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Theoretical and Mathematical Physics, 132(1): 983­999 (2002)

DYNAMICS OF INHOMOGENEOUS CHAINS OF COUPLED QUADRATIC MAPS
A. Yu. Loskutov, A. K. Prokhorov, and S. D. Rybalko


A new effective local analysis method is elab orated for coupled map dynamics. In contrast to the previously suggested methods, it allows visually investigating the evolution of synchronization and complex-b ehavior domains for a distributed medium describ ed by a set of maps. The efficiency of the method is demonstrated with examples of ring and flow models of diffusively coupled quadratic maps. An analysis of a ring chain in the presence of space defects reveals some new global-b ehavior phenomena.

Keywords: distributed media, space­time chaos, coupled-map lattices

1. Introduction
Investigating lattices ofcoupled maps as mo dels of distributed media has recently aroused great interest (see, e.g., [1]­[5]). In this approach, the lattice elements are the points of the physical medium, and the character of the coupling between them determines the interaction in accordance with the basic physical principles. The main problems among those arising in this case are replacing the continuous physical space with an adequate discrete analogue and cho osing the correct coupling. But even with a favorable choice, the resulting lattice mo del rarely describes the abundance of phenomena observed in distributed systems. Therefore, one often resorts to considering these systems with an element evolution and coupling such that under the variation of the control parameters, they demonstrate a broad spectrum of phenomena characteristic of a distributed media [1], [2]. Lattice systems appear not only in considering distributed media but also in describing pro cesses developing in systems having an essentially discrete structure in both space and time. Many problems in synchronization theory for radio generators, in biology, and in medicine and also the study of the behavior of cellular automata and neural networks lead to such mo dels [4]. Depending on the dimensionality of the original mo deled system, coupled-map lattice systems can be one-, two-, or three-dimensional. In this paper, we study only one-dimensional lattices, i.e., linear chains of the type of a set of elements interacting according to a definite law. In constructing coupled-map lattices, we must first cho ose the form of the map determining the time evolution in each element. This choice of the map defines the lo cal phase space X on which the map f : X X acts, x f (x), x X. (1)

The phase space of the entire lattice is the direct pro duct of all lo cal phase spaces of the individual elements. In the one-dimensional case, all elements can be arranged on the line and enumerated with a single index.


Moscow State University, Moscow, Russia.

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 132, No. 1, pp. 105­125, July, 2002. Original article submitted Decemb er 17, 2001. 0040-5779/02/1321-0983$27.00 c 2002 Plenum Publishing Corp oration
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Then the phase space of such a chain including N elements can be written as X = i=1 Xi . The state of the chain is therefore determined by the vector = (x1 ,...,xN ) X . Another important problem in constructing map lattices is defining the coupling between the elements. Clearly, the coupling must specify the effect of the values of the elements of the entire lattice on the subsequent values of an individual element. In the one-dimensional situation, i.e., in the case of chains, the coupling can be defined by a map xi g (x1 ,... ,xN ),

N

i = 1,... ,N .

(2)

Hence, the dynamics ofa coupled-map chain can be represented as the compositions oftwo maps (1) and (2). In this case, the value of the ith element varies during one time step according to the law xi g f (x1 ),... ,f (xN ) ,

i = 1,... ,N ,

(3)

and the behavior ofthe entire chain decomposes into temporal and spatial parts determined by the respective map f (x) and transformation g (x1 ,... ,xN ). Although most of the results relating to coupled maps are obtained for this decomposition, it would be entirely incorrect to assert that this constraint is general. For example, in the temporal discretization of some systems of coupled ordinary differential equations and partial differential equations describing a real distributed medium, map systems appear whose evolution is defined by the transformation xi f (xi )+ g(x1 ,...,xN ), ¯

i = 1,... ,N ,

(4)

