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Periodic orbits and long transients in coupled map lattices
Physica D 45 (1990) North-Holland
PERIODIC
452-460
ORBITS
AND
LONG
TRANSIENTS
IN COUPLED
MAP
LATTICES
R. LIVI”> b, G. MARTiNEZ-MEKLER b, ’ and S. RUFFO be d a Dipartimento di Fisica, Universitd di Firenee, Large E. Fermi 2, 50125 Florence, Italy b Istituto Nationale di Fisica Nucleate, Sezione di Firenze Large E. Fermi 2, 50125 ’ Institute de Fisica, UNAM, Apdo. Postal 20-364, 01000 Mexico D.F., Mexico d Facoltd di Scienze M.F.N., Universitd. della Basilicata, Potenza, Italy
Received 17 January 1990 Revised manuscript received 27 February
Florence,
Italy
1990
A coupled map lattice model is studied which presents transient and asymptotic chaotic states depending on the value of the control parameters. A first-order approximation detects the presence of asymptotic periodic attractors. The statistics of transient times as well as their dependence on lattice size are investigated.
1. Introduction
regimes can be described sition
The
study
of complex
dynamical
behavior
is
currently of interest in many research fields ranging from physics to chemistry and biology. This
mechanism.
jecture
in terms of a phase tran-
In a recent
has been supported
paper
[2] this con-
by numerical
analysis
and
for a purely expanding linear piecewise map and for the logistic map. Both cases are characterized by the appearance of new ground states as the
simple models, which reproduce the main features of such dynamical regimes. Coupled map lattices (CML) have been proposed by various au-
strength of the coupling is varied. A similar crossover between “turbulent” and %oliton” phases was studied in a one-dimensional
thors [l] as a promising approach in this direction. They are usually defined as discrete (both in space
cell dynamical model for chemical turbulence [3]. The aim was to describe a typical reaction-diffus-
and time) dynamical systems, where replicas of a map are coupled via a nearest-neighbor interac-
ion process by the introduction of a three-state cellular automaton (CA) cell dynamics. This ap-
tion,
peared
has induced
a demand
reproducing
for sufficiently
a discrete
version
general
of a diffusive
coupling. Temporal and spatial discreteness allow natural computer implementation of such models. On the other hand, due to the diffusive coupling, CML are expected to be a simplified version of systems, where complex evolution may arise as a product of a collective behavior originated by coherent interactions among degrees of freedom. For instance, this is a typical situation for the transition to turbulence via spatiotemporal intermittency in fluid systems. Moreover, recent results (third paper of ref. [l]) have shed new light on the possibility of a statistical description of CML. It is expected that the crossover between different global dynamical
to be a necessary
@ 1990 - Elsevier Science Publishers
because
com-
In this paper we study a CML model which describes reaction-diffusion processes. Its definition and main features are presented in section 2. In the following sections the reader will realize that the proposed model is characterized by a very interesting and rich variety of dynamical behaviors. It is worth stressing that knowledge of the spatially homogeneous dynamics discussed in section 3 represents a sort of “first-order” dynamical description of our model. More precisely, the bifurcation diagram characteristic of the superstable
. 0167-2789/90/$03.50
ingredient,
plex behaviors in reaction-diffusion processes are expected to appear from the interaction of a great number of orbitally stable limit cycles.
B.V. (North-Holland)
Livi et al. / Periodic orbits in C M L s
version of our m a p shows t h a t only periodic orbits are present. This provides a guide for the description of the asymptotic dynamics in the inhomogeneous case, as discussed in section 4. In fact, the main feature of this model is the presence of periodic attractors (typical of the homogeneous dynamics) towards which the CML may relax after some transient time. We have observed numerically that the statistics of the transient times is typically Poisson-like. This appears to be a generic feature of the model. This indicates the presence of characteristic escape times, depending on the values of the parameters. As discussed in section 4, Lyapunov characteristic exponent (LCE) analysis 'has provided a sound description for the case of inhomogeneous initial conditions. In particular the dynamical analysis of the LCE spectrum clearly characterizes: - the various dynamical mechanisms which lead to periodic attractors (when they are present); the "chaoticity" of the transients towards these attractors. Here the word "chaoticity" is used in order to stress the fact that, for any finite array of maps, the CML dynamics relaxes to a LCE s p e c t r u m with a positive component, before reaching over longer times its asymptotic value. We want to stress that this is a well-defined concept. In analogy with low-dimensional repellers, the positive part of the L C E s p e c t r u m can be interpreted as a measure of the chaotic component of the invariant set characterizing the transient dynamics. We also report some results on the exponential growth of the average transient time for increasing lattice dimension, thus showing the relevance of transient dynamics to the properties of the system in the t h e r m o d y n a m i c limit. Finally, we report some simple analytic calculations of the LCE s p e c t r u m when the dynamics is a t t r a c t e d towards a homogeneous fixed point. -
2. T h e m o d e l CML models are defined on a regular lattice where the state x of a variable at site (or cell) i evolves according to a m a p F applied to its neighborhood vi:
453
(1)
i jEv~
where t represents the discrete time step and i -1, 2, .., N with periodic b o u n d a r y conditions. In one dimension the ~ j ' s are usually chosen as follows: ai----1-c,
ai+1=~/2,
c E [0,1].
(2)
This rule mimics a s p a c e - t i m e discrete version of a nonlinear diffusive process as observed, for instance, in reaction-diffusion systems. In such cases for c = 0 (isolated cells) the dynamics (1) should generate a globally stable limit cycle. The simplest choice t h a t complies with this constraint is the reduction of dynamics of an isolated cell to a cyclic visitation of discrete levels [3]. This actually reduces CML to CA. One of the motivations behind this work is to study a CML model in which the periodic behavior is a consequence of the coupling and not intrinsic to the single m a p dynamics. Keeping this in mind we have chosen a m a p of the interval with a fixed-point dynamics, in which an expanding and a contracting component can be identified. We shall call these components "turbulent" and "quiescent" phases, respectively. The m a p p i n g is given by Fa,b(X) = x / a ,
x < a,
= a + b(x - a),
z > a,
(3)
with 0 < a, b < 1. Any initial condition (apart from x = 0) will eventually be m a p p e d into the stable fixed point x = a. For the coupled problem our choice of parameters a j is o~i = 1,
O~i:t: 1 =
e/2,
e E [ - 1 , oo).
(4)
Notice that this p a r a m e t e r selection requires the introduction of a modulo 1 operation in the argument of F. This corresponds to a shift operation, which, as we shall see in the following sections, gives rise to different global periodic behaviors. Our coupled m a p dynamics m a y now be s u m m a rized by the following fornmla:
xt-F1 i
1
t
xt
= G ~ , b ( ~ c D x i + rl i),
(5)
454
.Livi et al. / Periodic orbits in CMLs
where Ga,b(+) = Fa,b([2] mod l), D is the discrete Laplace operator and n is the reaction constant. The
argument
of G makes
with a reaction-diffusion
explicit,
system;
the
relation
in our case, both 0.7
the reaction and the diffusion constants are related to the coupling paramet,er E , the former being e/2 and the latter
1 + E. We will see that
0.6
the model
0.5
so defined exhibits ext,remely rich dynamical behavior, with periodic and chaotic regimes arising
0.4
as collective
0.3
phenomena.
