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One-dimensional map lattices: Synchronization, bifurcations, and chaotic structures
Belykh, Vladimir N.; Mosekilde, Erik
Published in: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics
Link to article, DOI: 10.1103/PhysRevE.54.3196
Publication date: 1996
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Citation (APA): Belykh, V. N., & Mosekilde, E. (1996). One-dimensional map lattices: Synchronization, bifurcations, and chaotic structures. Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 54(4), 3196-3203. https://doi.org/10.1103/PhysRevE.54.3196
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PHYSICAL REVIEW E VOLUME 54, NUMBER 4 OCTOBER 1996
One-dimensional map lattices: Synchronization, bifurcations, and chaotic structures
Vladimir N. Belykh Mathematics Department, Volga State Academy of Water Transportation, 5 Nesterov Street, 603600 Nizhny Novgorod, Russia
Erik Mosekilde Physics Department, Center for Chaos and Turbulence Studies, The Technical University of Denmark, DK-2800 Lyngby, Denmark ͑Received 17 October 1995; revised manuscript received 1 May 1996͒ The paper presents a qualitative analysis of coupled map lattices ͑CMLs͒ for the case of arbitrary nonlin- earity of the local map and with space-shift as well as diffusion coupling. The effect of synchronization where, independently of the initial conditions, all elements of a CML acquire uniform dynamics is investigated and stable chaotic time behaviors, steady structures, and traveling waves are described. Finally, the bifurcations occurring under the transition from spatiotemporal chaos to chaotic synchronization and the peculiarities of CMLs with specific symmetries are discussed. ͓S1063-651X͑96͒05309-3͔
PACS number͑s͒: 05.45.ϩb, 05.90.ϩm, 87.10.ϩe
I. INTRODUCTION probabilities of different durations of the laminar phases along the chain. Near the forced site, this probability was As simplified models of spatially extended systems under found to decay exponentially. For sites further away, how- nonequilibrium conditions, the dynamics of coupled map lat- ever, the exponential decay was replaced by a power law tices ͑CMLs͒, i.e., of systems with discrete time, discrete with an exponent of Ϫ3/2. space, and a continuous state, has attracted a rapidly growing In order to examine their thermodynamic properties, interest in recent years ͓1–6͔. Computer simulations have Bourzutschky and Cross ͓6͔ have considered the long- revealed a variety of behaviors from the very simple to the wavelength limit of the behavior of simple CMLs. Using a very complex and the core problem of the transition from generalization of the fluctuation-dissipation theorem, they one-dimensional chaos associated with the temporal behav- have tried to define a temperature that could be useful in the ior of the local element to multidimensional spatiotemporal description of spatially extended, nonequilibrium systems. chaos in the coupled map lattice has been elucidated for dif- Such a temperature could provide constraints, for example, ferent maps and different types of coupling. on the effective noise term in the corresponding Langevin In particular, Kaneko ͓1͔ has investigated the develop- equation. ment of spatiotemporal intermittency in the form of a lami- The purpose of the present paper is to proceed with a nar motion interrupted by bursts. This study was performed more analytic approach to the description of coupled map with a class of coupled map lattices for which the individual lattices. Previous contributions in this direction are due, for map was close to a transition to temporal intermittency and instance, to Afraimovich and Nekorkin ͓7͔, who analyzed the the observed geometric structures in space-time resembled stability of chaotic waves propagating in a discrete chain of the structures found in cellular automata. As the coupling diffusively coupled maps, and to Kuznetsov ͓8͔, who applied was increased, the number of positive Lyapunov exponents renormalization-group theory to study universality and scal- also increased and a kind of fully developed turbulence ap- ing in CML dynamics. The scaling behavior of coupled map peared. Under certain conditions, localized chaos was ob- lattices has also been studied by Kook, Ling, and