where, as before, the function f (x) determines the temporal dynamics and the transformation g (x1 ,...,xN ) defines the variation of the value of the element xi depending on the current state of the system. In this approach, the temporal and spatial transformations act simultaneously. In the literature devoted to this topic, the most popular systems are one-dimensional chains of diffusively coupled maps [1]­[5]. A diffusive coupling means that the states of an element depend on only the values of their nearest neighbors. In this situation, the dynamics of a chain of type (3) is determined by a map in the form xi f (xi ,a)+ f (xi-1 ,a) - 2f (xi ,a)+ f (xi+1 ,a) , 2 (5)

where is the diffusion co efficient. In some works (see, e.g., [1]­[4] and the references therein), the function f (x, a) was taken in the form of a quadratic map f (x, a) = 1 - ax2 . Depending on the parameter values a [0, 2] and [0, 1], this transformation can manifest a very broad spectrum of types of global behavior ranging from synchronization of all elements and perio dic temporal and spatial dynamics to space­time chaos. Along with chains of diffusively coupled maps (5), the behavior of maps of the form xi f (xi ,a)+ (xi-1 - 2xi + xi+1 ) 2

(6)

is often investigated, where x [0, 1], a is the nonlinearity parameter, and the co efficient characterizes the interaction between the elements. This choice of the coupling means that the spatial and temporal actions under map (6) o ccur simultaneously as in the case of transformation (4).
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Apart from the indicated types of interaction, the so-called flow mo del approximating a free fluid flow is sometimes considered. In this case, the chain is a system of unilaterally coupled maps [5]­[8], xi f (xi ,a)+ f (xi-1 ,a) - f (xi ,a) , (7)

where f (x, a) defines the evolution of every chain element. Depending on the parameter values a and , this transformation of the function f (x, a) = 1 - ax2 demonstrates a broad spectrum of types of space­time behavior such as space chaos and perio dicity in time, spatial (and temporal) perio dic and quasiperio dic structures, and also some definite space­time forms. These forms include different kinds of formations ranging from individual structures surrounded by chaos to developed space­time chaoticity. It is clear that for any chain, it is necessary to indicate its behavior on the boundary. The most frequently studied systems are coupled maps with free or perio dic boundary conditions. Free boundary conditions are defined by the relations x0 = xN +1 0 (or x0 0 for systems of form (7)). For periodic boundary conditions, it is necessary to set xk = xN +k , k = 1,... ,N . If the map parameter values differ in a lattice, then the related system is no longer homogeneous, and investigating it becomes a much more complicated problem [9]. In practically all works devoted to studying diffusively interacting maps, homogeneous lattices are investigated (as a rule, numerically). But from the physical standpoint, the homogeneity of a space (understo o d as the identity of all elements or the equality of parameter values for the elements forming the lattice in the case under consideration) is an idealization intro duced to simplify the analysis. It is therefore very interesting to discover what qualitative changes in the system dynamics are caused by the appearance of inhomogeneities. There can be quite various forms of inhomogeneity ranging from individual defects to perio dic inhomogeneity throughout the space. In this paper, defects are understo o d as a difference in parameter values for some of the elements forming the lattice. Some examples of these inhomogeneous distributed systems are considered in Sec. 4. From the mathematical standpoint, coupled-map lattices with a finite number of elements are dynamic systems with a finite number of degrees of freedom. For these systems, an apparatus for analytic and numerical investigation is developed. But if N is very large, then the calculation of the main characteristics of the dynamic system is either very cumbersome or impossible in principle. Moreover, many of the parameters that must be calculated for dynamic systems (such as the metric entropy, the spectrum of Lyapunov exponents, the rate of decrease of temporal correlations, etc.) reflect the asymptotic behavior of the system as t and throughout the space as a whole [10], [11]. But knowing these parameters for coupled-map systems is essentially uninteresting. It is more important to find the characteristics determining the lo cal properties of these systems and their evolution in time. Some authors suggested various means for observing the evolution of lattice mo dels ranging from visual analysis metho ds to calculation of the local Lyapunov exponents (see, e.g., [12], [13] and the references therein). In this paper, we develop a new analysis metho d for coupled-map lattices that permits determining the behavior of the individual elements and the dynamics of the system as a whole. In contrast to the other metho ds, is allows investigating the lo cal behavior of individual elements of a distributed system and the evolution of the global dynamics of the entire phase space. The metho d is based on decomposing the entire time interval into short subintervals on which the degree of nonperio dicity of the tra jectories of all lattice elements is analyzed. This makes it possible to determine the synchronization domains, their temporal perio d, their transformation in time, and their break under spatial "chaotization." This paper is a continuation of the investigations of inhomogeneous coupled map lattices initiated in [9].