In this respect
vance of the model goes beyond reaction-diffusion process.
the rele-
the modeling
of a
0.2 0.1
0.0
1 0.1
0‘0
3.
Homogeneous
dynamics
Homogeneous
initial conditions,
5’ E [0,11, reduce
i.e. Z: = 2’ Vi,
eq. (5) to
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig. 1. The thick solid line segments are the graph of eq. (6) for (a,~,b) = (0.4,0.3,0.0); when b = b, = 0.61538 the horizontal line should be substituted by the dashed line; when b = b, = 0.76923 it should be substituted by the dash-dotted one.
,‘+l = Ga,6 [(l + E)z’], where the explicit
dependence
i is lost but the E dependence
on the space index is maintained.
This
persistence of the E dependence is a feature of our model, at variance with the usual CML treatments reported so far in the literature [1] #l. It allows us, as we shall show further on, to construct a “firstorder” approximation of the collective behavior of eq.
(5)
with
generic
of the analysis We shall devote
initial
conditions,
of the homogeneous this section
in terms
dynamics
to the study
(6). of the
homogeneous dynamics (6), for which some exact results can be established, with the aim of gaining some insight of the behavior for random initial conditions. 3.1.
The case b =0
In section 2 we mentioned that the modulo-1 operation is the origin of multiperiodic dynamics. The study of eq. (6) for b = 0, which corresponds to the map with a superstable fixed point i = a (see fig. l), provides a first understanding of this ‘statement. It is straightforward to observe that when E E [-I, a - l), for any initial condition, #r Let us observe that such a dependence is maintained also for diffusive coupling (see ref. [2]), when spatially periodic solutions of period larger than 1 are considered.
2’, the system will relax to the stable fixed point 2 = 0, while at E = a - 1, the system remains frozen in %O. For E > a - 1, there is always a stable periodic orbit following:
present.
This can be proved by the
Lemma 1 For any z”, 0 < a < 1 and e E (a - 1, co) the preimages in eq. (6) of Z = a have Lebesgue measure PL = 1, i.e. they coincide with the whole interval Lebesgue Proof:
[0, l]
apart from
a set of zero
measure. When
E E (a -
1, oo) the first preimage
of the point z = a is the set A = (a/(1 + E), l/(1 + E)); the following preimages are then some intervals
Ik such that
pi
# 0 and Ik nA
= 0,
i.e. the II, belong to the set where Ga,e is expanding. Let us suppose now that there exists a set J, FL(J) # 0 such that JnnA = 0 Vn = 1,2,... (where J” E G,“,,(J) stands for the nth image of J). By definition, J” n(u, Ik) = 0 tin, but this is impossible. According to the hypothesis, J and its images must belong to the expanding component of Go,0 , and there will exist an N < co such cl that JN n(& Ik) # 0. Theorem 1 For any 0 < a < 1 and E E (a - 1, co) the map (6) has only periodic orbits for pi -aEmost
all
3’.
Livi et al. //Periodic orbits in CMLs (] 1 0,9 0.8 0,7 0.6 0,5 0,4 0,5 0
0,2
0.4
0.6
0.8
'l
1.2
1.4
16
1,8 ~- 2
Fig. 2. Partial (a, e) bifurcation diagram for eq. (6), showing some of the regions of stable periodic dynamics (horizontal stripes indicate period 1, vertical period 2 and black period 3). To the left of line D1 (1-reinjection zone) all period-l,2,3 tongues are indicated. In between D1 and D2 all period-1 tongues and all period-2 and -3 tongues bounded by the first two period-1 tongues are shown. To the right of D~, all period-2 and some of the period-3 delimited by the first two period-1 tongues are depicted.
As a consequence of l e m m a 1 for #Lalmost all ~0 the dynamics will eventually reach the point $ = a, whose image #L-almost surely belongs to the set of preimages of a. This proves the theorem and moreover implies that • = a belongs to every stable periodic orbit, the period of which depends on the values of a and ~. [] Proof:
As we vary ~ from a - 1 to 0 we find stable solutions of period n ÷ 1 for a n/('~+l) - 1 ~ e < a (~-l)/n-1, n = 1, 2, . . .. When e = 0 we have the decoupled case with the fixed point solution 5~ = a. Once E is positive then the coupling is diffusive, which is physically more interesting. In this case the (a, ~) p a r a m e t e r space is partitioned into tongues of different periodicity. As a consequence of theorem 1, these tongues fill the (a,c) space and do not overlap. This last p r o p e r t y is due to the non-intersection of the preimages of a for mapping (6), i.e., multistability is ruled out for b=O.
In fig. 2, the plot of just some of the low-period tongues already gives an indication of the complexity of the full bifurcation diagram. A useful classification of this diagram can be made in terms of the n u m b e r of reinjections, i.e. the n u m b e r of plateaus in the graph of m a p p i n g (6), for a given value of (a, e). For a > v, we are in the 1-reinjection zone of the (a, e) diagram (region on the left of D1 in fig.
455
2). Within this region, the tongues are disposed hierarchically: each tongue of period n > 1 has on both sides one tongue of period n + 1, with the separatrices of each tongue corresponding to infinitely periodic solutions. As we enter the 2-reinjection zone bounded by the straight lines a -- e and a = 1 + e (D1 and D2 of fig. 2, respectively), the above structure is strongly enriched by the generation of new tongues. Moreover, new features appear: for example, a period-3 tongue shares a b o u n d a r y with the second period-1 tongue. In general, this trend toward increasing complexification is observed as the number of reinjections grows. It is not our aim in this work to give a detailed description of the bifurcation diagram but rather focus our attention on the dynamical behavior of the CML in some regions of the 1- and 2-reinjection zones between the first and second period-1 tongues.
3.2. T h e case b ¢ 0
When b ¢ 0, the overall periodic behavior is modified by the appearance of chaotic dynamics. Some features of the bifurcation diagram of the now three-dimensional p a r a m e t e r space ( a , c , b ) can be studied by looking at the evolution of the b --- 0 plane for increasing values of b. As an example of this procedure we shall consider the first period-1 tongue. In fig. 1, we plot the m a p p i n g (6) for the (a, e, b) p a r a m e t e r values: (0.4, 0.3, 0), (0.4, 0.3, be) and (0.4, 0.3, br), where bc(a,~) = [ 1 / ( l ÷ c ) - a ] / ( 1 - a ) is the value of b at which the mapping looses its fixed point and br(a, c) -- 1 / ( I + E ) is the m i n i m u m value of b for which the m a p p i n g becomes fully unstable. Since we have chosen (a,E) within the b = 0 first period-1 tongue, as long as 0 < b < be we will always have a stable fixed point Sfp -(b 1)a/[b(1 + E) - 1] within the interval A of l e m m a 1. Moreover, the fixed point is stable and its domain of attraction is the whole interval, since the preimages of A are the same as in the b -- 0 case. However, in this case the inverse image of a is no longer the interval A. This shows that the system evolves asymptotically to a stable fixed point -
456
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et al. / Periodic
orbits
in CMLs
4. Inhomogeneous
1.0
initial
conditions
0.9
0.8 0.7 0.6
0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.2
0.4
0,6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Fig. 3. Diagram of the first period-l tongue in the (a,~) parameter plane for different b values: when b = 0 the tongue is the sum of regions PRl and CRl; for 6 = 0.2 it has been compressed to region PRl. In this last case region CR1 corresponds to a chaotic behavior.