Schmidt served, i.e., the burst regions were confined to specific areas ͓9͔, by Kaspar and Shuster ͓10͔, and by Bohr and Chris- and could not propagate throughout the whole space. Kaneko tensen ͓11͔. ͓2͔ has also studied the information flow in coupled map Amritkar et al. ͓12͔ have investigated the stability of spa- lattices and has introduced the concept of comoving tially and temporally periodic orbits in one- and two- Lyapunov exponents to characterize convective instabilities dimensional coupled map lattices. Using the fact that the in open flow systems. More recently, Kaneko ͓3͔ has studied stability matrices for such solutions are block circulant and the dynamics of globally coupled maps and developed a hence can be brought onto a block diagonal form through a mean-field description of the fluctuations in such systems. unitary transformation, they derive conditions for the stabil- Willeboordse ͓4͔ has studied the problem of pattern selec- ity of periodic solutions in terms of the criteria for smaller tion in chains of diffusively coupled logistic maps. Besides orbits. Druzhinin and Mikhailov ͓13͔ have considered a par- globally incoherent patterns in the form of frozen random ticular type of CML where the coupling affects only the bi- lattices or spatiotemporal intermittency, slowly moving co- furcation parameter of the local maps. In the continuum limit herent structures were observed and Willeboordse proposes a this corresponds to a reaction-diffusion equation for which scheme to encode standing as well as traveling waves into the diffusion coefficient depends on the state of the system. the CML dynamics. Xie, Hu, and Qu ͓5͔ have investigated They show that the formation of stable solitonlike patterns is so-called on-off intermittency for a coupled map lattice op- possible and, assuming the local map to be logistic, they erating near a spatiotemporal period-2 solution. When apply- obtain an expression for the solution close to the period- ing noise at a single site, they have determined the relative doubling bifurcation where the fixed point turns unstable. In
1063-651X/96/54͑4͒/3196͑8͒/$10.0054 3196 © 1996 The American Physical Society 54 ONE-DIMENSIONAL MAP LATTICES: SYNCHRONIZATION... 3197 a detailed numerical study of bounded one- and two- stable chaotic time behaviors, steady structures, and traveling dimensional CML with diffusively coupled logistic maps, waves are studied by applying the detailed rigorous analysis Giberti and Vernia ͓14͔ have followed the change in the of an associated two-dimensional map proposed by Belykh steady states as the coupling parameter is increased. Ascrib- et al. ͓17͔. In contrast to Afraimovich and Nekorkin ͓7͔,we ing to the single map a bifurcation parameter slightly above prove that the coordinates of traveling chaotic waves need the Feigenbaum accumulation point, the dynamics of the un- not be confined to some particular states of the lattice ele- coupled lattice is totally chaotic. As the coupling parameter ments. For weak coupling, the ability of the map T to pro- is increased, more and more periodic orbits of period 2k start duce an extremely complex dynamics as a result of the cha- to arise, with the orbits with longer periods appearing first. otic time behavior of the local maps is discussed in Sec. III. Above a certain coupling strength, almost all randomly cho- There we also state Theorem 5 on the bifurcation set corre- sen initial conditions lead to a stable period orbit. By means sponding to the disappearance of a complex limiting set un- of continuation techniques, Giberti and Vernia ͓14͔ also der the transition to synchronization. Finally, we discuss the show how the stable periodic orbits develop from the un- symmetric solutions to the map T in the case of pure diffu- stable periodic orbits of the uncoupled system. sive coupling. Giacomelli, Lepri, and Politi ͓15͔ have examined the sta- tistical properties of the bidimensional patterns generated II. SYNCHRONIZATION from delayed and extended maps and Losson and Mackey IN COUPLED ONE-DIMENSIONAL MAPS ͓16͔ have considered coupled map lattices as models of de- terministic and stochastic differential-delay equations. By ro- Consider the diffusively coupled 1D map array T given ϩ tating the time-space reference