2. Local criterion for coupled-map dynamics
Although the present paper is devoted to studying one-dimensional coupled-map lattices, the metho d presented in this section can be used to investigate the multidimensional case as well. By construction, it
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permits analyzing the state of an element in the space of the distributed system on a small time interval based on knowing the time sequence at the given point. The values of each lattice element at consecutive instants in the evolution pro cess form a series x1 ,x2 ,x3 ,...,xn ,..., i i i i (8)

where i is the index of the element and n is the discrete time. To reveal the perio dicity of the series and determine its variation in time, it is obviously necessary to take an arbitrary segment of the series and to try to determine to what extent it is close to a perio dic one. Decomposing the entire series into short segments of the same length, we admit a certain coarsening with respect to time. Analyzing the perio dicity of the series in each of these segments and expressing it by a number, we can trace the evolution of the given characteristic in the transition from one segment to another. We now describe this pro cedure in more detail. Let T be the number of elements in the series, i.e., let the series have the form x1 ,x2 ,...,xT (for brevity, we omit the index i). We decompose it into identical time intervals and cho ose T and such that T = k , where k is a positive integer. Next, we test each of the series segments of length for perio dicity. For this, we perform the following pro cedure for each segment x1 ,... ,x of the series. 1. We compare it with a specially constructed series x1 ,x1 ,x1 ,...,x1 such that all its elements are equal to x1 and then express the result by a number 1 . The pro cedure for calculating 1 is presented below. At the moment, it is important that the number 1 should have the property that the smaller 1 is, the closer the series segment in question is to the specially constructed series x1 ,x1 ,x1 ,... ,x1 . 2. The given series is compared with the series x1 ,x2 ,x1 ,x2 ,... ,x1 ,x2 ,... consisting of periodic subsequences of perio d two. The comparison result is expressed by a number 2 . 3. A similar operation is performed for the series composed of three elements x1 ,x2 ,x3 , four elements x1 ,x2 ,x3 ,x4 , etc., up to x1 ,x2 ,... ,x/2 . The comparison results are expressed by numbers 3 ,4 ,... ,/2 , which again reflect some degree of deviation of series (8) under study from the corresponding mo del series composed of individual consecutive elements. 4. We find the parameter min = min{1 ,2 ,...,/2 }. (9)

The comparative characteristic p is calculated in the general form for each p = 1,... , /2 by the formula p =
i+1

xi - xi (m -p

2 o d (p+1))

.

(10)

It can be seen easily that p is the mean square deviation of the elements of the residual xp+1 ,...,x of the original series from the mo del perio dic series x1 ,x2 ,... ,xp ,x1 ,x2 ,...,xp ,... . According to (10), the inequality p 0 holds for each value of p. It is clear that if the relation min = p = 0 holds for some p, then the series x1 ,...,x is perio dic with a perio d multiple of p. To determine the perio d of the original series exactly, we must stop the procedure for calculating the characteristics p at the first value of p such that p 0. But if the calculation pro cess results in = min{1 ,2 ,... ,/2 } > 0, then we can definitely assert that the series x1 ,...,x is not perio dic. The value of p for which the given minimum is attained can be called a perio d that approximates the given aperio dic series. Calculating the value of for the individual segments of length , we can trace their dynamics throughout the entire interval of length T . Calculating
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and the perio ds on which the minimum is attained for all points in the chain, we can effectively reveal the synchronization domains in the space of the coupled-map system in question and study the dynamics of the individual elements. It is easy to understand that, rigorously speaking, the suggested analysis metho d for coupled-map lattices is not exact. We must first cho ose an adequate value of , i.e.., the length of the analyzed segment. The most suitable value of can be found by calculating the sets of p for different values of . Another disadvantage is that the characteristic min is positive for elements whose dynamics is quasiperiodic. Nevertheless, it is sufficiently close to zero because quasiperio dic motion can be approximated arbitrarily accurately by perio dic motion. But because we are interested in the qualitative pattern of the distribution of min rather than in concrete parameter values, this disadvantage is not very essential.