instead of reaching in a finite number of steps a period-l orbit. When b > b, the mapping is unstable since all the slopes are larger than one. For the remaining case of b, > b > b, we conjecture, on the basis of numerical evidence, that stable periodic orbits are not present. So far our calculation of the Lyapunov exponent within this region suggests a chaotic dynamics. The above information is summarized in fig. 3. For b # 0, the b = 0 first period-l tongue delimited above by a(c,O) = l/(1 + &) is compressed to the striped region PRl bounded above by a(~, b) = [(l/(1 + E) - b]/(l - b). Within this region the stable period-l dynamics is preserved. Region CR1 on the other hand corresponds to a chaotic behavior. The mechanism we have so far described for the evolution of the first period-l tongue takes place for any tongue of the b = 0 bifurcation diagram. The tongue reduces in size as b grows, leaving behind a region with no stable periodic orbits. Namely, if we take any set of parameters (a, E, b = 0), it must belong to a tongue, say of period n; we may then define the values b: and b: for the graph of the nth iteration of the mapping in analogy to b, and b, of fig. 1. Note that b,(E) > b:(a,c), which is in agreement with the fact that for any E > (1 - b)/b the dynamics is globally unstable.
The dynamical scenarios which characterize the homogeneous map (6) are ruled by different kinds of attractors. For any fixed value of b the (a,~) space is partitioned into zones dominated by periodic or chaotic dynamics. Here we want to study how these dynamical regimes change when the coupled map dynamics (5) is considered in full generality. Even when relatively simple local dynamics is considered, the coupling of a large number of spatial degrees of freedom may produce complex collective behaviors. We will stress in this paper the presence of chaotic transients which grow with system size (this feature has already been noted [4] for other models). The knowledge of the partition of the (a, E, b = 0) space into periodic attractors for the homogeneous state provides a good approximation for the orbits of lower period towards which an inhomogeneous or random initial state eventually converges. A more complicated situation arises for b # 0, when chaotic attractors and chaotic transients may coexist, which require a more refined description of the invariant and transient measures. For fixed N (2 20) and fixed parameter values a, E by varying b we have observed a value 6(N, a, E), below which the initially inhomogeneous state relaxes to a periodic attractor, showing a transient chaotic behavior. Above b numerical experiments indicate the presence of a chaotic attractor. We will not analyze this case in this paper, but rather concentrate on the study of chaotic transients leading to periodic attractors. From the analysis of the previous section we know that the necessary condition for this to happen is b < b,. Most of the results that we are going to discuss have been obtained by numerical simulations. The analysis of a chaotic transient requires two ingredients. First, one must compute a transient mean time over different initial conditions, when the dynamics relaxes on an attractor. We shall see that the probability distribution of transient times decreases exponentially at large times. Hence, a typical escape time can be defined, as in the theory of low-dimensional repellers [5]. Second, one must characterize the chaoticity within the transient. For this purpose we have chosen to study the finite time spectrum of Lyapunov characteristic ex-
L i v i et al.
/
ponents (LCE), defined as follows: Ai+I(T) = r/i+l(T)
-
-
457
P e r i o d i c orbits i n C M L s
co0 - ' '
'1 ....
I ....
I ....
I ....
1'-
7h(T),
where 1
t+l II,~lt +- -lt - -A,~2 A...A,~+lll -~....
In
400 ,
A being the external product and ~j,3 t . 1,...,N a set of N-dimensional linearly independent vectors chosen to be initially an orthonormal basis in the tangent space of the dynamical system (5). The linear dynamics of the ~. is therefore =
~t.+l :
t t
.
.
~0
.
. 40
.
80
I .... O0
],, 100
= ~:,
with
( A)'u
0
-
OF Oy~ Ox;
t '
Fig. 4. U n n o r m a l i z e d frequency d i s t r i b u t i o n of t r a n s i e n t times for N = 50 and ( a , e , b ) = (0.75,0.3332,0) inside the first period-1 t o n g u e of fig. 2. T h e statistical s a m p l e has been obtained by considering 104 different r a n d o m initial conditions.
jEvl
The existence of a positive component of the L C E s p e c t r u m as T --- oo guarantees the presence of a chaotic a t t r a c t o r [5]. In the presence of a chaotic transient we observe the relaxation within a finite time to a LCE s p e c t r u m having a positive component, before the latter asymptotically converges to the full negative L C E s p e c t r u m typical of a periodic attractor.
4.1. The case b = 0 In the superstable case b = 0 , for finite N , the LCE s p e c t r u m collapses to - o o . This is easily revealed numerically by the convergence to the null vector of the set of tangent vectors ~i- We have observed two main mechanisms for this convergence, depending on the chosen initial condition. If the L C E spectrum uniformly collapses to - o o following the largest LCE, then, the dynamics converges to the corresponding homogeneous periodic orbit in the (a, c) p a r a m e t e r space. New periodic solutions, unexpected from the analysis based on the homogeneous state, arise when a subset of the L C E spectrum first collapses to - o ~ , followed later by the largest LCEs. In this case the asymptotic state of the lattice is inhomogeneous in space and periodic in time. This multistability is not present in the first period-1 tongue. In fact 0 < ~ < (1 - a ) / a is a sufficient condition for the
dynamics to relax towards the superstable fixed point for any finite N. This is a straightforward consequence of the fact that the argument of Ga,0 in eq. (5) is at most 1, rendering the modulo-1 operation ineffective. This implies that, no m a t t e r what the values of x~=l are, some x~+~ will certainly reach the fixed point for a finite v. From then on, whatever ~t +i ±~l is, the argument of Ga,o always belongs to the attracting superstable fixed point, thus producing a dynamically frozen situation. We have numerical evidence that as N --- oo, r --~ oo in a complicated way (for small N a logarithmic tail is observed in the first period-1 tongue). Therefore, in this limit the transient becomes the a s y m p t o t i c state and the lattice will never satisfy the conditions discussed above which guarantee the convergence to the fixed point. Returning to the finite N case, fig. 4 shows the frequency of transient times over a sample of 104 different random initial conditions for (a, c, b) = (0.75,0.3332,0) inside the first period-1 tongue. Similar frequency distributions have been observed for increasing N (up to N = 400). During this process the most probable transient time value increases and the variance is approximately constant. The frequency distributions a p p e a r to be Poisson-like, with an exponential tail for large transients. Repeating the same analysis just outside this
458
Livi et al. / Periodic
tongue, (a,~, b) = (0.75,0.3335,0), we observe again Poisson-like distributions of transient times and a very slow increase of the average transient with N for small N. However, at variance with the previous case, multistability is present, i.e. the lattice evolves towards different periodic attractors varying the initial condition. In any case, the existence of a typical escape time characteristic of the statistics is a common feature of our model for any (a, E, 0) values, although the mechanisms of convergence to asymptotic attractors may be different. This is reminiscent of chaotic repellers, e.g. Cantor unstable invariant sets found in some low-dimensional maps. However, while such sets are in general difficult to reveal numerically, in our case most of the dynamics is ruled by this kind of transient chaos if N is sufficiently large. In the language introduced in ref. [4] the observed transients look “quasi-stationary”, although in our model we observe only an exponential increase of transient times with system size, at variance with the superexponentials found in ref.
