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Theoretical and Mathematical Physics, Vol. 124, No. 3, 2000

MODEL ACTIVE

OF A SPATIALLY MEDIUM

INHOMOGENEOUS

ONE-DIMENSIONAL

K. A. Vasil'ev, ~ A. Yu. Loskutov,

1 S. D. Rybalko, 1 and D. N. Udin 1

We investigate the dynamics of one-dimensional discrete models of a one-component active medium analyticalh~. The models represent spatially inhomogeneous diffusively concatenated systems of one-dimensional piecewise-continuous maps. The discontinuities (the defects) are interpreted as the differences in the parameters of the maps constituting the model. Two classes of defects are considered: spatially periodic defects and localized defects. The area of regular dynamics in the space of the parameters is estimated analytically. For the model with a periodic inhomogeneity, an exact analytic partition into domains with regular and with chaotic types of behavior is found. Numerical results are obtained for the model with a single defect. The possibility of the occunence of each behavior type for the svstem as a whole is investigated.

1.

Introduction

Tile approximate representation of continuous media by their discrete analogues is a rather effe.ctive research method. The conversion to tile discrete form call be either purely spatial or both spatial and temporal. In the spatial discretization, the initial system is approximated by a finite or countable set of elements with a certain forin of coupling between them. Every elelnent represents a dynamic system with a small number of variables. If, in addition, the dynamic system is deternfined by a map, i.e., by a transformation with discrete time, then the discretization is called spatial temporal. The spatial teinporal discrete models are called net or lattice models. Many problems in the nonlinear theory of a continuous medium (:an be reduced to problems with discrete models. For example, some problems in statistical physics can be effectively solved through the representation of the medium by its discrete space tinm approximation [1-4], excitable continuous systems are often described in terms of their discrete analogues [4, 5], etc. (cf. [6 13] and the references therein). Moreover, any sort of numerical analysis of continuous systems is always connected with discrete space time systems because any nunmrical procedure is based on a finite difference scheme. There is a variety of space-time lattice forms. The most common are tile lattices where every element somehow interacts only with its nearest neighbors (see, e.g., [8-13] and the references therein). Another form of coupling is global interaction in which every pair of elements is connected [14--16]. In the ease of a local interaction, diffusion coupling between the elements is normally considered [1, 2, 4, 8-13] (see also the references therein). This form of coupling is mainly used to model the phenomena related to space time chaos, structure generation, and self-organization. If the value of a governing parameter of the family of maps constituting a lattice is constant for tile entire lattk:e, tile lattice is called homogeneous. Varying this paranmter and the vahle of the diffusion (or the coupling constant), we can investigate the phase diagram, the possible phase transitions, and tile related space time patterns. If the parameters of the maps are different, then tile net is inhomogeneous. Such nets are nmch more ditficult to study. Ahnost all previous works dedicated to diffusively interacting maps 1Moscow State University, Moscow, Russia. Translated ['tom Teoreticheskaya i Matematichesk~wa Fizika, Vol. 124, No. 3, pp. 506-519, September, 2000. Original article submitted September 20, 1999; revised April 13, 2ooo. 1286 0040-5779/00/1243-1286525.00 9 2000 Kluwer Academic/Plenum Publishers


were restricted to a study (usually a numerical study) of homogeneous nets. From the physical standpoint, however, it is obvious that tile space homogeneity (or, ill this case, tile identity of all elements) is a strong idealization adopted to simplify the analysis. It is therefore interesting to investigate tile qualitative changes in tile system dynamics related to the inhomogeneity. The latter Call be of different types ranging from individual defects to a periodic intlomogeneity of tile entire space. The present work is focused oil spatially inhomogeneous one-dimensional systems (i.e., chains) of diffusively connected piecewise-linear one-dilnensional mat)s. Tile term map here refers to a transformation of an interval I into itself: To: I ---, I, where I = [a, [3] E ]Rl, x H G(x, a), x E I, G(x, a). is a function, and a is a parameter (or, generally, a set of parameters). In terms of the iterations k, this call be written as x(k + 1) = G(x(k), a), x E I. The inhomogeneities are given as maps with different parameters. Tile nlaps themselves (i.e., the functions G) are chosen such that the systems represent models of a (me-component active medium [17]. Based on all exact calculation of the Lyapunov indices, we investigate different regimes in the behavior of the system with a periodic inhomogeneity and of the system with a single defect. The phase space structure of these systems is also described.