3. Examples of homogeneous systems
We demonstrate the efficiency of the suggested metho d with several well-known examples. 3.1. Flow mo del. We investigate different types of dynamics for flow mo del (7). As already mentioned, chain (7) has a broad spectrum of different behavior mo des depending on the nonlinearity degree determined by the parameter a and by the values of the diffusion co efficient . Moreover, the simplest types of limiting behavior can be calculated analytically. We consider this in more detail. The mo del under investigation can be expressed in the form of an iterative relation, xi +1 = f (xi ,a)+ f (xi-1 ,a) - f (xi ,a) , n n n n i = 1,...,N , (11)

where n is the discrete time and i is the spatial co ordinate. Let f (x, a) = 1 - ax2 and x0 0 be taken as the boundary conditions, although their choice do es not affect the qualitative investigation results. The general state of system (11) at the instant n is obviously defined by the N -dimensional vector n = (x1 ,x2 ,... ,xN ). Consequently, the dynamics of the entire chain is expressed by some transformation n n n F : X X , n+1 = F (n ), of the phase space X into itself. It can be clearly seen from the construction of the model that F = g f (see (1) and (2)). Formally (see above), lattice (11) is a discrete-time dynamic system with N degrees of freedom. Therefore, studying stationary and perio dic states reduces to analyzing the spectrum of eigenvalues of the Jacobi matrix DF and its powers DF (p) , where p is the length of the perio d under investigation. We write the expression for the matrix DF , DF ( ) = -(1 - )2ax1 0 0 . . . 0 0 -(1 - )2ax2 0 . . . 0 0 0 -(1 - )2ax3 . . . 0 ··· ··· ··· ··· .. . ··· 0 0 0 0 . . . -(1 - )2axN .

(12)

We note that DF is a triangular matrix. Consequently, its powers DF (p) are also triangular matrices. We now calculate the homogeneous stationary states of mo del (7). The homogeneity means that the relation xi = x holds for any i = 1,... ,N , and the stationarity expresses the time independence of the n n homogeneous state, i.e., xi = x x . Hence, the values of x must satisfy the condition n n x = f (x ,a)+ f (x ,a) - f (x ,a) = f (x ,a). In other words, the x are fixed points of the map defined by f (x, a), x,2 1 = -1 ± 1+4a . 2a (14)
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(13)


of chain (7). Here, S 1 and S 2 are the resp ective existence domains of the stable p oints x and x , and 1 2 Z is the existence domain of zigzag solutions (18).

Fig. 1. Existence and stability domains for the stationary states x and x and for the zigzag b ehavior 1 2

Thus, system (7) has two homogeneous states, 1 = (x ,x ,... ,x )and 2 = (x ,x ,... ,x ). Their stability 1 1 1 2 2 2 is determined by the magnitudes of the eigenvalues of the matrices DF (1 ) and DF (2 ). Because DF ( ) is a triangular matrix, we can use (12) to find the eigenvalues i 1 = -(1-)2ax and i 1 = -(1-)2ax 1 2 for the respective matrices 1 and 2 . The states 1 and 2 are stable if |1,2 | 1, i.e., -1 < -(1 - )2ax < 1, 1 -1 < -(1 - )2ax < 1. 2 (15)

Substituting x and x results in a relation between the parameters a and , 1 2 1- 1 1 < < 1+ 1+ 1+4a 1+ 1+4a for x , 1 (16) for x 2 .

