent on both δ = ∂B′ (cid:12)∗. (31b) A and B, i.e., xt =xt(A) and yt =yt(A,B). Then their 2,n ∂λ2,n (cid:12)(cid:12) linearized maps at z and z′ are given by ∂B (cid:12)∗ t t (cid:12) (cid:12) 2n As n δ1,n and δ2,n approach δ1 and δ2, which are DT(n) = DT (z ), (25a) just th→e p∞arameterscaling factorsin the firstand second (A,B) (A,B) t tY=1 subsystems, respectively. Note also that as in the 1D 2n+1 case, the local rescaling factors of the state variables are DT((An′+,B1)′) = DT(A′,B′)(zt′). (25b) simply given by tY=1 dx δ 1,n α = = , (32a) (Here DT is the linearized map of T.) Let 1,n dx′(cid:12) t (cid:12)∗ 1,n their eigenvalues, called the stability multipliers, be (cid:12) (λ1,n(A),λ2,n(A,B)) and (λ1,n+1(A′),λ2,n+1(A′,B′)). α2,n = ddyy′(cid:12)(cid:12) = δt2,n, (32b) The recurrence relations for the old and new parame- (cid:12)∗ 2,n ters are then given by equating the stability multipliers (cid:12) (cid:12) where of level n, λ (A) and λ (A,B), to those of the next 1,n 2,n ′ ′ ′ level n+1, λ (A) and λ (A,B ), i.e., 1,n+1 2,n+1 dx′ ∂y′ λ (A)=λ (A′), (26a) t = dA′(cid:12)∗, t = ∂B′(cid:12)∗. (33) λ2,n(1A,n,B)=λ21,,nn++11(A′,B′). (26b) 1,n ddAx(cid:12)(cid:12)(cid:12)∗ 2,n ∂∂By(cid:12)(cid:12)(cid:12)∗ (cid:12) (cid:12) (cid:12) Here α and α also converge to the orbital scaling ∗ ∗ 1,n 2,n The fixed point (A ,B ) of the renormalizationtrans- factors, α and α , in the first and second subsystems, 1 2 formation (26), respectively. ∗ ∗ We numericallyfollowthe orbits withperiod2n inthe λ (A )=λ (A ), (27a) 1,n 1,n+1 unidirectionally-coupledmapsT ofEq.(2)andmakethe ∗ ∗ ∗ ∗ λ (A ,B )=λ (A ,B ). (27b) 2,n 2,n+1 RG analysis of the bicritical behavior. Some results for the intermediate level n are shownin Fig. 3. Figure 3(a) gives the bicritical point (A ,B ). By linearizing the c c shows the plots of the first stability multiplier λ (A) renormalization transformation (26) at the fixed point 1,n ∗ ∗ versus A for the cases n = 6,7. We note that the in- (A ,B ), we have tersection point, denoted by the solid circle, of the two (cid:18)∆∆BA (cid:19) =(cid:18) ∂∂∂∂AABA′′(cid:12)(cid:12)∗∗ ∂∂∂∂BBBA′′(cid:12)(cid:12)∗∗ (cid:19)(cid:18)∆∆BA′′ (cid:19) (28) scwtuharebvrieelsitAyλ1∗6m,6aunaldntidpλl∗1λie,16r,,7arrgeeisvtpehesecttchivreietlpiyco,ailnintpot(hAine∗6t,fiaλrn∗1s,dt6)stuhobefslcyervsitteeilcm6a.,l (cid:12) ′ (cid:12) =∆n(cid:18)∆∆(cid:12) BA′ (cid:19),(cid:12) (29) AvaslutheselAev∗elanndinλc∗re,arseessp,eAct∗nivaenldy.λ∗1N,notaepparlsooacthhathtetirhelimrait- 1 where ∆A = A A∗, ∆B = B B∗, ∆A′ = A′ A∗, tio of the slopes of the curves, λ1,6(A) and λ1,7(A), for ∗ ∆B′ =B′ B∗,−and − − A = A6 gives the parameter scaling factor δ1,6 of level − 6 in the first subsystem. Similarly, Fig. 3(b) shows the ∆n =Γ−n1Γn+1; (30a) plotsofthesecondstabilitymultiplierλ2,n(A∗6,B)versus B for the casesn=6,7. The intersectionpoint, denoted dλ1,n 0 ∗ Γn = ∂λ∂dAA2,n(cid:12)(cid:12)(cid:12)(cid:12)∗∗ ∂∂λB2,n(cid:12)∗ , (30b) aλBl26s∗,o7a(bAnyd∗6,tλhB∗2e),6sgoailrviedestchitrehclecer,pitooifcinathltep(oBtwin6∗ot,λca∗2un,r6dv)etoshfeλ2lce,r6vi(etAilc6a6,l,Bswt)ahabenirlde- (cid:12) (cid:12) dλ1,n+(cid:12)1 (cid:12)0 ity multiplier, respectively, in the second subsystem. As Γn+1 = ∂λ∂d2AA,n′′+1(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)∗∗ ∂λ∂2B,n′+1(cid:12)(cid:12)∗ . (30c) tlsihymsetitelemvvae,lltuhneesi,nrBactr∗ieoaasnoefds,tλhB∗2e,n∗srleaosnppdeescλto∗2if,vnethlayel.scoAucsrovinnevst,ehrλeg2e,fi6rt(osAt∗6tsh,uBebir)- Here Γ−1 is the inverse o(cid:12)f Γ and t(cid:12)he asterisk denotes and λ (A∗,B), for B =B∗ gives the parameter scaling n n 2,7 6 6 ∗ ∗ the fixed point (A ,B ). After some algebra, we obtain factor δ of level 6 in the second subsystem. 