2.

Models of spatially inhomogeneous active media

The main result in this section is tile description of the dynamics of two types of inhomogeneous active media. Vr consider a model representing a one-dimensional anmflar lattice of concatenated maps with a periodic inhomogeneity and a chain with a single defect. 2.1. Homogeneous
medium.

We examine a system of N diffusively concatenated maps of the fornl

xn(k + 1) = G(xn(k), (~, 7) + s

(k) - 2Xn(k ) -}- Xn_l(k))

(1)

with the periodic boundary conditions Xn+N(k) =-- xn(k), where n = 1,2,...,N is the discrete space coordinate, k = 0, 1,2,... is the discrete time coordinate, c > 0 is the diffusion coefficient (which we assume to be constant), and a > 0 and 7 are parameters. For the fmlction G, we set
(1 - 2c~)x - 7,

x < 1/2, x > 1/2.

G(x,a,7)=x+aF(x)-7=

(1

2a)x

7+2a,

(2)

This form of the flmction G(x, a, 7) was chosen because system of maps (1) with elements of form (2) is a discrete realization of the basic model of a one-component one-dimensional active medimn described by an equation of the Kohnogorov Petrovskii Piskunov form:

OU(x, t) _ D i:)2U(x, t) Ot Oz 2 + ~(U).
As is known, this equation is commonly met in problems ill biophysics, combustion theory, chemical kinetics (e.g., the description of the Belousov Zhabotinskii reaction), solid state electronics, etc. (see, e.g., [18-21]). It is easy to see that G(x, c~, 7) is a piecewise-linear function whose graph has a constant slope equal to (1 - 2~). Depending on tile parameter a, the nmp generated by tile function G, i.e., x H G(x, a, 7), call manifest qualitatively different types of behavior. For 0 < a < 1 (Fig. la), its behavior is regular, and depending on 7, tile map call have one or two stable points (the points A and B) attracting ahnost all points of the phase space. For c~> 1, tile absolute value of tile slope of the graph of tile map exceeds unity (see Fig. lb), which corresponds to an exponential divergence of neighboring trajectories. Ill this case, if the motion of a phase point is finite, the dynamics is chaotic. H()mogeneous model (1) composed of concatenated functions of form (2) was described in detail in [17]. We recall the principal results of this study. System (1), (2) Call be interpreted as an N-dimensional
1287


a(x)

a(.)
I

[

/

2

/

I
/

rf

i"o

/

I
0.5 1 (a) 2 x /tO

/

I
0.5

,
i 2

X

(b)
(a) for (t = 0.25 and T = -0.2; (b) lbr ct = 125 andT=-l.

Fig. 1.

The mapx~G(x,a,7):