frame, a delayed map can be by ͑1͒ with xi,xi( j). As before, jZ is a discrete time k transformed into a CML with asymmetric coupling. The coordinate, f (xi)C ͑kу1͒ is a nonlinear mapping function, equivalence between these systems is formal, however, and and i[1,N]oriZis a discrete space coordinate. and ␥ different causality conditions apply to the two cases. Some are non-negative coupling parameters to be referred to as future events in the delayed map are past events in the CML diffusion and shift coefficients. For finite arrays we shall representation and vice versa. As a result, certain statistical consider zero-flux properties are the same for the two representations, while others are not ͓15͔. In several cases of interest, the Hopf x0ϵx1 , xNϵxNϩ1 ͑3a͒ equation describing the evolution of the ensemble density in phase space for a delayed differential equation may be ap- or periodic proximated by a Perron-Frobenius equation in RN for a CML x ϵx , x ϵx ͑3b͒ system. This can be used to explain the statistical cycles 0 N Nϩ1 1 observed numerically in delayed differential equations in boundary conditions. The single map S: x f (x) is assumed terms of stable density limit cycles ͓16͔. 1 to have an attractive interval I ʚR . The→ purpose of the In the present paper we consider the N-dimensional or 0 ͑ present section is to discuss some general properties of the infinite-dimensional map ͒ map T relating to its stable and regular behaviors. Let us first determine the attracting domain of the map T. Here we may T: xi f ͑xi͒ϩ͓xiϩ1Ϫ͑1ϩ␥͒xiϩ␥xiϪ1͔, ͑1͒ → state the following. 1 ϩ where x x ( j)R . jZ is a discrete time and i[1,N] Theorem 1. ͑a͒ Let the map S␣ : x ␣ϩ f (x) have an i, i → or iZ, depending on the boundary conditions, is a discrete attracting domain I␣ for ͉␣͉р␣0 and let I*ϭ(x1*рxрx2*)be space coordinate. Obviously, with g(x)ϭ f (x)Ϫx, one can an interval such that for any xI* and any ␣͓Ϫ␣0 ,␣0͔, derive ͑1͒ from the partial differential equation S␣I*ʚI*. ͑b͒ If there exists a value r satisfying the condi- tions Ϫrрx*Ͻx*рr, rр␣ /2͑1ϩ␣͒, then the map T has 2 1 2 0 x an attracting domain Dϭ͕ x Ͻr, iϭ1,2,...,N͖. The proof of ץ xץ xץ ϭg͑x͒ϩ͑1Ϫ␥͒ ϩ ͑2͒ ͉ i͉ .s2 this theorem can be found in the Appendixץ sץ tץ Let us now consider the possibility of synchronization of by approximating the time and space derivatives with differ- the individual maps into an overall uniform behavior. The ences and by rescaling the nonlinearity. This relates our coupled map T has a one-dimensional invariant manifold model to the dynamics of extended, nonequilibrium media. ͑diagonal͒ Dϭ͕x1ϭx2ϭ•••ϭxNϪ1ϭxN͖. Indeed, if Our aim is to study some global aspects of the dynamics of ϩ xi͑0͒D, iϭ1,2,...,N, then xi( j)D for jZ , such that the map T such as the relation between the dynamics of the T͉D͉ϭS. To analyze the stability of D let us introduce the single map S: x f (x) and of the coupled map lattice for variables different values of→ the coupling parameter , and the general features of the bifurcations that take place as is increased. y i͑ j͒ϭxi͑ j͒Ϫxiϩ1͑ j͒, iϭ1,2,...,n, n,NϪ1 ͑4͒ We first consider the sufficient conditions for the map T to have an attractive domain ͑Theorem 1͒. This allows us to and the differences estimate the limits of variation for the state variables. In Theorem 2, one of the main properties of the map T is stated, f „xi͑ j͒…Ϫ f „xiϩ1͑ j͒…ϭ f Ј͑„xi͑ j͒,xiϩ1͑ j͒…͒y i͑ j͒, ͑5͒ namely, its ability to generate simple behavior as a result of synchronization when all elements of the CML, indepen- where (xi ,xiϩ1). According to the mean value theorem, dently of the initial conditions, acquire a uniform behavior f Ј͑͒ is a piecewise smooth function of xi and xiϩ1. Let us determined by the local one-dimensional ͑1D͒ map. The denote i( j)ϭ f Ј͑„xi( j),xiϩ1(j)…͒Ϫ␥i , where ␥1ϭ1, 3198 VLADIMIR N. BELYKH AND ERIK MOSEKILDE 54
␥nϭ␥, and ␥iϭ1ϩ␥, iϭ2,3,...,nϪ1, and let us introduce III. STEADY STATES AND STABLY TRAVELING CHAOTIC WAVES the vector Y( j)ϭcolumn „y 1( j),y2(j),...,yn(j)… and the nϫn matrix Q  ( j) , where „ i … Let us hereafter study the problem of the existence and stability of the steady states of the map T. The fixed points of  0 ••• 000 1 T are defined by the conditions ␥2 ••• 000 xi͑ jϩ1͒ϭxi͑ j͒ xi , iϭ1,2,...,N ͑11a͒ 0␥3 ••• 000 , ӇӇӇ Ӈ Ӈ Ӈ or Q͑i͒, .