2,6 6 With increasing the level up to n = 15, we numer- ically make the RG analysis of the bicritical behavior. We first solve Eq. (27) and obtain the bicritical point ∗ ∗ (A ,B )oflevelnandthepairofcriticalstabilitymulti- n n ∗ ∗ pliers(λ ,λ )ofleveln. Next,weusethe formulasof n=7 1,n 2,n -1.4 Eqs.(31) and(32) and obtainthe parameterand orbital scaling factors of level n, respectively. These numerical n=6 RG results for the first and secondsubsystems are listed 1,n *l1 iinntThaebnleusmIVeriacnadlRVG, rreesspuelctstiivserlye.mNaroktaebtlhyaitmtphreoavcecduwraicthy l the level n and their limit values agree well with those -1.8 obtained by a direct numerical method. (a) 1.40114 A* 1.40117 A n=7 -1.174 n=6 2,n *2 l l -1.184 (b) 1.09009 B* 1.09010 B FIG. 3. Plots of (a) the first stability multipliers λ1,n(A) ∗ versus A and (b) the second stability multipliers λ2,n(A6,B) versusB forthecasesn=6,7. In(a),theintersectionpoint, denoted by the solid circle, of the two curves λ1,6 and λ1,7 ∗ ∗ ∗ ∗ gives the point (A6,λ1,6) of level 6. As n → ∞, (An,λ1,n) ∗ ∗ converges to its limit point (A ,λ1). Similarly, in (b), the intersection point,denotedalsobythesolid circle,ofthetwo ∗ ∗ successive curves λ2,6(A6,B) and λ2,7(A6,B) gives thepoint ∗ ∗ ∗ ∗ (B6,λ2,6) of level 6. As n → ∞, (Bn,λ2,n) also approaches ∗ ∗ its limit point (B ,λ2). For otherdetails, see thetext. TABLE IV. Sequences of the critical point, the first critical stability multiplier, the parameter ∗ ∗ andorbitalscalingfactors,{An},{λ1,n},{δ1,n}and{α1,n},inthefirstsubsystem. Forcomparison, we also list theresults obtained by a direct numerical method in thelast row. n A∗n λ∗1,n δ1,n α1,n 6 1.4011551890889291 -1.6011912111212 4.6692030721 -2.5026204595 7 1.4011551890921332 -1.6011913425171 4.6692014285 -2.5028459883 8 1.4011551890920484 -1.6011913262887 4.6692016314 -2.5028946520 9 1.4011551890920507 -1.6011913282943 4.6692016063 -2.5029050377 10 1.4011551890920506 -1.6011913280464 4.6692016094 -2.5029072678 11 1.4011551890920506 -1.6011913280770 4.6692016091 -2.5029077449 12 1.4011551890920506 -1.6011913280732 4.6692016091 -2.5029078472 13 1.4011551890920506 -1.6011913280737 4.6692016091 -2.5029078691 14 1.4011551890920506 -1.6011913280736 4.6692016091 -2.5029078738 15 1.4011551890920506 -1.6011913280736 4.6692016091 -2.5029078748 1.4011551890920506 -1.6011913280736 4.6692016091 -2.5029078751 7 Foundation of China. I also thank Profs. Chen and Liu TABLEV. Sequencesofthecriticalpoint,thesecondcrit- and other members for their hospitality during my visit. icalstabilitymultiplier,theparameterandorbitalscalingfac- ∗ ∗ tors, {Bn}, {λ2,n}, {δ2,n} and {α2,n}, in the second subsys- tem. For comparison, we also list the results obtained by a direct numerical method in thelast row. ∗ ∗ n Bn λ2,n δ2,n α2,n 6 1.090092490313 -1.177467 2.39507 -1.502785 7 1.090094351702 -1.178671 2.39358 -1.503173 [1] M. J. Feigenbaum, J. Stat. Phys. 19, 25 (1978); 21, 669 8 1.090094321847 -1.178625 2.39359 -1.504426 (1979). 