piecewise-linear map f: ]RN ~ R N. Because G(x,(t,7) is a piecewise-linear flmction with a constant derivative, the differential D/ of the map f is a matrix with constant coefficients. It is therefore easy to find the Lyapunov indices Ai of this map. If p,~ are the eigenvalues of D f, then A, = log IPsl. If any eigenvalue p~ lies outside tile unit circle on the complex plane, then all trajectories of the map f are unstable. Otherwise, the model dynamics is regular. The problem of the system behavior is therefore connected with investigating the location of the roots Ps. To cah:ulate Ps, we express the characteristic polynomial for the N-dinlensional map differential in terms of determinants of three-diagonal matrices with different dimensions. We can obtain recursive relations for these matrices. Interpreting these relations as finitedifference equations with given initial conditions, it is possible to find the solutions, which are proportional to the Chebyshev polynonfials of a linear function of the eigenvalues of Dr. It can be shown that because of the properties of Chebyshev polynomials, the characteristic polynomial reduces to a quadratic third-order polynomial of the known function ps. Calculating the values of Ps and analyzing their locations on the complex plane lead to a partition of the parameter space of map system (1), (2) into two domains (Fig. 2): 1. Domain D1 is (tetermined by parameters ct and ~ such that the absolute values of all eigenvalues of the map f differential are smaller than rarity, and the model dynamics is therefore regular. 2. Domain D2 is determined through the condition that the set of eigenvalues Ps contains at least one root satisfying the inequality IP~I > 1. In this case, generally, the dynamics can be infinite. However, if the motion is finite, the behavior of system (1), (2) is said to be chaotic [9 11]. Systetn (1), (2) is spatially homogeneous in the sense that all its elements (i.e,, the maps x ~ G(x, ct, 7)) are equal. However, our goal is to investigate the spatially inhomogeneous diffusive model. The inhomogeneity can be related either to different map types acting in different space points or (if the maps tbr all the elements are equal) to ditIbrent map parameters. In this analytic study, we restrict ourselves to the latter case, i.e., we assume that every element is a piecewise-linear map x,i ~ G(x~, (~, 7) (see Eq. (2)) but, in contrast to the homogeneous case, there are two clement types distinguished by the values of the parameters (ti of the function G(x,i; ai, 7). 2 We let the parameter c~ correspond to one type of the elements and the parameter/3 correspond to the other type. 2Generally, the elements can also differ by the vahm of 7- However, it is shown below that the dynamics of both the homogeneous and the inhomogeneous systems does not depend on % 1288


0.5

Fig. 2.

Parameter space of a homogeneous annular chain of diffusively concatenated maps (1), (2).

As in honiogeneous case (1), we take the chain in the form of a ring, i.e., Xn+N(k) = xn(k). We investigate two qualitatively different cases in detail: the model with spatially periodic defects and the model with a single defect. 2.2. Annular model with a spatially periodic inhomogeneity. The term spatially periodic inhomogeneity refers to the case where the values of the parameter of the flmction G change from one element to another periodically, i.e., the parameters vary as a,/3, c~, fl,.... The dynanfics of a chain with such an inhomogeneity can be flflly described analytically. W> consider a concatenated one-dimensional net of the form

xn(l~+l)=[G(xn(k),o~,~)+s(.~Sz+l(k)-2xn(lr
c(.n(k),/3,

l(k))

for odd n,

(a)

+ 4x.+,(a.) - 2.n(k) + .,,_,(k))

for even

where Xn+N(k) ~ xn(k), the function G is still given by relation (2), and N is even (otherwise chain (3) cannot be converted to the form of a ring). Vv> seek the values of the parameters a,/3, and s corresponding to qualitatively different regimes of behavior of periodically inhomogeneous ammlar model (3), (2). W'e calculate the Lyapunov indices of system (3), (2) using the technique developed in [17]. Taking the boundary conditions into account, we express the differential of the N-dimensional map generated by a system of form (3) as
12c2a ~ 0 --E

Df =_ QN =

c 0
9

1 - 2c - 2/3 ~
.

c 1-2c--2o~
,

... ...
,

0 0 1 -2~-2

c

0
QN,

0
we

.--

/3

)
\

.

To find tile eigenvalues Ps, s = 1, 2, ..., N, of tile matrix 2qz
0 :

calculate tile determinant 0
0 ~ :

~
2z2~ S :

0
~ 2ZlC :

...
-'' --".. 0 0 :

det(QN --

pIN) =

0

0 0

0 0

0 0

... ...

2z2c
1289


where z 1 and z2 are determined fl'om the relations 1 - 2e - 2ct - p = 2Zle and 1 - 2c - 2fl - p = 2z2c. Expanding the determinant with respect to the first row, we obtain det(QN -- pIN) = 2Zl~BN-1 -- g2BN-2 -- 2CN -- g2BN_2, where 2zle ~ 2z2c c
,

0 e 2zlc
,

...