WI.
We have already observed that the modulo-1 shift operation should be at the origin of the complex behaviors of this model. We expect, for instance, that increase in the number of reinjections should increase transient times. In order to verify this hypothesis we have repeated the same numer-
orbits in CMLs
ical simulations in some points inside the second period-l tongue in the 2-reinjection zone. In fig. 5 we show the increase of the mean transient time with N, which is typically well fit by an exponential at large N. Another effect of increasing number of reinjections is the convergence towards multiple periodic attractors. For instance in a period-3 tongue of the 2-reinjection zone for (a, E, b) = (0.7692307,1.3, 0) we have observed convergence to three distinct attractors of periods 3 (homogeneous), 45, 243 (periodic inhomogeneous states), over a number of 30 trials for N = 9. Before converging to these attractors the dynamics develops on a chaotic transient. In fig. 6 we show the transient LCE spectrum for N = 50 (crosses) and N = 100 (full triangles) for the same parameter values, in both cases we plot the largest N/2 exponents Xi as a function of i/N. Two features are relevant: the convergence to a limit distribution of the LCE spectrum as N increases (typical of chaotic dynamical systems [6,7]) and the presence of a positive component. Another interesting effect is that the limiting distribution of LCE extends to -oo. This can be proved rigorously since the determinant of the matrix & in eq. (7) is zero due to the presence of a zero derivative attractive interval. We have seen that the greater the number of reinjections, the longer the transients. This effect
Fig. 5. Logarithm of the average transient time as a function of the lattice size N for (a,~, b) = (0.81,1.3,0). obtained by taking the mean over 100 different random initial conditions.
Each point is
Livi et al. / Periodic orbits in CMLs
459
1
0.5 0
t
I
0.45
0.5
-0.5
!
~A
-1
~A
-1.5
-2
~A
-2.5
A
3
A A
-3.5 4
Fig. 6. Transient spectrum of Lyapunov characteristic exponents for (a, e, b) : (0.7692307, 1.3, 0) for N : 50 (crosses) and N = 100 (full triangles), in both cases we plot the largest N / 2 exponents as a function of i/N. Observe the convergence to a limiting transient LCE distribution as N --* c¢. The spectrum is independent of the initial Condition.
remains in the e --* oc limit since the Lebesgue measure of the plateaus in the graph of the m a p corresponding to the homogeneous solution is finite and tends to 1 - a. A simple qualitative characterization of s p a c e time patterns is obtained by partitioning the interval into two regions, corresponding to the expandi n g and contracting components, respectively. Chaotic transients show symbolic patterns typical of spatiotemporal intermittency [8] (see fig. 3 in ref. [9] for a typical pattern), both for cases leading to homogeneous and inhomogeneous periodic states. 4.2. A simple analytical result for b 5 0 We have not yet analyzed numerically in detail the evolution of the lattice from an inhomogeneous initial state for b =~ 0. However a simple result is obtained when the dynamics converges to a homogeneous fixed point. In this state the eigenvalues of matrix A t in (7) are given by pk:b
l÷ccos
~:
,
k=l,...,N.
One can therefore obtain rigorously the asymptotic LCE spectrum, by taking the logarithms of the moduli of the #h's. The previous formula was
obtained also in re£ [10] for constant expansion rates. This is for instance the case in the first periodic region PR1 of fig. 3 when b < be. If this happens one can extend the definition of chaotic transient used in section (4.1) to the b ~ 0 case, when the asymptotic solution is a homogeneous fixed point. In fact one can define transient time as the time needed to reach the known a s y m p t o t i c L C E spectrum. Little is known about the b > be case. As we have anticipated in section 4 the value b > be, above which a chaotic a t t r a c t o r exists, depends on N in a nontrivial way; a preliminary analysis shows a slow decrease of b with N , presumably reaching be as N ~ oc. While the study of transient chaos by the analysis of the LCE s p e c t r u m can be reproduced for b < b, the characterization of chaoticity for b > b represents a much more difficult task. In the latter case the LCE spectra do not allow a clear qualitative discrimination between transient chaos and chaotic attractors, since the shape of L C E spectra does not change significantly with respect to that obtained in a regime of chaotic transient. Interesting effects should be present at the border of the region PR1 with the region CR1 of fig. 3, and in general at the transition from a region where a chaotic transient is present to one where the dynamics evolves towards a chaotic attractor.
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Spatiotemporal symbolic patterns reproduce those of fully developed turbulence [ll] only far above 6, while near 6 a large degree of spatial cor-
Acknowledgements
relation
CNR/CONACYT
is still present
(see fig. 4a in ref. [9] for an
GM
thanks
the
DGAPA
of Mexico
of
the
UNAM,
City and the INFN
in
example).
Florence for financial support. Moreover RL and SR thank the CNR/CONACYT support to their visits in Mexico City. We all thank F. Bagnoli, S.
5.
Isola and A. Politi
Conclusions
In this paper we have investigated the dynamical properties of a CML model for reactiondiffusion processes. The presence of chaotic behavior characterizing the transient dynamics towards periodic states has been analyzed in detail. In particular
a study of the homogeneous
dynamics
References 111 I. Waller and Il. Kapral, Phys. Rev. A 30 (1984) 2047;
has
provided, through the identification of some properties of the asymptotic periodic state, a “first or-
PI
der” description
131 (41
of the dynamical
system.
second paper of ref. [l] the stability spatially periodic state with coupling cally investigated;
our approach
In the
of an initial was numeri-
is complementary,
[51
in the sense that we study analytically the periodic time evolution of a homogeneous initial state. The main results that we have obtained concern: _ the statistics of transient times, which has been observed to be Poisson-like; this suggests the existence of characteristic escape times from the
PI 171 PI PI
chaotic transient; _ the presence over typical transient times of a relaxed spectrum of LCE with a positive component detecting chaotic states; _ an exponential growth with lattice size of the average transient times. This demonstrates the relevance of chaotic model
transients
in the thermodynamic
to the properties limit.
of our
for useful discussions.