1 1 < < 1+ 1- 1+4a - 1 1+4a - 1

Figure 1 demonstrates the calculation results for domains (16) with [0, 1] and a [0, 2]. The domains marked by S 1 and S 2 correspond to the stable points x and x . 1 2 Coupled-map chain (7) has another remarkable type of perio dic behavior in a wide range of the parameters (a, ). Namely, in the parameter range a [1.6, 2.0], the dynamics of the entire system manifests the so-called zigzag motion in which the behavior of all elements (or of the ma jority of the elements) has a cyclicity of period two in both space and time. In this behavior, the tra jectories of the individual points are two-perio dic, but the neighboring elements are in antiphasic oscillations.
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We investigate such a state analytically. The perio d two of each element is determined by the two values x1 and x2 . The cyclicity in space and time is expressed by the relations x1 = (1 - )f (x2 )+ f (x1 ), x2 = (1 - )f (x1 )+ f (x2 ). Solving Eqs. (17) and eliminating the homogeneous states x1 = x2 , we obtain 1± 4(1 - 2)2 a +4 - 3 . 2(1 - 2)a (17)

x1,2 = The stability of the resulting periodic ^ ^ product DF (1 )DF (2 ), where DF is x1 ,... ). It is clear that the product of DF 2 . The elements on the diagonal ^ DF (2 ). Therefore, the eigenvalues of

(18) eigenvalues of the matrix ^ and 2 = (x2 ,x1 ,...,x2 , ) is the triangular matrix ^ elements of DF (1 ) and

state can be investigated by analyzing the ^ given by (12), 1 = (x1 ,x2 ,... ,x1 ,x2 ,... ), ^ ^ the triangular matrices DF (1 ) and DF (2 2 of DF are the pro ducts of the diagonal DF 2 can be written as

i = (1 - )2 f (x1 )f (x2 ) = (1 - )2 4a2 x1 x2 , whence, in view of (18), it follows that 4(1 - )2 - 4(1 - )2 a. (1 - 2)2

=

Finally, to construct the domain where the zigzag behavior of (x1 ,x2 ) is observed, we must take the system of inequalities 4(1 - 2)2 a +4 - 3 0, -1 < < 1 (19)

into account. System (19) reflects the existence and stability conditions for this behavior. The calculation with inclusion of the additional constraints on the parameters (a [0, 2] and [0, 1]) leads to the relations 4(1 - )3 - (1 - 2)2 4(1 - )3 +(1 - 2)2
(20)

The resulting construction of domain (20) is shown in Fig. 1, where it is denoted by Z . As follows from Fig. 1, the domain Z can be overlapped with the existence domains S 1 and S 2 for stable points. This means that the limit state of chain (11) depends on the initial distribution {xi }N for some fixed values of 0 i=1 the parameters a and belonging to the intersection domains. The dynamics of these chains is said to be multistable. The boundary condition x0 0 has not been taken into account in all calculations relating to stationary and perio dic states and their stability (see (13), (15), (17), and (19)). In other words, all the results are obtained in the absence of a boundary, i.e., on the condition that N . But these results adequately reflect the qualitative pattern of the dynamics of chain (7). Moreover, in this case, the boundary condition can be regarded as some constant external perturbation of the coupled-maps that corrects the dynamics of the limit state.
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The system behavior at large values ofthe dimensionality N is essentially metastable. This manifests itselfin the tra jectory behavior strongly depending on the domain from which the initial data (x1 ,x2 ,...,xN ) 0 0 0 are chosen. The above analysis of stable solutions in no way reflects the related attraction domains. Therefore, the determination of these states may encounter some difficulties even in an exact calculation (with regard to the boundary conditions). Clearly, the variety of the possible behavior mo des of system (7) is not at all confined to solutions (14) and (18). To demonstrate the variety of the dynamics for chain (11) with f (x, a) = 1 - ax2 , we present the numerical experiment results for the given mo del based on the criterion described in Sec. 2. 1. a = 1.7, = 0.45. In the chain, we observe temporal perio dicity and doubling of the period in element-to-element motion from left to right. But the values of the elements xi are distributed in space chaotically, which is demonstrated in Fig. 2. The smaller the values of and p are, the darker the corresponding point is in the graph. The value = 0 corresponds to the darkest points in the diagram for . It is easy to see that the values of for all elements and at all instants are zero, and the lengths of the perio ds in this case are doubled from left to right up to p = 32. To analyze the spatial pattern, the values of xi at some consecutive instants are also shown in Fig. 2. It is intuitively clear that this chain shows spatial chaos. This can be confirmed more rigorously based on an analysis using a specially constructed map (see [5]). A more detailed investigation of this state showed that the value of the maximum perio d up to which the doubling continues depends only on a and , and the index of the element after which no further doubling o ccurs increases as N increases. 2. a = 1.7, = 0.11595. For the given values of the degree of nonlinearity and the coupling force, the authors of [8] revealed a type of behavior under which the ma jority of the elements are in zigzag motion, whereas some equidistant individual points are in chaotic motion. In this case, the distance between the chaotic defects depends logarithmically on the difference - c , where c 0.11525 is some critical value. The described phenomenon is reflected in Fig. 3. The chaotic defects are expressed by clear "flickering" lines, and the distance between them can be calculated easily. 3.2. Ring chain of quadratic maps. We consider a one-dimensional lattice of form (5), 1 xi +1 f (xi ,a)+ f (xi-1 ,a) - 2f (xi ,a)+ f (xi+1 ,a) , n n n n n 2