0 0 0 2z2cl

2z2e ,
BN

BN =

0
9

...

0
9

c 2Zle e
.

0 e 2z2c
.

- 99 -- 99 9

0 0 0

.

9
---

9
...

".
2z1E
Z 5 +--+

0 It is easy to see
that

0

0

0

0

0

BN(Z1,Z2)

Call be obtained from BN(Zl, z2) through the substitution

Z2, and we

therefore use a recursive relation to find BN(Zl, z2). Expanding BN with respect to the first row, we obtain
2ZlSBN-I BN = --22BN-2 2

for even N, for odd N.

2Z2CBN-1 -- g2BN

It is now easy to express tile odd-order determinants through tile even-order determinants:
B2N+2 + y2B2N

B2N+I =

2ZlS

(4)

W'e then obtain a recursive relation for the even-order determinants:
B2N = C2(4ZlZ2 -- 2)BzN-2 - g4B2N-4.

(5)

Relation (5) can be interpreted as a difference equation with the initial conditions Bo = 1, 132 = J(4zlz2 - 1). (6)

Further, solving system (5), (6) and taking Eq. (4) into account, we obtain for even N, BN(Zl, Z2) =

c

N

Z2

UN(~.)

for odd N,

where Ux(z) is a Chebyshev polynonlial of the second kind. In turn, the relations for BN(q, z2) have the fornl ENUN(Z) for even N,
J~N(ZI'Z2) =

"EN ~UN(Z)

for odd N.

Using tile relations for BN-1, BN-2, and /3N-2, we find (|et((~N -- piN) = 2EN (zUN_I(Z) -- UN_2(Z) -- 1). 1290

(r)


It is now easy to find the eigenvalues of the differential of map (3), (2). It follows from Eq. (7) that zUN_I(Z) --UN_2(z)--I=O or TN(Z) -- 1 = 0, where we use tile relation between the second-kind Chebyshev polynomials the first-kind Chebyshev polynomials TN(Z). Because

(s)

UN(Z)

and

T

(z) =

-

+

2

-

(where z 2 = zlz~_), if we set t = z - v/z 2 - 1, we obtain 1 ( tN+N ) ~ --1=0

or (t N - 1) 2 = 0 from Eq. (8). Consequently, ts = e i~'~/N and zs = cos(27rs/N), s = 1, 2,..., N. Recalling that 2zle - 2z2c = 4z2c 2, we obtain the simple equation (1-2e-2ct-p)(1-25-2.~-p) =4e

2

cos"

, (2Tr) ~-s

.

Solving this equation with respect to p, we find the set of eigenvalues of the differential of map (3), (2):

PL2=l-2~--cx-/34-

c~-/3) 2+4e 2cos 2

s

,

(9)

where s = 1, 2 .... , N/2. The dynamics of system (3), (2) is fiflly regular if all values of Ps lie inside the unit circle on the complex plane. All P~,2 are real; therefore, the regular-dynamics condition is satisfied if 2c + c~ +/3 > X/4c 2 + (c~ -/3) 2 , z + ---~ + 2+ <1.

Because c, a, 3 > 0, the first inequality of this system is always true. In turn, the condition c + (a + ~)/2 + ~a 2 + (ct - fl)2/4 > 1 is satisfied in the parameter space domain where A~ > 0. The boundary between the domains of qualitatively different (regular or chaotic) dynamics is a surface in the three-dimensional space of the parameters (c~,/3, e) given by the expression e+----~+ 2+__

-

1.