1101 IllI
K. Kaneko, Progr. Theor. Phys. 72 (1984) 480; L.A. Bunimovich and Ya.G. Sinai, Nonlinearity 1 (1989) 491. L.A. Bunimovich, A. Lambert and Il. Lima, J. Stat. Phys., in press. Y. Oono and C. Yeung, 3. Stat. Phys. 48 (1987) 593. J.P. Crutchfield and K. Kaneko, Phys. Rev. Lett. 60 (1988) 2715. L.P. Kadanoff, Applications of scaling ideas to dynamics, in: Regular and Chaotic Motions in Dynamic Systems, eds. G. Velo and A.S. Wightman, NATO AS1 Series, Vol. 118 (Plenum Press, New York, 1985). J.P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57 (1985) 617. R. Livi, A. Politi and S. Ruffo, J. Phys. A 19 (1986) 2033. K. Kaneko, Prog. Theor. Phys. 74 (1985) 1033. F. Bagnoli, S. Isola, Ft. Livi, G. Martinez-Mekler and S. R&o, Periodic orbits in a coupled map lattice model, in: Cellular Automata and Modeling of Complex Systems, eds. P. Manneville, N. Boccara, G.Y. Vichniac and R. Bideaux, Springer Proceedings in Physics, Vol. 46 (Springer, Berlin, 1990) p. 282. K. Kaneko, Physica D 23 (1986) 436. K. Kaneko, Physica D 34 (1989) 1.
PERIODIC
452-460
ORBITS
AND
LONG
TRANSIENTS
IN COUPLED
MAP
LATTICES
R. LIVI”> b, G. MARTiNEZ-MEKLER b, ’ and S. RUFFO be d a Dipartimento di Fisica, Universitd di Firenee, Large E. Fermi 2, 50125 Florence, Italy b Istituto Nationale di Fisica Nucleate, Sezione di Firenze Large E. Fermi 2, 50125 ’ Institute de Fisica, UNAM, Apdo. Postal 20-364, 01000 Mexico D.F., Mexico d Facoltd di Scienze M.F.N., Universitd. della Basilicata, Potenza, Italy
Received 17 January 1990 Revised manuscript received 27 February
Florence,
Italy
1990
A coupled map lattice model is studied which presents transient and asymptotic chaotic states depending on the value of the control parameters. A first-order approximation detects the presence of asymptotic periodic attractors. The statistics of transient times as well as their dependence on lattice size are investigated.
1. Introduction
regimes can be described sition
The
study
of complex
dynamical
behavior
is
currently of interest in many research fields ranging from physics to chemistry and biology. This
mechanism.
jecture
in terms of a phase tran-
In a recent
has been supported
paper
[2] this con-
by numerical
analysis
and
for a purely expanding linear piecewise map and for the logistic map. Both cases are characterized by the appearance of new ground states as the
simple models, which reproduce the main features of such dynamical regimes. Coupled map lattices (CML) have been proposed by various au-
strength of the coupling is varied. A similar crossover between “turbulent” and %oliton” phases was studied in a one-dimensional
thors [l] as a promising approach in this direction. They are usually defined as discrete (both in space
cell dynamical model for chemical turbulence [3]. The aim was to describe a typical reaction-diffus-
and time) dynamical systems, where replicas of a map are coupled via a nearest-neighbor interac-
ion process by the introduction of a three-state cellular automaton (CA) cell dynamics. This ap-
tion,
peared
has induced
a demand
reproducing
for sufficiently
a discrete
version
general
of a diffusive
coupling. Temporal and spatial discreteness allow natural computer implementation of such models. On the other hand, due to the diffusive coupling, CML are expected to be a simplified version of systems, where complex evolution may arise as a product of a collective behavior originated by coherent interactions among degrees of freedom. For instance, this is a typical situation for the transition to turbulence via spatiotemporal intermittency in fluid systems. Moreover, recent results (third paper of ref. [l]) have shed new light on the possibility of a statistical description of CML. It is expected that the crossover between different global dynamical
to be a necessary
@ 1990 - Elsevier Science Publishers
because
com-
In this paper we study a CML model which describes reaction-diffusion processes. Its definition and main features are presented in section 2. In the following sections the reader will realize that the proposed model is characterized by a very interesting and rich variety of dynamical behaviors. It is worth stressing that knowledge of the spatially homogeneous dynamics discussed in section 3 represents a sort of “first-order” dynamical description of our model. More precisely, the bifurcation diagram characteristic of the superstable
. 0167-2789/90/$03.50
ingredient,
plex behaviors in reaction-diffusion processes are expected to appear from the interaction of a great number of orbitally stable limit cycles.
B.V. (North-Holland)
Livi et al. / Periodic orbits in C M L s
version of our m a p shows t h a t only periodic orbits are present. This provides a guide for the description of the asymptotic dynamics in the inhomogeneous case, as discussed in section 4. In fact, the main feature of this model is the presence of periodic attractors (typical of the homogeneous dynamics) towards which the CML may relax after some transient time. We have observed numerically that the statistics of the transient times is typically Poisson-like. This appears to be a generic feature of the model. This indicates the presence of characteristic escape times, depending on the values of the parameters. As discussed in section 4, Lyapunov characteristic exponent (LCE) analysis 'has provided a sound description for the case of inhomogeneous initial conditions. In particular the dynamical analysis of the LCE spectrum clearly characterizes: - the various dynamical mechanisms which lead to periodic attractors (when they are present); the "chaoticity" of the transients towards these attractors. Here the word "chaoticity" is used in order to stress the fact that, for any finite array of maps, the CML dynamics relaxes to a LCE s p e c t r u m with a positive component, before reaching over longer times its asymptotic value. We want to stress that this is a well-defined concept. In analogy with low-dimensional repellers, the positive part of the L C E s p e c t r u m can be interpreted as a measure of the chaotic component of the invariant set characterizing the transient dynamics. We also report some results on the exponential growth of the average transient time for increasing lattice dimension, thus showing the relevance of transient dynamics to the properties of the system in the t h e r m o d y n a m i c limit. Finally, we report some simple analytic calculations of the LCE s p e c t r u m when the dynamics is a t t r a c t e d towards a homogeneous fixed point. -
2. T h e m o d e l CML models are defined on a regular lattice where the state x of a variable at site (or cell) i evolves according to a m a p F applied to its neighborhood vi:
453
(1)
i jEv~
where t represents the discrete time step and i -1, 2, .., N with periodic b o u n d a r y conditions. In one dimension the ~ j ' s are usually chosen as follows: ai----1-c,
ai+1=~/2,
c E [0,1].
(2)
This rule mimics a s p a c e - t i m e discrete version of a nonlinear diffusive process as observed, for instance, in reaction-diffusion systems. In such cases for c = 0 (isolated cells) the dynamics (1) should generate a globally stable limit cycle. The simplest choice t h a t complies with this constraint is the reduction of dynamics of an isolated cell to a cyclic visitation of discrete levels [3]. This actually reduces CML to CA. One of the motivations behind this work is to study a CML model in which the periodic behavior is a consequence of the coupling and not intrinsic to the single m a p dynamics. Keeping this in mind we have chosen a m a p of the interval with a fixed-point dynamics, in which an expanding and a contracting component can be identified. We shall call these components "turbulent" and "quiescent" phases, respectively. The m a p p i n g is given by Fa,b(X) = x / a ,
x < a,
= a + b(x - a),
z > a,
(3)
with 0 < a, b < 1. Any initial condition (apart from x = 0) will eventually be m a p p e d into the stable fixed point x = a. For the coupled problem our choice of parameters a j is o~i = 1,
O~i:t: 1 =
e/2,
e E [ - 1 , oo).