000••• nϪ2 0 g͑xi͒ϩ͓xiϩ1Ϫ͑1ϩ␥͒xiϩ␥xiϪ1͔ϭ0, ͑11b͒
000••• ␥nϪ1 where as before g(x), f (x)Ϫx. Equation ͑11b͒ may be con- ͩ 000••• 0 ␥nͪsidered as a spatial map and the steady states for T arise as ͑6͒ the solutions to this map satisfying the boundary conditions ͑3͒ or, in the case of unbounded array, without boundary By subtracting from each equation in ͑1͒ the subsequent conditions. equation, we get the map Introducing the new variables
L͑ j͒; Y͑ jϩ1͒ϭQ i͑ j͒ Y͑ j͒, ͑7͒ „ … uiϭ␥͑xiϪxiϪ1͒, uiϩ1ϭ␥͑xiϩ1Ϫxi͒, ͑12͒ where the dependence on j is determined by the original map we obtain the two-dimensional map F: ͑1͒. Theorem 2. Let the map T have an attractive domain D.If x ϭx ϩu Ϫ Ϫ1g x , ϩ iϩ1 i i ͑ i͒ the matrix Q„i( j)… for xi( j)D, jZ has the eigenvalues sk( j), kϭ1,2,...,n, inside the unit circle in the complex Ϫ1 uiϩ1ϭ␥͓uiϪ g͑xi͔͒, iϭ1,2,...,N or iZ. ͑13͒ plane ͓͉sk( j)͉Ͻ1͔, then the manifold D is absolutely asymp- totically stable. This map has the Jacobian Jϭ␥Ͼ0 and hence F is a one-to- Indeed, as each iterate of the map ͑7͒ is contracting, the one map. Moreover, in the case where ␥ϭ1 and the function composition of maps Ll,L( jϩl)L( jϩlϪ1)•••L(j) is con- g(x) is periodic, the map F reduces to the standard map tracting as well. The asymptotic stability of the manifold D ͓18,19͔. Depending on the boundary conditions, the trajecto- implies the global synchronization of the map T. Hence, ries of F, representing the steady states of the map T, are as when j ϱ, each cell acquires a uniform behavior with re- → follows. In the case of zero-flux boundary conditions, each spect to the map S, independently of the initial conditions fixed point of T is a trajectory of F satisfying the condition ͓17͔. In order to calculate the eigenvalues of Q i( j) one can „ … u1ϭ0, uNϩ1ϭ0. ͑14͒ estimate the trace elements i( j) relating the derivatives f Ј͑͒ to the attracting domain D and use the recurrent for- In the case of periodic boundary conditions, each fixed point mula for ⌬n,det Q( j), of T is a period-N cycle of the map F, i.e.,
2 ⌬kϭk⌬kϪ1Ϫ␥ ⌬kϪ2 ͑8͒ uϭFNu, u ϭu , u Þu , i ͓1,Nϩ1͔. 1 Nϩ1 i1 i2 1,2 ͑15͒ with the initial conditions ⌬Ϫ1ϭ0 and ⌬0ϭ1. A similar ap- proach was used by Afraimovich and Nekorkin ͓7͔ to exam- In the limit N ϱ, each bounded trajectory of ͑13͒ ine the stability of the steady states. If one is to use the norm k → (xk ,uk)ϭT (x0 ,u0), kZ, corresponds to a steady state of of a matrix Pϭ(pkl), ʈPʈϭ͚k,l͉pkl͉, another criterion of the map T. synchronization is ʈQʈϽ1. From this criterion follows the The trajectories of the map F were studied by Belykh ͓20͔ inequality for an arbitrary nonlinear function g(x) having l zeros and lϪ1 extrema alternating between one another. In particular, 2͑1ϩ␥͒͑NϪ2͒ϩ͑NϪ1͓͒max͉f Ј͉͔Ͻ1. ͑9͒ the case of an odd sinelike periodic function was considered. xD The fixed points of F are alternatingly of saddle type and of elliptical ͑or reverse saddle͒ type. Applying these results to Hence the sufficient condition for the manifold D to be ab- the present case, we obtain the following. solutely stable is Theorem 3. ͑a͒ For any Nу2 there exist values of the parameters ,␥ such that the map F has a trajectory satisfy- 1Ϫ2͑1ϩ␥͒͑NϪ2͒ Ͻ ing the boundary conditions ͑14͒ or ͑15͒. ͑b͒ The map F max͉f Ј͉ . ͑10͒ Ϫ1 xD NϪ1 displays the bifurcation curve ͕␥ϭ␥*͑ ͖͒, ␥*͑0͒ϭ1, ␥*(Ϫ1)ϭ0 at which the homoclinic orbit is in the tangency * Example 1. Consider Nϭ2 ͑nϭ1͒. The map L( j)in͑7͒ of the stable and unstable manifolds of a saddle point. For takes the form y( jϩ1)ϭ[f Ϫ (1ϩ )]y(j), which is stable the range of parameters Hϭ͕␥ ͑˜␥,1͔, ˜␥ϭ␥* for Ͼ , Ј ␥ * if Ϫ1ϩ͑1ϩ␥͒Ͻf ЈϽ1ϩ͑1ϩ␥͒, x D. ˜␥ϭ0 for р ͖, the map F has a structurally stable ho- * 54 ONE-DIMENSIONAL MAP LATTICES: SYNCHRONIZATION... 3199 moclinic orbit in the neighborhood of which the trajectories of F are topologically conjugated to the Bernoulli shift over pу2 symbols. Example 2. At ␥ϭ0 the coupled map lattice ͑1͒ is unidi- rectional and the map F is reduced to the one-dimensional mapping x xϪϪ1g(x), which becomes the short map x ϪxϪ→x2 for Ϫ1g(x)ϭϪ(Ϫ2xϪx2). For this func- tion,→ under the coordinate transformation (x,u) [(1ϩ)Ϫ1/2(xϩ1),(1ϩ)Ϫ1/2u], the map F for ␥Ͼ0 takes→ the form
