PHYSICAL REVIEW LETTERS week ending PRL 96, 034105 (2006) 27 JANUARY 2006 Symbolic Dynamics of Coupled Map Lattices Shawn D. Pethel,1 Ned J. Corron,1 and Erik Bollt2 1U.S. Army Aviation and Missile Command, AMSRD-AMR-WS-ST, Redstone Arsenal, Alabama 35898, USA 2Department of Math and Computer Science, Clarkson University, Potsdam, New York 13699-5805, USA (Received 28 February 2005; published 27 January 2006) We present a method to reduce the RN dynamics of coupled map lattices (CMLs) of N invertibly coupled unimodal maps to a sequence of N-bit symbols. We claim that the symbolic description is complete and provides for the identification of all fixed points, periodic orbits, and dense orbits as well as an efficient representation for studying pattern formation in CMLs. We give our results for CMLs in terms of symbolic dynamical concepts well known for one-dimensional chaotic maps, including generating partitions, Gray orderings, and kneading sequences. An example utilizing coupled quadratic maps is given. DOI: 10.1103/PhysRevLett.96.034105 PACS numbers: 05.45.Jn, 05.45.Ra The study of complex motion is greatly simplified by I a; b and with local dynamics fi: I ! I. A typical investigating models that employ a coarse space-time dis- diffusively coupled map lattice (CML) as introduced by cretization. Such models, typified by coupled map lattices Kaneko is written as (CMLs), have been shown to reproduce the essential fea- i i iÿ1 i1 tures of turbulence in physical, chemical, and biological xn1 1 ÿ fi xn=2fiÿ1 xn fi1 xn ; systems [1]. A further simplification can be achieved by (1) considering iterates of the resulting map through a partition which reduces the chaotic motion to a purely symbolic along with rules for the boundary sites [17]. The behavior signal with associated transition rules. The study of such of the lattice with respect to the coupling parameter is of signals is called symbolic dynamics [2]. An effective state- primary interest here; the dynamics of the local maps fi are space-time discretization connects dynamical systems the- well understood. In particular, we choose f with a good ory to the study of formal languages, and hence, to com- symbolic representation. puter science, information theory, and automata [3]. Such a function can be taken from the family of uni- Symbolic dynamical methods have been studied as a pos- modal (single-humped) maps over the interval. We label sible way to understand space-time chaos [4] and have the location of the maxima as the critical point xc and recently been used [5] to bound entropy and determine divide the interval into two sets P fP0;P1g, where P0 ergodic properties of CMLs [6]. These attempts rely on a; xc and P1 xc;b. The point x0 2 I is mapped to the specially constructed CMLs or on showing that Markov semi-infinite symbol sequence x0s s0s1s2s3 ... partitions of local, uncoupled dynamics are preserved in where the presence of weak interactions [7,8]. 0 if fn x 2P ; s 0 0 (2) In this Letter we describe the symbolic dynamics due to n 1 if fn x 2P : generating partitions, which we claim exist in a much 0 1 broader class of CMLs, including those with strong cou- The critical point can be assigned either symbol. In this pling. For CMLs of N unimodal maps the implication is formulation the dynamics are characterized by a set of that evolution through the RN state space is reducible to a rules that determine which sequences are allowed by f. sequence of N-bit symbols. We conjecture that the sym- The set of allowable sequences can be related to the state bolic description is complete and provides for the rigorous space through the inverse mapping identification of all fixed points, periodic orbits, and dense \1 orbits as well as an efficient representation for studying r s fÿn P : (3) pattern formation in CMLs [9]. We also show that the sn n0 symbol ordering properties of 1D maps extend naturally to CMLs and allow the calculation of topological features For a faithful representation of chaotic dynamics, we re- without the need for extensive time series data. Impor- quire that r s converge to a single point in the interval, or tantly, the results presented here make straightforward to the null set if s is not permitted by f. A partition that the application to CMLs of symbolic dynamical methods satisfies this property is referred to as generating. Note that used recently for the targeting and control of chaos [10– for noninvertible maps fÿ1 has multiple solutions, one on 12], for probing the limits of synchronization [13], and for each monotonic segment of f. For a partition to be gen- efficient chaos communication schemes [14–16]. erating each of these branches must be uniquely labeled, We consider a map lattice with N sites labeled i otherwise points that share a common image would not be i 1; ...;N. Each site is described by a state xn in the interval distinguishable in sequence space. 