(4)
Notice that this p a r a m e t e r selection requires the introduction of a modulo 1 operation in the argument of F. This corresponds to a shift operation, which, as we shall see in the following sections, gives rise to different global periodic behaviors. Our coupled m a p dynamics m a y now be s u m m a rized by the following fornmla:
xt-F1 i
1
t
xt
= G ~ , b ( ~ c D x i + rl i),
(5)
454
.Livi et al. / Periodic orbits in CMLs
where Ga,b(+) = Fa,b([2] mod l), D is the discrete Laplace operator and n is the reaction constant. The
argument
of G makes
with a reaction-diffusion
explicit,
system;
the
relation
in our case, both 0.7
the reaction and the diffusion constants are related to the coupling paramet,er E , the former being e/2 and the latter
1 + E. We will see that
0.6
the model
0.5
so defined exhibits ext,remely rich dynamical behavior, with periodic and chaotic regimes arising
0.4
as collective
0.3
phenomena.
In this respect
vance of the model goes beyond reaction-diffusion process.
the rele-
the modeling
of a
0.2 0.1
0.0
1 0.1
0‘0
3.
Homogeneous
dynamics
Homogeneous
initial conditions,
5’ E [0,11, reduce
i.e. Z: = 2’ Vi,
eq. (5) to
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig. 1. The thick solid line segments are the graph of eq. (6) for (a,~,b) = (0.4,0.3,0.0); when b = b, = 0.61538 the horizontal line should be substituted by the dashed line; when b = b, = 0.76923 it should be substituted by the dash-dotted one.
,‘+l = Ga,6 [(l + E)z’], where the explicit
dependence
i is lost but the E dependence
on the space index is maintained.
This
persistence of the E dependence is a feature of our model, at variance with the usual CML treatments reported so far in the literature [1] #l. It allows us, as we shall show further on, to construct a “firstorder” approximation of the collective behavior of eq.
(5)
with
generic
of the analysis We shall devote
initial
conditions,
of the homogeneous this section
in terms
dynamics
to the study
(6). of the
homogeneous dynamics (6), for which some exact results can be established, with the aim of gaining some insight of the behavior for random initial conditions. 3.1.
The case b =0
In section 2 we mentioned that the modulo-1 operation is the origin of multiperiodic dynamics. The study of eq. (6) for b = 0, which corresponds to the map with a superstable fixed point i = a (see fig. l), provides a first understanding of this ‘statement. It is straightforward to observe that when E E [-I, a - l), for any initial condition, #r Let us observe that such a dependence is maintained also for diffusive coupling (see ref. [2]), when spatially periodic solutions of period larger than 1 are considered.
2’, the system will relax to the stable fixed point 2 = 0, while at E = a - 1, the system remains frozen in %O. For E > a - 1, there is always a stable periodic orbit following:
present.
This can be proved by the
Lemma 1 For any z”, 0 < a < 1 and e E (a - 1, co) the preimages in eq. (6) of Z = a have Lebesgue measure PL = 1, i.e. they coincide with the whole interval Lebesgue Proof:
[0, l]
apart from
a set of zero
measure. When
E E (a -
1, oo) the first preimage
of the point z = a is the set A = (a/(1 + E), l/(1 + E)); the following preimages are then some intervals
Ik such that
pi
# 0 and Ik nA
= 0,
i.e. the II, belong to the set where Ga,e is expanding. Let us suppose now that there exists a set J, FL(J) # 0 such that JnnA = 0 Vn = 1,2,... (where J” E G,“,,(J) stands for the nth image of J). By definition, J” n(u, Ik) = 0 tin, but this is impossible. According to the hypothesis, J and its images must belong to the expanding component of Go,0 , and there will exist an N < co such cl that JN n(& Ik) # 0. Theorem 1 For any 0 < a < 1 and E E (a - 1, co) the map (6) has only periodic orbits for pi -aEmost
all
3’.
Livi et al. //Periodic orbits in CMLs (] 1 0,9 0.8 0,7 0.6 0,5 0,4 0,5 0
0,2
0.4
0.6
0.8
'l
1.2
1.4
16
1,8 ~- 2
Fig. 2. Partial (a, e) bifurcation diagram for eq. (6), showing some of the regions of stable periodic dynamics (horizontal stripes indicate period 1, vertical period 2 and black period 3). To the left of line D1 (1-reinjection zone) all period-l,2,3 tongues are indicated. In between D1 and D2 all period-1 tongues and all period-2 and -3 tongues bounded by the first two period-1 tongues are shown. To the right of D~, all period-2 and some of the period-3 delimited by the first two period-1 tongues are depicted.
As a consequence of l e m m a 1 for #Lalmost all ~0 the dynamics will eventually reach the point $ = a, whose image #L-almost surely belongs to the set of preimages of a. This proves the theorem and moreover implies that • = a belongs to every stable periodic orbit, the period of which depends on the values of a and ~. [] Proof:
As we vary ~ from a - 1 to 0 we find stable solutions of period n ÷ 1 for a n/('~+l) - 1 ~ e < a (~-l)/n-1, n = 1, 2, . . .. When e = 0 we have the decoupled case with the fixed point solution 5~ = a. Once E is positive then the coupling is diffusive, which is physically more interesting. In this case the (a, ~) p a r a m e t e r space is partitioned into tongues of different periodicity. As a consequence of theorem 1, these tongues fill the (a,c) space and do not overlap. This last p r o p e r t y is due to the non-intersection of the preimages of a for mapping (6), i.e., multistability is ruled out for b=O.
In fig. 2, the plot of just some of the low-period tongues already gives an indication of the complexity of the full bifurcation diagram. A useful classification of this diagram can be made in terms of the n u m b e r of reinjections, i.e. the n u m b e r of plateaus in the graph of m a p p i n g (6), for a given value of (a, e). For a > v, we are in the 1-reinjection zone of the (a, e) diagram (region on the left of D1 in fig.
455
2). Within this region, the tongues are disposed hierarchically: each tongue of period n > 1 has on both sides one tongue of period n + 1, with the separatrices of each tongue corresponding to infinitely periodic solutions. As we enter the 2-reinjection zone bounded by the straight lines a -- e and a = 1 + e (D1 and D2 of fig. 2, respectively), the above structure is strongly enriched by the generation of new tongues. Moreover, new features appear: for example, a period-3 tongue shares a b o u n d a r y with the second period-1 tongue. In general, this trend toward increasing complexification is observed as the number of reinjections grows. It is not our aim in this work to give a detailed description of the bifurcation diagram but rather focus our attention on the dynamical behavior of the CML in some regions of the 1- and 2-reinjection zones between the first and second period-1 tongues.