¯x ϭxϩuϪͱ1ϩ͑1Ϫx2͒, ¯uϭ␥͓uϪͱ1ϩ͑1Ϫx2͔͒, ͑16͒ i.e., the form ͑13͒ with Ϫ1ϭͱ1ϩ and g(x)ϭ1Ϫx2. Here an overbar is used to denote the next iterate. Figure 1͑a͒ illustrates the homoclinic orbit bifurcation curve ␥ ϭ␥*(ͱ1ϩ) for the map ͑16͒ and Figs. 1͑b͒ and 1͑c͒ show the stable and unstable manifolds of the saddle point for the values ϭ0.3, ␥ϭ0.6 in the region H after tangency and for the values ϭ0.5, ␥ϭ0.1 before tangency, respectively. Note that the bifurcation curve for Ͼ0 follows the rough approximation ␥*ϭ͑15Ϫ12͒/20. A corollary of Theorem 3 is that the coupled map T has fixed points in the form of a regular stationary space distri- bution of coordinates for bounded arrays and in the form of chaotic distributions for unbounded arrays. Note that in the degenerate case ␥ϭ0 when the map T becomes unidirec- tional and the stationary distributions of coordinates ͑fixed points͒ are determined by the one-dimensional map x xϪϪ1g(x), one also has the possibility of observing complex→ behavior of T. Following Afraimovich and Nekorkin ͓7͔, let us now con- sider solutions to the coupled map lattice ͑1͒ in the form of waves traveling at a constant speed and with unchanged shape xi( j)ϭ⌿(iϩ j). The equations for such solutions be- come
xkϩ1ϭ f ͑xk͒ϩ͓xkϩ1Ϫ͑1ϩ␥͒xkϩ␥xkϪ1͔, ͑17͒ where kϭiϩ j is a traveling coordinate and the x notation xkϭ⌿(k) has been preserved. Introducing in analogy with ͑12͒ the new coordinates
␥ FIG. 1. Homoclinic orbit bifurcation for a two-dimensional map u ϭ͑x Ϫx ͒ , ͑18͒ i i iϪ1 1Ϫ corresponding to two diffusively coupled short maps. ͑a͒ Bifurca- tion diagram. ͑b͒ Stable and unstable manifolds after bifurcation we obtain a 2D mapping ͑ϭ0.3, ␥ϭ0.6͒. ͑c͒ Phase picture before bifurcation ͑ϭ0.5, ␥ϭ0.1͒. g͑xi͒ ␥ g͑xi͒ x ϭx ϩu ϩ , u ϭ u ϩ iϩ1 i i 1Ϫ iϩ1 Ϫ1 ͩ i 1Ϫ ͪ earities. Let us consider small values of the coupling param- ͑19͒ eter . The stability of a fixed point (x1* ,x2* ,...,xN*)ofT with respect to a perturbation ZiϭxiϪxi* is defined by the of the same form as ͑13͒. By virtue of Theorem 3, the map T linear map generates traveling waves of chaotic profile for parameters corresponding to the homoclinic orbits and, consequently, Smales horseshoes exist for each previously defined sine- Z͑ jϩ1͒ϭAZ͑j͒, ͑20͒ wave type of function g(x) having not less than two zeros. The stability analysis of the steady states and traveling where Z( j) is the column matrix „Z1( j),Z2(j),...,Zn(j)…. A waves performed by Afraimovich and Nekorkin ͓7͔ for cubic is the Jacobian of T at (x1* ,x2* ,...,xN*) with the boundary functions g(x) in the finite as well as the infinite- conditions ͑3͒. For zero-flux boundary conditions, the Jaco- dimensional case may now be extended to arbitrary nonlin- bian matrix becomes AϭQ(ai), where 3200 VLADIMIR N. BELYKH AND ERIK MOSEKILDE 54
N aiϭ f Ј͑x*͒Ϫ␥i , ␥1ϭ1, ␥Nϭ␥, i 2 ͑i͒ „ai͑1͒ai͑2͒ϩ ␥ Ϫs…ϭ0, ͑26͒ i͟ϭ1 ␥iϭ1ϩ␥ for iϭ2,3,...,NϪ1. ͑21͒ (i) where ͕␥ ͖ϭ͑␥,2␥,...,2␥,␥͒ and ai( j)ϭ f Ј„xi*( j)…Ϫ␥i . For practical calculations of the stability conditions the fol- It is easy to verify that for functions g(x) that are propor- lowing formula for ⌬N,detA turns out to be useful tional to , the condition of stability for ͑26͒ may be fulfilled in the same way as in Example 3. Using the property of the ⌬ ϭa ⌬ Ϫ␥2⌬ , ⌬ ϭ0, ⌬ ϭ1, k k kϪ1 kϪ2 Ϫ1 0 matrix ͑25͒ that all its elements outside the trace are of O͑͒, the stability conditions for any k-cycle with kϾ2 can be * akϭ f Ј͑xk ͒Ϫ␥k , obtained in a similar way as for the 2-cycle. kϭ1,2,...,N, ␥ ϭ1, ␥ ϭ␥, 1 N IV. NONWANDERING SET OF THE WEAKLY COUPLED MAP ␥kϭ1ϩ␥ for kϭ2,...,NϪ1. Consider now the dynamics of the map T for small values 2 In the limit of weak coupling, when terms of O͑ ͒ may be of . Let ⍀s be a limiting set of a single map S and let us neglected, we obtain associate this set with each cell of the uncoupled map lattice T such that ⍀iϭ⍀s , iϭ1,2,...,N. Obviously, for the N coupled map lattice, the limiting set in RN has a topological ⌬1ϭa1 , ⌬2ϭa2⌬1 ,..., ⌬Nϭ ak . ͑22͒ limit k͟ϭ1 N Hence the characteristic equation can be written in the form lim⍀͑͒ϭ⍀͑0͒ϭ͟ ⍀i , ͑27͒ 0 iϭ1 N →