(lo)

As an illustration, we consider the sections of tile phase space of system (3), (2) by planes 6 = const, where (~ = /3 - o' is tile iMlomogeneity parameter (Fig. 3). Let D1 be tile regular-dynamics domain, and let D2 be the domain where A~ > 0. The boundary between these two domains is deternfined by an equation that follows from Eq. (10). For comparison, the boundaries separating the domains D1 and D2 for the homogeneous ((~ = 0) system of concatenated maps (see Fig. 2) are shown in Fig. 3a,b (dashed lines). As follows from Fig. 2, the model with 5 > 1 exhibits a chaotic dynamics for any values of c and c~. For 6 < 1; there exists a domain D, where the dynamics is regular. In a certain range of values of the parameters 0 < ~ < 1, this domain is a subarea of the regular behavior domain of the homogeneous model. For 1291


E

(1-6)/(2--6) 0.5

I

0.5 (1-6)/(2-6) I~D,2 D "- ...
1-($ 1 C~

-6

1

a

0

(a)

(b)

Fig. 3. Sections of the parameter space for model (2), (3) by the planes 6 = const: (a) for -1 < 6 < 0; (b) for O< 6 < 1. -1 < 6 < 0, there exist values of the parameters e and a such that the dynamics of the inhomogeneous model is regular, while the dynamics of the homogeneous one is chaotic. The restriction ct > -~ corresponds to the condition (3 > 0. For ~ < -1, the domain D1 valfishes, and model (3), (2) exhibits only chaotic properties for all possible values of e and a. As directly follows fi'om Fig. 3, there are two qualitatively different possibilities: either the parameter a belongs to the regularity domain, the parameter fl belongs to the chaos domain with ks > 0, and the general dynamics is regular or the dynamics of the periodically inhomogeneous model can be chaotic 9 The realization of one of these two possibilities depends on tile value of the diffusion parameter c. 2.3. Annular model with a single defect. We now consider another interesting spatially inhomogeneous model of diffusively concatenated maps, the system with a single defect. In this case, the parameter f4 (see formula (2)) corresponds to one of the N elenmnts of the system, while the parameter (t corresponds to the remaining N-1 elements. Without loss of generality, we can set this single different element to correspond to n = 1. Therefore, we exanfine a model of the form

{

o,,

+

Xn(k

i) =

a(xl(k),Z,"/) +s

.... ,(k) - 2x.(k) + -- 2Xl

n = 2,3,...,N, n=l,

(n)

where Xn+ N ~ xn(k ). As above, let the function G(x, a, ~) be given by expression (2). W> deternfine the ditfbrential of the corresponding map for system (11), (2):
12e - 2~ c 0

DI--QN=

a 0
9

1 - 2c - 2a ~
9

c 1--2~--2a
.

.... .....
-9

'~ 0 0 /" (12)

e If the technique used niodel (11), (2), then roots of a polynomial (titti~rent approach for We estimate the matrix (12) based on A = {aij}~· belong

0

0

.--

1 -- 2~- 2(~

above to find the Lyapunov indices (see Sees. 2.1 and 2.2 and also [17]) is applied to the characteristic equation for map dittbrential (12) leads to the necessity to find the of order 2N+2. This can only be done mnnerically. We therefore use a somewhat this model. regular-dynamics domain of system (11), (2) using the estimate for the eigenvalues of the Guershgorin theorem [22]. According to this theorem, all eigenvalues of a matrix to a union of circles on the complex plane:

ps 0{zec:ll -a. jl i=1 1292

s=l,2,...,X,


/

Fig. 4. Estiinate of the regular-dynamics domain for inhomogeneous chains with two element types in the parameter space (a, j3, c). where R'i is either the row quasi-norm of tile matrix A, R i = ~j=l,j#i ]aij], or the columnar quasi-norm of the matrix A, R} = ~'i~l,i#j [a,~j]. Because map differential (12) is a real symmetric matrix, the row and the colunmar quasi-norms coincide, and the eigenvahms of Df are real. Therefore, the Guershgorin circles are transformed into intervals on the real axis:
' ! n

n

i=l

Because 6 > 0, it is easy to find from expression (12) that

R'~ = ~
j=l j#i

I",~Jl = Icl + I~I = 26

We note that R~ does not depend on i and there are only two different types of diagonal elements in the map differential D] (see Eq. (12)); therefore, the eigenvalues of Df belong to the union of two intervals {z E ~: I1 - 26- 2az I _ 26} U {z E Ii~: I1 - 2s- 2/~- z I _< 2c}.