3.2. T h e case b ¢ 0
When b ¢ 0, the overall periodic behavior is modified by the appearance of chaotic dynamics. Some features of the bifurcation diagram of the now three-dimensional p a r a m e t e r space ( a , c , b ) can be studied by looking at the evolution of the b --- 0 plane for increasing values of b. As an example of this procedure we shall consider the first period-1 tongue. In fig. 1, we plot the m a p p i n g (6) for the (a, e, b) p a r a m e t e r values: (0.4, 0.3, 0), (0.4, 0.3, be) and (0.4, 0.3, br), where bc(a,~) = [ 1 / ( l ÷ c ) - a ] / ( 1 - a ) is the value of b at which the mapping looses its fixed point and br(a, c) -- 1 / ( I + E ) is the m i n i m u m value of b for which the m a p p i n g becomes fully unstable. Since we have chosen (a,E) within the b = 0 first period-1 tongue, as long as 0 < b < be we will always have a stable fixed point Sfp -(b 1)a/[b(1 + E) - 1] within the interval A of l e m m a 1. Moreover, the fixed point is stable and its domain of attraction is the whole interval, since the preimages of A are the same as in the b -- 0 case. However, in this case the inverse image of a is no longer the interval A. This shows that the system evolves asymptotically to a stable fixed point -
456
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et al. / Periodic
orbits
in CMLs
4. Inhomogeneous
1.0
initial
conditions
0.9
0.8 0.7 0.6
0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.2
0.4
0,6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Fig. 3. Diagram of the first period-l tongue in the (a,~) parameter plane for different b values: when b = 0 the tongue is the sum of regions PRl and CRl; for 6 = 0.2 it has been compressed to region PRl. In this last case region CR1 corresponds to a chaotic behavior.
instead of reaching in a finite number of steps a period-l orbit. When b > b, the mapping is unstable since all the slopes are larger than one. For the remaining case of b, > b > b, we conjecture, on the basis of numerical evidence, that stable periodic orbits are not present. So far our calculation of the Lyapunov exponent within this region suggests a chaotic dynamics. The above information is summarized in fig. 3. For b # 0, the b = 0 first period-l tongue delimited above by a(c,O) = l/(1 + &) is compressed to the striped region PRl bounded above by a(~, b) = [(l/(1 + E) - b]/(l - b). Within this region the stable period-l dynamics is preserved. Region CR1 on the other hand corresponds to a chaotic behavior. The mechanism we have so far described for the evolution of the first period-l tongue takes place for any tongue of the b = 0 bifurcation diagram. The tongue reduces in size as b grows, leaving behind a region with no stable periodic orbits. Namely, if we take any set of parameters (a, E, b = 0), it must belong to a tongue, say of period n; we may then define the values b: and b: for the graph of the nth iteration of the mapping in analogy to b, and b, of fig. 1. Note that b,(E) > b:(a,c), which is in agreement with the fact that for any E > (1 - b)/b the dynamics is globally unstable.
The dynamical scenarios which characterize the homogeneous map (6) are ruled by different kinds of attractors. For any fixed value of b the (a,~) space is partitioned into zones dominated by periodic or chaotic dynamics. Here we want to study how these dynamical regimes change when the coupled map dynamics (5) is considered in full generality. Even when relatively simple local dynamics is considered, the coupling of a large number of spatial degrees of freedom may produce complex collective behaviors. We will stress in this paper the presence of chaotic transients which grow with system size (this feature has already been noted [4] for other models). The knowledge of the partition of the (a, E, b = 0) space into periodic attractors for the homogeneous state provides a good approximation for the orbits of lower period towards which an inhomogeneous or random initial state eventually converges. A more complicated situation arises for b # 0, when chaotic attractors and chaotic transients may coexist, which require a more refined description of the invariant and transient measures. For fixed N (2 20) and fixed parameter values a, E by varying b we have observed a value 6(N, a, E), below which the initially inhomogeneous state relaxes to a periodic attractor, showing a transient chaotic behavior. Above b numerical experiments indicate the presence of a chaotic attractor. We will not analyze this case in this paper, but rather concentrate on the study of chaotic transients leading to periodic attractors. From the analysis of the previous section we know that the necessary condition for this to happen is b < b,. Most of the results that we are going to discuss have been obtained by numerical simulations. The analysis of a chaotic transient requires two ingredients. First, one must compute a transient mean time over different initial conditions, when the dynamics relaxes on an attractor. We shall see that the probability distribution of transient times decreases exponentially at large times. Hence, a typical escape time can be defined, as in the theory of low-dimensional repellers [5]. Second, one must characterize the chaoticity within the transient. For this purpose we have chosen to study the finite time spectrum of Lyapunov characteristic ex-
L i v i et al.
/
ponents (LCE), defined as follows: Ai+I(T) = r/i+l(T)
-
-
457
P e r i o d i c orbits i n C M L s
co0 - ' '
'1 ....
I ....
I ....
I ....
1'-
7h(T),
where 1
t+l II,~lt +- -lt - -A,~2 A...A,~+lll -~....
In
400 ,
A being the external product and ~j,3 t . 1,...,N a set of N-dimensional linearly independent vectors chosen to be initially an orthonormal basis in the tangent space of the dynamical system (5). The linear dynamics of the ~. is therefore =
~t.+l :
t t
.
.
~0
.
. 40
.
80
I .... O0
],, 100
= ~:,
with
( A)'u
0
-
OF Oy~ Ox;
t '
Fig. 4. U n n o r m a l i z e d frequency d i s t r i b u t i o n of t r a n s i e n t times for N = 50 and ( a , e , b ) = (0.75,0.3332,0) inside the first period-1 t o n g u e of fig. 2. T h e statistical s a m p l e has been obtained by considering 104 different r a n d o m initial conditions.
jEvl
The existence of a positive component of the L C E s p e c t r u m as T --- oo guarantees the presence of a chaotic a t t r a c t o r [5]. In the presence of a chaotic transient we observe the relaxation within a finite time to a LCE s p e c t r u m having a positive component, before the latter asymptotically converges to the full negative L C E s p e c t r u m typical of a periodic attractor.
4.1. The case b = 0 In the superstable case b = 0 , for finite N , the LCE s p e c t r u m collapses to - o o . This is easily revealed numerically by the convergence to the null vector of the set of tangent vectors ~i- We have observed two main mechanisms for this convergence, depending on the chosen initial condition. If the L C E spectrum uniformly collapses to - o o following the largest LCE, then, the dynamics converges to the corresponding homogeneous periodic orbit in the (a, c) p a r a m e t e r space. New periodic solutions, unexpected from the analysis based on the homogeneous state, arise when a subset of the L C E spectrum first collapses to - o ~ , followed later by the largest LCEs. In this case the asymptotic state of the lattice is inhomogeneous in space and periodic in time. This multistability is not present in the first period-1 tongue. In fact 0 < ~ < (1 - a ) / a is a sufficient condition for the
dynamics to relax towards the superstable fixed point for any finite N. This is a straightforward consequence of the fact that the argument of Ga,0 in eq. (5) is at most 1, rendering the modulo-1 operation ineffective. This implies that, no m a t t e r what the values of x~=l are, some x~+~ will certainly reach the fixed point for a finite v. From then on, whatever ~t +i ±~l is, the argument of Ga,o always belongs to the attracting superstable fixed point, thus producing a dynamically frozen situation. We have numerical evidence that as N --- oo, r --~ oo in a complicated way (for small N a logarithmic tail is observed in the first period-1 tongue). Therefore, in this limit the transient becomes the a s y m p t o t i c state and the lattice will never satisfy the conditions discussed above which guarantee the convergence to the fixed point. Returning to the finite N case, fig. 4 shows the frequency of transient times over a sample of 104 different random initial conditions for (a, c, b) = (0.75,0.3332,0) inside the first period-1 tongue. Similar frequency distributions have been observed for increasing N (up to N = 400). During this process the most probable transient time value increases and the variance is approximately constant. The frequency distributions a p p e a r to be Poisson-like, with an exponential tail for large transients. Repeating the same analysis just outside this
458
Livi et al. / Periodic
tongue, (a,~, b) = (0.75,0.3335,0), we observe again Poisson-like distributions of transient times and a very slow increase of the average transient with N for small N. However, at variance with the previous case, multistability is present, i.e. the lattice evolves towards different periodic attractors varying the initial condition. In any case, the existence of a typical escape time characteristic of the statistics is a common feature of our model for any (a, E, 0) values, although the mechanisms of convergence to asymptotic attractors may be different. This is reminiscent of chaotic repellers, e.g. Cantor unstable invariant sets found in some low-dimensional maps. However, while such sets are in general difficult to reveal numerically, in our case most of the dynamics is ruled by this kind of transient chaos if N is sufficiently large. In the language introduced in ref. [4] the observed transients look “quasi-stationary”, although in our model we observe only an exponential increase of transient times with system size, at variance with the superexponentials found in ref.