͑akϪs͒ϭ0 which is the topological product of the limiting sets ⍀i for k͟ϭ1 each cell in the array. Recall that ⍀sϭഫx(0)R1x(0) , where ϭlim f k x͑0͒ depends on x͑0͒, unless ⍀ is a * x(0) k ϱ „ … s and the eigenvalues are given by skϭ f Ј(xk )Ϫ␥k , unique fixed→ point of f . Hence, allowing for the dependence kϭ1,2,...,N. The condition of stability ͉sk͉Ͻ1, related to the on the initial conditions, the limiting set of the uncoupled function g(x), then attains the form map can be expressed as Ϫ2ϩ͑1ϩ␥͒ϽgЈϽ␥, xD, ͑23͒ N ⍀͑0͒ϭ ഫ ͓ limX͑ j͔͒ϭ͟ ⍀i„xi͑0͒…. ͑28͒ where D is the attracting domain of ͑1͒. Note that no accu- N j ϱ iϭ1 X͑0͒R → mulation of terms of O͑2͒ occurs when N ϱ. Hence, with- out performing a detailed analysis of the transition→ N ϱ,we This implies that any combination of N initial conditions of can state that the condition of stability ͑23͒ applies→ for any the single map interpreted as an initial condition X͑0͒ of the NZϩ. By means of an example we shall show that the map map T leads to a certain component of the limiting set ⍀͑0͒. For example, if is the k -cycle of the map S, then the ͑13͒ has chaotic trajectories under the condition ͑23͒. This xi(0) i proves the existence of stable chaotic stationary distributions component of ⍀͑0͒ corresponding to that initial condition is of T. the k-cycle with k being the lowest common multiple of Example 3. For ␥ϭ1 and g(x)ϭa sin x, the map ͑13͒ k1 ,k2 ,...,kN . Another important feature of the limiting set reduces to the standard map ⍀͑0͒ of the uncoupled maps is that each of the k-cycles of T has real multipliers. ¯x ϭxϩuϪa sin x, ¯uϭuϪa sin x. ͑24͒ In terms of the structural stability theory we may define the hyperbolic subset of ⍀͑0͒,⍀h͑0͒, including both the In this case homoclinic orbits exist for any aϾ0. Moreover, trivial unity of nondegenerate k-cycles ͑with multipliers for aϳ1 the map ͑24͒ has no closed invariant curves and ͉si͉Þ1, iϭ1,2,...,N͒ and a nontrivial set with a Bernoulli arbitrary homoclinic connections exist. On the other hand, shift over some symbols. In this sense the general theory of the stability condition ͑23͒ holds for gϭa sin x. hyperbolic systems is applied for the map T in the neighbor- The stability of k-cycles of the map is given by ͑20͒ hood of ⍀h and the following assertion is valid. where the matrix Theorem 4. There exists such a small value of 0 that, for ͑0,0͒, the coupled map T has a structurally stable hyper- k bolic component ⍀h͑͒ of a limiting set ⍀͑͒ with the topo- Aϭ Q͑ f Ј„xi*͑ j͒…Ϫ␥i͒. ͑25͒ logical limit j͟ϭ1
lim⍀h͑͒ϭ⍀h͑0͒. Here xi*(1),xi*(2),...,xi*(k)ϭxi*(1) , iϭ1,2,...,N, are the 0 „ … → coordinates of the k-cycle and ͕␥i͖ϭ͑1,1ϩ␥,...,1ϩ␥,␥͒.In the particular case of a 2-cycle of the map T ͓for small , The immediate corollary of this assertion is that all the non- where terms of O͑2͒ can be neglected except when they degenerate k-cycles of the uncoupled map T are preserved occur in the trace terms͔, the characteristic equation becomes under a small increase of from zero. Another conclusion is 54 ONE-DIMENSIONAL MAP LATTICES: SYNCHRONIZATION... 3201
that, magnified by multiplying the dimensions, the most sig- ¯x lϭ f ͑xl͒ϩ͑y lϩ1Ϫ2xlϩy lϪ1͒, nificant features of the 1D map ͑such as its chaotic dynamics and the complex bifurcation structure͒ are preserved for a ¯y lϭ f ͑y l͒ϩ͑xlϩ1Ϫ2y lϩxlϪ1͒, ͑29͒ small range of the coupling parameter . Next we compare the case of large , where the coupled lϭ1,2,...,m, x0ϵx1 , y0ϵy1 . map T is synchronized and the limiting set ⍀͑͒ lies on the one-dimensional manifold D ͑Theorem 2͒, and the case of If Nϭ2mϩ1, T may be written in a similar form only with ¯ small , where the component of ⍀͑͒ outside D is nontrivial the additional equation y 0ϭ f (y 0)ϩ(x2Ϫ2y 0ϩx1) and ͑Theorem 4͒. Observing that the map T depends continu- without the restriction x0ϵx1 , y 0ϵy 1 . ously on the parameter and on the parameters of the single Let us denote x as the column matrix (x1 ,x2 ,...,xm), y as map, we obtain the following. the column matrix (y 1 ,y 2 ,...,ym), h(x,y) as the column ma- trix (y Ϫ2x ϩy ,y Ϫ2x ϩy ,...,y Ϫ2x ϩy ), Theorem 5. Let the single map have a parameter a as a 2 1 0 3 2 1 mϩ1 m mϪ1 and F(x) as the column matrix f (x ),f(x ),...,f(x ) . The multiplier S: x af(x), such that for aϾa it has a non- „ 1 2 m … → 1 system ͑29͒ then takes the form trivial limiting set ⍀s and let the invariant manifold D of the coupled map T(,a) be absolutely stable for some region of ¯x ϭF͑x͒ϩh͑x,y͒, ¯yϭF͑y͒ϩh͑y,x͒. ͑30͒ parameters ͑aϽa2 , Ͼ1͒. Then the map T has an