(21)

where f (x, a) = 1 - ax2 . System (21) was intensively investigated earlier in [1]­[3], where it was shown that this system has an extensive variety of behavior mo des ranging from synchronization of spatial structures to fully developed turbulence. As in the case of chain (7), the dynamics of system (21) essentially depend on the values of the parameters a and . We investigate some of the types of behavior for this chain using an analysis of point tra jectories by calculating the characteristics (see Sec. 2) and the perio d values p in different time intervals. 1. a = 1.44, = 0.1. Depending on the initial conditions, time-stable domains with regular and irregular dynamics are formed in space. Despite the diffusive coupling between the elements, the pattern of the system behavior is stable. This behavior of system (21) can be called a state of "frozen" structures. A typical pattern of these structures is presented in Fig. 4. This figure demonstrates a stable pattern of variation of the characteristic and the related perio d values p. 2. a individua dynamic time and
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= 1.88, = 0.3. An increase of the degree of nonlinearity determined by the parameter a in an l map f (x, a) leads to the destruction of all spatial synchronization domains. In this case, the behavior of the entire chain, as well as that of the individual elements, rapidly changes in both space [1]. This is well reflected by the irregularity parameter , whose behavior is demonstrated


Fig. 2.

Dynamics of system (11) for the parameter values a = 1.7 and = 0.45. Shown on the left

and right are the values of the characteristic dep ending on the index of the element (the horizontal axis) and on time in the scale (the vertical axis) and the corresp onding values of the p eriods p. At the b ottom, the distribution of xi at some consecutive instants is shown.

in Fig. 5. A similar pattern is also observed confirmation ofthis property, the same figure instants n. This behavior ofcoupled-map cha of the theory of nonequilibrium media, fully

for the distribution of perio ds p of individual elements. As a shows the distribution of the values of xi at several consecutive n ins is usually called space­time chaos or, using the terminology developed turbulence [1].

3. a = 1.8, = 0.3. In this, case, the domains of regular and chaotic dynamics alternate for randomly chosen initial conditions. This behavior is observed for a rather long time. But almost all chain elements
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Fig. 3.

The same as in Fig. 2 for a = 1.7 and = 0.11595.

ultimately start behaving regularly. An analysis shows that the chain tends to decompose into two-element cells in each of which the dynamics has the perio d two. This is demonstrated in Fig. 6, which shows the behavior of system (21) after 1 527 000 preliminary iterations. The general analysis of systems (7) and (21) in the above cases and for some other parameter values shows that the elaborated visualization metho d for the dynamics of a distributed medium approximated by coupled-map lattices is an effective means of investigating space­time evolution.