WI.
We have already observed that the modulo-1 shift operation should be at the origin of the complex behaviors of this model. We expect, for instance, that increase in the number of reinjections should increase transient times. In order to verify this hypothesis we have repeated the same numer-
orbits in CMLs
ical simulations in some points inside the second period-l tongue in the 2-reinjection zone. In fig. 5 we show the increase of the mean transient time with N, which is typically well fit by an exponential at large N. Another effect of increasing number of reinjections is the convergence towards multiple periodic attractors. For instance in a period-3 tongue of the 2-reinjection zone for (a, E, b) = (0.7692307,1.3, 0) we have observed convergence to three distinct attractors of periods 3 (homogeneous), 45, 243 (periodic inhomogeneous states), over a number of 30 trials for N = 9. Before converging to these attractors the dynamics develops on a chaotic transient. In fig. 6 we show the transient LCE spectrum for N = 50 (crosses) and N = 100 (full triangles) for the same parameter values, in both cases we plot the largest N/2 exponents Xi as a function of i/N. Two features are relevant: the convergence to a limit distribution of the LCE spectrum as N increases (typical of chaotic dynamical systems [6,7]) and the presence of a positive component. Another interesting effect is that the limiting distribution of LCE extends to -oo. This can be proved rigorously since the determinant of the matrix & in eq. (7) is zero due to the presence of a zero derivative attractive interval. We have seen that the greater the number of reinjections, the longer the transients. This effect
Fig. 5. Logarithm of the average transient time as a function of the lattice size N for (a,~, b) = (0.81,1.3,0). obtained by taking the mean over 100 different random initial conditions.
Each point is
Livi et al. / Periodic orbits in CMLs
459
1
0.5 0
t
I
0.45
0.5
-0.5
!
~A
-1
~A
-1.5
-2
~A
-2.5
A
3
A A
-3.5 4
Fig. 6. Transient spectrum of Lyapunov characteristic exponents for (a, e, b) : (0.7692307, 1.3, 0) for N : 50 (crosses) and N = 100 (full triangles), in both cases we plot the largest N / 2 exponents as a function of i/N. Observe the convergence to a limiting transient LCE distribution as N --* c¢. The spectrum is independent of the initial Condition.
remains in the e --* oc limit since the Lebesgue measure of the plateaus in the graph of the m a p corresponding to the homogeneous solution is finite and tends to 1 - a. A simple qualitative characterization of s p a c e time patterns is obtained by partitioning the interval into two regions, corresponding to the expandi n g and contracting components, respectively. Chaotic transients show symbolic patterns typical of spatiotemporal intermittency [8] (see fig. 3 in ref. [9] for a typical pattern), both for cases leading to homogeneous and inhomogeneous periodic states. 4.2. A simple analytical result for b 5 0 We have not yet analyzed numerically in detail the evolution of the lattice from an inhomogeneous initial state for b =~ 0. However a simple result is obtained when the dynamics converges to a homogeneous fixed point. In this state the eigenvalues of matrix A t in (7) are given by pk:b
l÷ccos
~:
,
k=l,...,N.
One can therefore obtain rigorously the asymptotic LCE spectrum, by taking the logarithms of the moduli of the #h's. The previous formula was
obtained also in re£ [10] for constant expansion rates. This is for instance the case in the first periodic region PR1 of fig. 3 when b < be. If this happens one can extend the definition of chaotic transient used in section (4.1) to the b ~ 0 case, when the asymptotic solution is a homogeneous fixed point. In fact one can define transient time as the time needed to reach the known a s y m p t o t i c L C E spectrum. Little is known about the b > be case. As we have anticipated in section 4 the value b > be, above which a chaotic a t t r a c t o r exists, depends on N in a nontrivial way; a preliminary analysis shows a slow decrease of b with N , presumably reaching be as N ~ oc. While the study of transient chaos by the analysis of the LCE s p e c t r u m can be reproduced for b < b, the characterization of chaoticity for b > b represents a much more difficult task. In the latter case the LCE spectra do not allow a clear qualitative discrimination between transient chaos and chaotic attractors, since the shape of L C E spectra does not change significantly with respect to that obtained in a regime of chaotic transient. Interesting effects should be present at the border of the region PR1 with the region CR1 of fig. 3, and in general at the transition from a region where a chaotic transient is present to one where the dynamics evolves towards a chaotic attractor.
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orbits
in CMLs
Spatiotemporal symbolic patterns reproduce those of fully developed turbulence [ll] only far above 6, while near 6 a large degree of spatial cor-
Acknowledgements
relation
CNR/CONACYT
is still present
(see fig. 4a in ref. [9] for an
GM
thanks
the
DGAPA
of Mexico
of
the
UNAM,
City and the INFN
in
example).
Florence for financial support. Moreover RL and SR thank the CNR/CONACYT support to their visits in Mexico City. We all thank F. Bagnoli, S.
5.
Isola and A. Politi
Conclusions
In this paper we have investigated the dynamical properties of a CML model for reactiondiffusion processes. The presence of chaotic behavior characterizing the transient dynamics towards periodic states has been analyzed in detail. In particular
a study of the homogeneous
dynamics
References 111 I. Waller and Il. Kapral, Phys. Rev. A 30 (1984) 2047;
has
provided, through the identification of some properties of the asymptotic periodic state, a “first or-
PI
der” description
131 (41
of the dynamical
system.
second paper of ref. [l] the stability spatially periodic state with coupling cally investigated;
our approach
In the
of an initial was numeri-
is complementary,
[51
in the sense that we study analytically the periodic time evolution of a homogeneous initial state. The main results that we have obtained concern: _ the statistics of transient times, which has been observed to be Poisson-like; this suggests the existence of characteristic escape times from the
PI 171 PI PI
chaotic transient; _ the presence over typical transient times of a relaxed spectrum of LCE with a positive component detecting chaotic states; _ an exponential growth with lattice size of the average transient times. This demonstrates the relevance of chaotic model
transients
in the thermodynamic
to the properties limit.
of our
for useful discussions.
1101 IllI
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