infinite number of bifurcations corresponding to the disappearance Systems of this type, exhibiting the symmetry (x,y) (y,x), ↔⍀͑͒പD͔ under a transition in parameter were considered in a recent paper by Reick and Mosekilde͓گof the set ⍀͑͒ space from the region ͑Ӷ1, aϾa1͒ to the region ͑Ͼ1 , ͓21͔. In this section we present some different properties of aϽa2͒. such symmetrically coupled systems with reference to maps In the general case, the only knowledge we have about of the form ͑29͒. We first observe that the general system these bifurcations is that they are bifurcations of periodic ͑30͒ has an m-dimensional invariant manifold D(m)ϵ͕xϭy͖ orbits and of homo- or heteroclinic orbits. As previously with the map on it, noted, Giberti and Vernia ͓14͔ have conducted a numerical study of the mechanisms by which stable periodic orbits ¯x ϭF͑x͒ϩh͑x,x͒, xRm. ͑31͒ arise in CMLs as the coupling between the local maps is increased. Considering a lattice of nine diffusively coupled In the case of ͑29͒, h(x,x)ϵ0 and the map ͑31͒ splits into m logistic maps with periodic boundary conditions and with the single maps S: xlϭ f (xl). local map operating slightly above the Feigenbaum accumu- Theorem 6. The map ͑29͒ has a two-dimensional manifold lation point, they find that the stable periodic orbits typically ͑2͒ emerge in inverse period-doubling bifurcations to subse- D ϭ͕x1ϭx2ϭ•••ϭxm , y1ϭy2ϭ•••ϭym͖ quently disappear in saddle-node bifurcations as they collide with the map on it, with unstable orbits. Secondary bifurcations in which peri- odic orbits are stabilized or destabilized while a pair of ei- ¯x ϭ f ͑x͒ϩ2͑yϪx͒, ¯yϭ f ͑y͒ϩ2͑xϪy͒, x,yR1. genvalues simultaneously pass the unit circle through ϩ1or ͑32͒ Ϫ1 may also occur. Giberti and Vernia ͓14͔ also point to the role played by the formation of normally attracting, one- This assertion follows in a straightforward manner from Eq. dimensional manifolds connecting 2N weakly hyperbolic or- ͑29͒ by subtracting xl(0)ϭx, yl(0)ϭy, lϭ1,2,...,m. Note bits, N being the number of lattice sites. Half of these peri- that the map ͑32͒ is the same as the initial map T for two odic orbits are stable and the other half are unstable. coupled maps with the substitution 2 and with zero-flux Relaxation of the trajectory towards such an invariant mani- boundary conditions. We also note that→ a symmetric period-2 fold is usually found to occur relatively fast. Once on the orbit of the map ͑32͒ (x(1),y(1)) (y(1),x(1)) represents a manifold, however, the dynamics becomes very slow, char- traveling wave of wavelength 2 moving→ along the array T at acterized, as it may be, by eigenvalues that deviate from 1 by unit velocity. In general, each period-k orbit may be inter- as little as 10Ϫ12. These one-dimensional manifolds may be preted as a stationary wave and as a generalization of Theo- involved in various global bifurcations in which, at the same rem 6 we state the following. time, the stability of the manifold and of the periodic orbits Theorem 7. For N being a multiple of p ͑for any p when are affected. In particular, Giberti and Vernia describe a spe- N ϱ͒ each space periodic solution of the system ͑1͒ → cial type of bifurcation that may occur in CMLs and in which xi( j)ϭy(i, j), y(iϩp,j)ϭy(i,j), y(i,j)ϭy(Ϫi,j) with a one-dimensional manifold of the type described above col- even period p lies on a p-dimensional manifold D(p) with a lapses with an unstable cycle of half the period. dynamics on it of the form ͑29͒ with mϭp. Indeed, denoting (2lϪ1,j)ϭxl( j) and (2l,j)ϭyl(j), instead of ͑1͒ we obtain Eqs. ͑32͒ with mϭp. Here (xl ,y l), (p) V. COUPLED MAPS WITH SYMMETRY lϭ1,2,...,p are the coordinates of D such that from (p) (p) „xl(0),yl(0)…D it follows that „xl( j),yl(j)…D , Consider the diffusively coupled maps ͑1͒ for ␥ϭ1 and jZϩ. Nϭ2m and assume that we have periodic boundary condi- Next, under the linear transformation tions ͑m ϱ when the array is unbounded͒. Using a new → notation y for the variables xi with odd ͑or even͒ i and a new xϭuϩv, yϭuϪv ͑33͒ numeration, we obtain an alternative form of Eq. ͑1͒ for the map T: the map ͑30͒ is transformed into the map G: 3202 VLADIMIR N. BELYKH AND ERIK MOSEKILDE 54