4. An analysis of inhomogeneous one-dimensional coupled-map chains
Investigation of nonlinear distributed systems shows that the presence of inhomogeneities (defects)
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Fig. 4.

Dynamics of ring chain (21) for the parameter values a = 1.44 and = 0.1.

mentioned in the introduction can substantially change the character of the system dynamics (see [9] and [14]). From the formal standpoint, the presence of at least a single element with a defect in a coupledmap lattice increases the number of control parameters. As a result, the dimensionality of the parameter space of the lattice regarded as a dynamic system increases, which in turn extends the set of possible behavior mo des because of the nonlinearity of the evolution law. It is clear that the investigation of these mo des in such a situation also becomes more complicated. We apply the elaborated metho d (see Sec. 2) to demonstrate the result of the presence of some defects in one-dimensional lattices of diffusively coupled maps. We consider chain (21) of length N = 100 with f (x, a) = 1 - ax2 in which all elements (except the
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Fig. 5.

The same as in Fig. 4 for the parameter values a = 1.88 and = 0.3.

central ones) have the same parameter value a = a1 = 1.44 and the central elements (with the indices i =47­53) have the different parameter value a = a2 = 1.97. For a = a1 = 1.44 and for a wide range of the values of , regular behavior in the form of small synchronized spatial structures is observed in homogeneous chain (21) (see above). In the case a = a2 = 1.97, homogeneous system (21) manifests all properties of space­time chaos. Thus, conditionally, the defect consists in that several "chaotic" elements are impregnated in a chain with regular behavior. The results of analyzing 5000 iterations of such a chain after 105 preliminary iterations are presented in Fig. 7. As is seen, the central elements are synchronized with the period two, and the chaos is moved to the other elements. But if 5 · 106 preliminary iterations are performed, then the chaos is localized in the elements with a2 = 1.97 and in some neighboring elements on
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Fig. 6.

The same as in Fig. 4 for the parameter values a = 1.8 and = 0.3.

the right and left. This shows that because of diffusion, the chaotic behavior propagates to elements with regular behavior. We now consider a different case of inhomogeneous chain (21). We recall that system (21) with the parameter value a = 1.8 and small values of the diffusion co efficient shows alternation of synchronization and also irregular behavior in both space and time (see Sec. 3.2). We construct a chain of form (21) with N = 100 such that 50 elements have the same value of the nonlinearity parameter (a = a1 = 1.8) and the other 50 elements have the different value a = a2 = 1.97, f (xi ,a1 )+ f (xi-1 ,a1 ) - 2f (xi ,a1 )+ f (xi+1 ,a1 ) n n n n 2 = f (xi ,a )+ f (xi-1 ,a ) - 2f (xi ,a )+ f (xi+1 ,a ) 2 2 n2 n n2 n 2 for o dd n, (22) for even n.
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xi +1 n


Fig. 7.

Dynamics of chain (21) (after 105 preliminary iterations) all of whose elements (except for the

central ones) have the same parameter value a = a1 = 1.44 and whose central elements (with the indices i =47­53) have the different parameter value a = a2 = 1.97. The diffusion coefficient is = 0.7.

Fig. 8.
7

Behavior of system (22) with alternation of the nonlinearity parameter in the elements after

5 · 10 preliminary iterations. The other parameter values are a1 = 1.8, a2 = 1.97, and = 0.7.

Thus, elements with different values of a in system (22) alternate in space. We set the value 0.7 for the diffusion parameter . This inhomogeneous system manifests space­time chaos on relatively short time intervals T 104 by analogy with the behavior of the homogeneous chain with a = 1.97. But
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Fig. 9.