Because ct, 3,r > 0, we then obtain an upper estimate for tile absolute values of the eigenvalues of IDf:

- [ a+2s, Therefore, the domain D1 in the parameter space

a>3. satisfying the conditions for a < 2, 1 fora>3,

(a, 3, 6)

b~:

/3+26<1 a+26<

(13)

corresponds to a regular dynamics of inhomogeneous model (11), (2), and the domain /)2: {/3+26>1 a + 26 > 1 fbr a < fl, for a > ~3, (14)

corresponds to positive As. The domains/)1 and/)2 are shown in Fig. 4. The domain L)1 is a lower estimate for the region of regular dynamics, because it was derived using the upper estimate for the eigenvalues of 1293


s

(1-6)/(2-6) 0.5

I
0.5 (1-~)/(2-6) (1-6)/2 -6 1 1-6

(a)

(b)

Fig. 5. Approximate estimate and exact result tbr the regular-dynamics domain of the model with periodic inhomogeneity: (a) for -1 < (~ < 0; (b) for 0 < 6 < 1. tile map ditthrential. Any point of the parameter space belonging to/91 corresponds to a regular dynamics of the model, while any point belonging to /92 corresponds to either regular or (if the motion is finite) chaotic dynamics. The derived estimate is valid not only for the single-detect model discussed but also for the entire class of aimular systems of maps with a diffusive concatenation of tbrm (1) characterized by the I)resence of two types of the elements (with different parameters c~ of the flmction G(z~, a~, ~.)). In this case, the number of elesnents of each type and their relative positions do ,lot matter. Indeed, tile qnasi-norms of the snap differentials for all such systems coineide, and the centers of two possible interwfls containing the eigenvalues of D/ (according to the Guershgorin theorem) are common as well. By a corollary of the Guershgorin theorem, if the two intervals where the estimate of p~ is made do not intersect, then there is a "clustering" of tim map differential eigenvalues. That is, q eigenvalues Ps, s = 1, 2,..., q, lie within one interval, and N-q eigenvalues Ps, s = q + 1, q + 2,..., N, lie within tile other interwfl, flere, q is the number of elements with the parameter (,:, and N-q is the number of elements with the paraineter ft. However, taking tiffs property into account does not improve the estimate of tim regular-dynamics domain. We compare estimate (13) with the exact result obtained above for the annular model with a spatially periodic inhomogeneity (see Sec. 2.2). It is clear that the model belongs to the class for which this estimate remains valid. For elucidation, we construct the sections 5 = const of the domain D1 corresponding to a regular dynamics of system (3), (2) and also those for the estinmte /9~ (Fig. 5). It can be seen in the figure that estimate (13) for the regular-dynamics domain approxilnates the true behavior of the model rather well. However, the estimate fails to reflect that for -1 < 6 < 0, the regular-dynamics domain for system (3), (2) is broader than the regular-dynamics domain of tim spatially homogeneous system. We now return to system of maps (11), (2) whose spatial inhomogeneity is connected with tile presence of only one defbet. We find the eigenvalues of map differential (12) numerically. In this case, the characteristic polynomial can be written as 2ze+2(a-fl) r 0
:

det(QN -- pIN) =

e 2zs e
:

0 ~ 2zc
:

----...
"..

e 0 0
:

a where 2zz

0

0

--9

2za

1 - 2c - 2(t - p. Expanding det(QN -- pIN) with respect to the first row, we obtain det(QN - pIN) = [2ze + 2((t -- fl)]
BN-1 --

2g2J~N-2

--

2(--1)No N,

(15)

1294


s

0.5 ["
0

...