1 1 ¯u ϭ 2 ͓F͑uϩv͒ϩF͑uϪv͒ϩ h͑uϩv,uϪv͒ ϩ1͒/6 for р „ 3 3 ͑ 8 Dϭ͕͉x͉Ͻ 2 ,͉y͉Ͻ 2 ͖, Ͻ 1 5 . ϩh͑uϪv,uϩv͒…͔,U͑u,v͒, ͭ ͑5Ϫ4͒/24 for ͑ 8 , 4 ͒ ͑38͒ 1 ¯v ϭ 2 ͓F͑uϩv͒ϪF͑uϪv͒ϩ„h͑uϩv,uϪv͒ The condition of absolute stability of the invariant manifold Ϫh͑uϪv,uϩv͒…͔,V͑u,v͒. ͑34͒ (xϭy) as determined by ͑10͒ and ͑38͒ has the form
Since V(u,0)ϵ0, ͕vϭ0͖ is an invariant manifold of ͑34͒. 0ϽϽ͑1Ϫ3a͒/2. ͑39͒ Moreover, the map G is invariant with respect to involution (u,v) (u,Ϫv) due to the obvious equalities Hence, according to Theorem 4, the transition of the param- ↔ eters ͑,a͒ from a region of small and a close to 1 ͑param- U͑u,Ϫv͒ϭU͑u,v͒, V͑u,Ϫv͒ϭϪV͑u,v͒. eter is supposed to match Theorem 4͒ to the region ͑39͒ produces an infinite number of bifurcations. This implies that the limiting set ⍀s of the trajectories of ͑34͒ For aϭ1, the map ͑37͒ has a period-2 symmetric orbit is symmetric with respect to the manifold ͕vϭ0͖, i.e., if a (x(2),y(2)) (y(2),x(2)), where point (u*,v*)⍀s then (u*,Ϫv*)⍀s. Note that each → symmetric trajectory s is mapped onto itself by an ⌫s⍀ x͑2͒ϭϪϪͱϩ2Ϫ2, y͑2͒ϭϪϩͱϩ2Ϫ2. involution transformation ⌫sϭI⌫s , where the matrix ͑40͒
E 0 2 Iϭ , ͑35͒ This orbit appears at ϭ2͑͒ϭϪ2ϩ and remains stable ͩ 0 ϪEͪ 2 for ͑2 , 4͒, where 4ϭ͑1Ϫ5͒/2ϩ is a torus bifurca- tion point. This agrees with the result obtained by Reick and E being a unit matrix. Similarly, each asymmetric trajectory ϩ s Ϫ Mosekilde ͓21͔ that the second period-doubling bifurcation ⌫ a ⍀ is mapped onto its reflection twin ⌫ a , i.e., 1 ϩ(Ϫ) Ϫ(ϩ) for ϭ0 ͑here at ϭ 2͒ is changed into a torus bifurcation for ⌫ a ϭI⌫ a . Ͼ0. A similar phenomenon was observed in the paper by Consider now the twin map Gt defined by Biragov, Ovsyannikov, and Turaev ͓22͔. ¯u ϭU͑u,v͒, ¯vϭϪV͑u,v͒ ͑36͒ ACKNOWLEDGMENTS and also displaying the symmetry with respect to ͑35͒. In this This study was performed with support from the Danish case we have the following. Research Academy and from the Russian Foundation of Fun- Theorem 8. The limiting set ⍀a of the twin map G as a t damental Research under Grant No. 93-013-16253. topological set coincides with the limiting set ⍀s of the map G. Indeed, I⍀sϭ⍀s and G⍀sϭ⍀s. Then IG⍀sϭ⍀s. But APPENDIX s s as IGϭGt so Gt⍀ ϭ⍀ . Consider the discontinuous map Corollary. ͑a͒ Let the map G have a symmetric period kϭ2k orbit ⌫ . Then, if k ϭ2k ϩ1 the map G has two ϩ 1 sk 1 2 t xi͑ jϩ1͒ϭ f „xi͑ j͒…ϩ␣i͑ j͒, jZ ͑A1͒ asymmetric period k orbits ⌫(1) and ⌫(2) , consisting of the 1 ak1 ak2 where points of ⌫sk .Ifk1ϭ2k2 the map Gt has a symmetric period t kϭ2k1 orbit ⌫ sk consisting of the points of ⌫sk , but with another route between them. ͑b͒ Let the map G have an ␣i͑ j͒ ϩ Ϫ asymmetric period-k orbit ⌫ ak ͑and its twin ⌫ ak͒; then the 2k ͓x ͑ j͒Ϫ͑1ϩ␥͒x ͑ j͒ϩ␥x ͑ j͔͒ for ͉x ͑ j͉͒Ͻr map Gt has a period-2k symmetric orbit ⌫ s consisting of iϩ1 i iϪ1 k Ϯ ϭ the points of ⌫ . The fixed point of G, for example, corre- ͭ ␣0 for ͉xk͑ j͉͒уr, kϭiϩ1,i,iϪ1. ak sponds to a period-2 orbit of Gt , the period-2 orbit of G to ͑A2͒ two fixed points of Gt , the period-3 orbit of G to a period-6 The function ␣ ( j) for ͉x ( j)͉Ͻr satisfies the inequality orbit of Gt , the period-4 orbit of G to a period-4 orbit of Gt , i k etc. Example 4. To illustrate the above results let us consider ͉„xiϩ1͑ j͒Ϫ͑1ϩ␥͒xi͑ j͒ϩ␥xiϪ1͑ j͒…͉Ͻ2r͑1ϩ␥͒ two diffusively coupled short maps and by virtue of the condition rϽ␣0/2͑1ϩ␣͒ we have 2 ¯ 2 ¯x ϭa͑ϪxϪx ͒ϩ͑yϪx͒, yϭa͑ϪyϪy ͒ϩ͑xϪy͒ ϩ ͑37͒ ͉␣i͑ j͉͒р␣0 , jZ , iϭ1,2,...,N. ͑A3͒ where an amplitude aϾ0 is introduced as a multiplier. The Then, from the first condition of the theorem and from the self-similar box-within-a-box structure of the bifurcations for condition Ϫrрx*, Ͻx2*рr it follows that for any trajectory ϩ the single map is described, for instance, by Mira ͓23͔. This of ͑A1͒,ifxi(0)I*, then xi( j )I*, iϭ1,2,3,...,N, jZ . structure is preserved for the coupled map ͑37͒ with an in- But, on the other hand, the initial map TI* coincides with duced structure of the trajectories in the phase space (x,y)as ͑A1͒. Hence the map T has an attractive domain D. In this described in Sec. II. The conditions for ͑37͒ to have an at- proof we did not use the boundary conditions and the theo- tracting domain for aр1 are ͑Theorem 1͒ rem holds in the case of unbounded array iZ. 54 ONE-DIMENSIONAL MAP LATTICES: SYNCHRONIZATION... 3203
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