Dynamics of ring chain (21) for the parameter values a = a1 = 1.8 and = 0.1.

if an asymptotic analysis of this dynamics is performed (e.g., by performing 5 · 107 iterations), then an unexpected phenomenon can be revealed, namely, the entire chain is synchronized. This manifests itself in a regular pattern being observed in the space of values of the parameter (see Fig. 8). Here, splashes of irregularity appear at equal time intervals and at the same distances from one another, i.e., it can be said that the chaos is in a kind of motion with a constant velo city along the chain. Figure 8 also demonstrates some small bent branchings from the straight lines. This means that the chaotic splashes move with a small acceleration rather than with a constant velo city. But they disappear after some time period. If this phenomenon is investigated thoroughly, then it can be found that the velo city of motion of the chaotic defects essentially depends on the values of a1 and a2 and also on the diffusion co efficient but is independent of the distribution of the initial conditions. Only the direction of their motion depends on the initial conditions. In conclusion, we investigate inhomogeneous ring chain (21) with a single defect. We let all chain elements have the same degree of nonlinearity (a = a1 = 1.8) except that a = a2 = 1.99 for the central element, and we cho ose the diffusion co efficient = 0.1. We first consider system (21) without defects. A typical dynamic pattern is show in Fig. 9. Most of the elements are synchronized during the entire evolution perio d, but, at the same time, there exist some small chaotic structures that wander in space. As is well seen from Fig. 9, they are mutually annihilated when they collide. Accordingly, they are also created in pairs. This behavior was compared to Brownian motion in [1] and [2]. We now add a defect with a = a2 = 1.97 at the center. Investigations show that the presence of an element with a higher degree of nonlinearity leads to a similar pattern except that there is always synchronization detuning in the central element and the number of wandering structures increases. Their dynamics completely repeat that of the structures in a homogeneous system with the only distinction that they can also disappear when colliding with the center (i = 50) and can be created at the center. Conditionally, the element with defect is a germ-absorber of "Brownian" particles. In this connection, it is interesting to study the state of this system at large times. It turns out that the limit state is symmetric irrespective of the initial conditions.
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Fig. 10.

The same as in Fig. 9 in the presence of a defect with a2 = 1.99 in one element (i = 75) for

= 0.1 after 5 · 107 preliminary iterations.

Figure 10 demonstrates a detailed behavior pattern after 5 · 107 iterations, where, for convenience, the defect with a2 = 1.97 is moved to the element with i = 75. We see that the chaotic particles are created on the defect in pairs, and they move in tra jectories symmetric with respect to the defect. They also disappear in collisions with each other and with the element having the defect. We note that the chaotic elements cause a slight synchronization detuning in all elements. As is seen in Fig. 10, this detuning propagates in space along different directions and converges at the point (on the ring) opposite the defect. The presented examples show that inhomogeneous map lattices have an absolutely unpredictable variety of possible mo des of motion, which can be effectively demonstrated using the characteristic intro duced in Sec. 2.

5. Conclusion
Distributed media can be rather well approximated by space- and time-discrete coupled-map systems or lattices. But knowing the parameters determining the global behavior of coupled-map lattices at asymptotically large times is not very interesting. It is more important to find characteristics determining the lo cal properties of these systems and their evolution in time. In this paper, a new analysis metho d is developed for coupled-map chains that permits visualizing the behavior of the individual elements and the dynamics of the entire system as a whole. It is used to investigate the behavior of both homogeneous systems of diffusively coupled one-dimensional quadratic maps and systems with spatial inhomogeneity. Different types of inhomogeneity have been thus considered, perio dic inhomogeneity throughout the space in the form of defects in several consecutive elements and a defect in a single element. We showed that the presence of different types of inhomogeneities in the mo del can substantially change the behavior of the entire system. In particular, for some definite values of the parameters and diffusion co efficients, the dynamics of a inhomogeneous chain with alternating defects are synchronized irrespective of the initial conditions. The peculiarity of this chain is that the homogeneous systems with parameter values determining the inhomogeneities show space­time chaos. We can say that
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the synchronization of the complex behavior develops by intro ducing a spatially perio dic inhomogeneity. In conclusion, we note that analyzing the dynamics of coupled-map systems is a very complicated problem and, as a rule, even the consideration of homogeneous discrete mo dels encounters essential difficulties. Nevertheless, the suggested criterion for lo cal analysis permits simplifying the investigation and visually demonstrating the behavior of lattices as a whole. REFERENCES
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