0031-9007=06=96(3)=034105(4)$23.00 034105-1 © 2006 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 96, 034105 (2006) 27 JANUARY 2006 Returning to the CML, denote by F the product function of fi onto each site, and by A an N N coupling matrix. Then (1) can be generalized as the composition H A F, where A is chosen consistent with H: IN ! IN. Assuming A is nonsingular, the elements of Hÿ1 are X ÿ1 ÿ1 ÿ1 j H x0i fi Aij x0 ; (4) j ÿ1 ÿ1 keeping in mind that fi is multivalued. Because fi acts alone on the ith element the associated generating partition i P of the solitary map fi suffices to distinguish the ÿ1 branches of Hi . Accordingly, we propose a partition for the CML that is the product of the partitions of the local maps. More precisely, _N i 1 2 N P CML P P _ P _ ..._ P ; (5) FIG. 1. Refinements of the square under a two element qua- i1 dratic map lattice, Eq. (6), with coupling strength 0:1. 0 0 (a) The primary partitioning of the square into four symbols, where P _ P fPk \ Pl: 0 k jP j1; 0 l (b) the second refinement into 2-block sequences, (c) 3-block 0 0 jP j1g is the mathematical join of P and P . The sequences (unlabeled) with region 313 darkened, and (d) region form of (4) implies that the preimages of H x0 are 313 subdivided into its 4-block words. The equivalent sequences uniquely labeled under this choice of partition. We con- under the inverse Gray transformation are shown parenthetically. jecture, then, that P CML is generating for chaotic behavior in the CML. Our reasoning is that if the ergodicity is the dynamics are considerably more complicated. Empiri- preserved by the coupling then the set of preimages cally, a positive Lyapunov exponent in this regime indi- fHÿn x : n 2 Ng, each labeled differently under P , 0 CML cates that the map is asymptotically expanding along typi- covers the chaotic attractor and thus no two points on the cal orbits and thus the inverse mapping (3) converges to a attractor can be represented by the same semi-infinite 2 symbol sequence. set of zero measure. Figure 1 illustrates the refining of I N into increasingly smaller regions for 0:1. When this In the case where all fi are unimodal, the I state space can be coarsely discretized into 2N regions without incur- process is extended indefinitely, every point on the attractor ring a loss of fine detail on the attractor. As an example, is assigned a different symbol sequence. consider the 2-element lattice H: I2 ! I2 written as The most powerful feature of the symbolic picture is that time evolution is reduced to a simple shift in sequence 1 1 2 xn1 1 ÿ f xnf xn; space. We denote the shift operator as and define its (6) action on a symbol sequence as s s s ...s s s ..., x2 f x1 1 ÿ f x2; 0 1 2 1 2 3 n1 n n so that the effect of is to discard the leading symbol. If n where f x1 ÿ 2x2 is the quadratic map on the interval we set s x0, then it follows from (2) that s I ÿ1; 1. The quadratic map has a generating partition xn. Consequently, the entire evolution of x0 is con- consisting of the regions ÿ1; 0 and 0; 1 which we shall tained in its symbolic representation. A point on a 1 1 1 period-m orbit, for example, is conjugate to a sequence denote as fP0;P1g, respectively, for coordinate x , and as 2 2 2 formed by infinitely repeating m symbols, or m blocks. fP0;P1g for x . Applying (5) the partition for the coupled system is This property has application in describing the spatiotem- poral patterns that are observed in large CMLs [9]. A static 2 1 P CML P _ P or repeating global pattern would be represented by an fP2 \ P1;P2 \ P1;P2 \ P1;P2 \ P1g: (7) eventually stationary or eventually periodic symbolic se- 0 0 0 1 1 0 1 1 quence. Spatially localized structures, such as traveling Reading the subscripts of each term as a binary number, we waves, appear in the bitwise decomposition of the symbols. relabel the regions as fP0;P1;P2;P3g and assign to them In all cases, spatiotemporal patterns are more efficiently the symbols 0, 1, 2, and 3.