9 1.090094328376 -1.178649 2.39310 -1.504894 [2] Kaneko, Phys. Lett. A 111, 321 (1985); I. S. Aranson, 10 1.090094347652 -1.178820 2.39280 -1.504993 A. V. Gaponov-Grekhov and M. I. Rabinovich, Physica 11 1.090094348817 -1.178844 2.39281 -1.505163 D 33, 1 (1988); K. Kaneko, Physica D 68, 299 (1993); 12 1.090094348536 -1.178830 2.39278 -1.505263 F.H.Willeboordse andK.Kaneko,Phys.Rev.Lett.73, 13 1.090094348675 -1.178847 2.39274 -1.505280 533 (1994); F. H. Willeboordse and K. Kaneko, Physica 14 1.090094348704 -1.178856 2.39273 -1.505296 D 86, 428 (1995); O. Rudzick and A. Pikovsky, Phys. 15 1.090094348701 -1.178853 2.39273 -1.505311 Rev. E 54, 5107 (1996); J. H. Xiao, G. Hu and Z. Qu, 1.090094348701 -1.17885 2.3927 -1.505318 Phys. Rev. Lett. 77, 4162 (1996); Y. Zhang, G. Hu and L. Gammaiton, Phys.Rev.E 58, 2952 (1998). [3] B. P. Bezruchko, V. Yu. Gulyaev, S. P. Kuznetsov, and E. P.Seleznev, Sov. Phys.Dokl. 31, 268 (1986). [4] A. P. Kuznetsov, S. P. Kuznetsov and I. R. Sataev, Int. IV. SUMMARY J. Bifurcation and Chaos 1, 839 (1991). [5] J.-M.MaoandJ.Greene,Phys.Rev.A35,3911(1987); We have studied the scaling behavior of period dou- S.-Y. Kim and H.Kook, Phys. Lett.A 178, 258 (1993). blingsnearthebicriticalpoint,correspondingtoathresh- [6] B. Derrida, A. Gervois and Y. Pomeau, J. Phys. A 12, oldofchaosinbothsubsystems. Forthisbicriticalcase,a 269 (1979); B. Derrida and Y. Pomeau, Phys. Lett. A newtype ofnon-Feigenbaumcriticalbehaviorappearsin 80, 217 (1980); B. Hu and J. M. Mao, Phys. Rev.A 25, the second (response) subsystem, while the first (drive) 1196 (1982); 25, 3259 (1982); Phys. Lett. A 108, 305 (1985); J.M.MaoandR.H.G.Helleman,Phys.Rev.A subsystem is in the Feigenbaum critical state. Employ- 35,1847(1987);S.-Y.KimandB.Hu,Phys.Rev.A41, ingthe truncationandeigenvalue-matchingmethods, we 5431 (1990). made the RG analysis of the bicritical behavior. For [7] B.P.Bezruhko,Yu.V.Gulyaev,O.B.Pudovochkin,and the case of the truncation method, we analytically ob- E. P.Seleznev, Sov. Phys.Dokl. 35, 807 (1991). tained the fixed point, associated with the bicritical be- [8] R.S.MacKay,Ph.D.thesis,PrincetonUniversity,1982. havior, and its relevant eigenvalues. These analytic RG See Eqs. 3.1.2.12 and 3.1.2.13. results are not bad as the lowest-order approximation. [9] O. E. R¨ossler, Phys. Lett. A 71, 155 (1979). To improve the accuracy, we also employed the numer- [10] S.-Y. Kim and W.C. Lim (tobe published). ical eigenvalue-matching RG method, and obtained the bicriticalpoint,theparameterandorbitalscalingfactors, andthe criticalstabilitymultipliers. Note thatthe accu- racyinthe numericalRGresultsisimprovedremarkably with increasing the level n. Consequently, these numer- ical RG results agree well with the results obtained by a direct numerical method. The results on the bicriti- cal behavior in the abstract system of unidirectionally- coupled 1D maps are also confirmed in the real system of unidirectionally-coupled oscillators [10]. ACKNOWLEDGMENTS ThisworkwassupportedbytheKoreaResearchFoun- dation under Project No. 1998-015-D00065 and by the BiomedlabInc. Somepartofthismanuscriptwaswritten during my visit to the Center of Nonliner Studies in the Institute of Applied Physics and Computational Mathe- matics, China, supported by the Korea Science and En- gineering Foundation and the National Natural Science 8