D.e --.~,
1

0.5, (1-~)/2

D2 D1 ~" 1-5 c~
1

I DI"
-5

(~

(a)

(b)

Fig. 6. Characteristic form of the sections 6 = const of the parameter space for a chain of N=20 elements with a single defect: (a) for 5 -0.3; (b) for 6 = (}.3. where

2z$
BN = 0 0

e 2ze c :
0

0 e 2zc :
0

... ... --"..
9 --

0 0 0 : 2z~

As above, we use the expansion of BN with respect to the first row to obtain tile recursive relation BN = 2zgBN-1 -- C2Bg-2. This relation can be interpreted as a difference equation with the initial conditions Ba = 2z& B2 = e2(4z 2 - 1), whose solution is of the form

BN = eNUN(z),
where UN(z) is a second-kind Chebyshev polynomial. Taking Eq. (16) and the expression for UN(Z),

(16)

UN(Z) = (Z+ ~ ) N + I

_ (Z-- ~ ) N + I 2~2~ 2 - 1

= (1/t)N+i __tN+i

i/t

-t

'

where t = z - v/z u - 1, into account, we obtain [t~-N+' + 1 (~)1--t2N det(QN -- pIN) = 2a N [ ~-~-ff + (1 -- t 2) t N-1 (--1)N 1

from Eq. (15). Hence, the characteristic equation for map differential (12) can be expressed as t 2N+2 +2 ( ~ ) t ~ --t2N--2(--1)Nt N+2 ++2(--1)NtN+t 2 --2 (-~) t-- 1 = 0. (17)

It is now easy to calculate Ps by finding the roots of Eq. (17) nmnerically. Comparing the Ps values to unity for all possible values of the parameters a, /~, and c, it is straightforward to determine the domains D1 and D2 of qualitatively different dynamics for an annular chain of diffusively colmected maps with a single defect. The sections of these donmins by the planes 5 = const (where 5 = ~ - (~) are shown in Fig. 6. Estimate (13) tbr the regular-dynamics domain is shown by the dashed line. In this case as it was for the model with a periodic inholnogeneity, this domain (for positive 5) is broader than that predicted by the estimate. However, in contrast to the periodic inhomogeneity, the estimate for 5 < 0 flflly coincides with the numerical result and also with the result tbr a homogelmous chain. Therefore, if the parameter correst)onding to one of the system elements is smaller than those corresponding to all other elements, then this defect has no influence on the dynamics of the entire ensemble. However, if the anomalous paralneter value is larger than the normal value, then the regular dynamics domain of the model is narrower than that in the homogeneous case. 1295


3.

Conclusion

We investigated the behavior of systems of diffusively conc.atenated one-dimensional maps that model a one-dimensional active medimn with a spatial inhomogeneity. Different types of iixhomogeneity were exanlined. For all systems whose spatial inhomogeiieity is related to the presence of two types of elements that differ by a parameter value, a lower estimate for the regular ensemble dynamics domain in the space of the parameters was derived analytically. This estimate was based on the localization of the differential eigenvalues of the map. Numerical and exact analytic results were obtained for two specific models of this class. The results for the three examined cases, i.e., estimate (13) of the regular-dynamics domain for the class of models with two types of elements, the exact results for the system with periodic inhomogeneity (Fig. 3), and the exact results for the single-defect model (Fig. 6), reveal that there exist two general types of system dynamics. For the first type, the parameter of one sort of elements belongs to the regular-dynamics domain while the parameter of the other sort belongs to the unstable-behavior domain, and the general dynamics remains regular. For the second type, in contrast, the model dynamics can be unstable or, in the case of finite motion, chaotic. The realization of either behavior type depends on the diffusion parameter and is possible only if the parameter is small. Obviously, the general dynamics of all chains belonging to the c.lass of our interest cannot be regular for sufficiently large values of ~. As far as we know, this work is the first attempt to describe the dynamics of inhomogeneous systems of concatenated maps analytically. The present knowledge is far from exhaustive even for the homogeneous discrete models of continuous media. Much is yet to be elucidated, nlany qualitative mid quantitative questions remain open. Moreover, this work did not address the problems of possible stationary states of the model in the regularity domain, of wave motions along the chain and their dependence on the defects, of the dynamics in the case of time-dependent parameters, etc. Some of these questions will be answered in the near future [23, 24]. Acknowledgments. This work was supported in part by INTAS (Grant No. 97-1094).

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