Symbolic dynamics and synchronization of coupled map networks with multiple delays
aa r X i v : . [ n li n . C D ] D ec Symbolic dynamics and synchronization of coupled mapnetworks with multiple delays
Fatihcan M. Atay
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany
Sarika Jalan
Department of Physics and Centre for Computational Science and Engineering, National University ofSingapore, 117456, Republic of Singapore
J¨urgen Jost
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, GermanySanta Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA.
Abstract
We use symbolic dynamics to study discrete-time dynamical systems with multiple timedelays. We exploit the concept of avoiding sets, which arise from specific non-generatingpartitions of the phase space and restrict the occurrence of certain symbol sequences relatedto the characteristics of the dynamics. In particular, we show that the resulting forbiddensequences are closely related to the time delays in the system. We present two applicationsto coupled map lattices, namely (1) detecting synchronization and (2) determining unknownvalues of the transmission delays in networks with possibly directed and weighted connectionsand measurement noise. The method is applicable to multi-dimensional as well as set-valuedmaps, and to networks with time-varying delays and connection structure.
Preprint . For final version, see:
Physics Letters A (2010) 130–135.doi: 10.1016/j.physleta.2010.10.044
1. Introduction
Symbolic dynamics is a versatile tool for describing the complicated time evolution ofdynamical systems, the Smale horseshoe being a famous prototype [1]. Here, instead of rep-resenting a trajectory by a continuum of numbers, one watches the alternation of symbolsfrom a finite alphabet. In the process some information is “lost” but certain important in-variants and robust properties of the dynamics may be kept [2, 3]. Most studies of symbolicdynamics are based on the so-called generating partition [4] of the phase space, for which
Email addresses: [email protected] (Fatihcan M. Atay), [email protected] (Sarika Jalan), [email protected] (J¨urgen Jost)
Preprint submitted to Elsevier November 15, 2018 opological entropy achieves its maximum [5]. Symbolic dynamics based on generating par-titions plays a crucial role in understanding many different properties of dynamical systems.However, finding generating partitions is generally a difficult problem [6, 7, 8, 9, 10]. Someconsequences of using misplaced partitions have been investigated in [11]. Nevertheless, cer-tain non-generating partitions have recently been shown to have particular uses. Specifically,appropriately chosen partitions that restrict the appearance of certain symbolic subsequenceshave been used for distinguishing random from deterministic time series [12], and for inves-tigating the collective behavior of coupled systems [13].On the other hand, time delays arise naturally in the modeling of many physical systems.In spatially extended systems, such as networks, delays are a consequence of the fact thatsignals cannot be transmitted instantly over distances. An additional source of delays canbe the time it takes for each unit to process the information it receives before it acts on it.Interestingly, networks of dynamical systems can still synchronize their actions under certainconditions despite time delays, although the synchronized solution can be very differentfrom an undelayed network [14, 15]. Sometimes the value of the delays are unknown ormay be changing in time, and the determination of the delay value is a problem in itself[16, 17, 18, 19, 20, 21, 22]. In studying the collective behavior of networks of dynamicalsystems, it is therefore both realistic and important to take time delays into account in themodeling and to develop techniques to handle the subsequent complications in the analysis.In the following, we use symbolic dynamics for the study of discrete-time systems withmultiple connection delays. Although significant time delays are common in physical andbiological systems, the effects of delays on the symbolic dynamics have not received muchattention so far. We extend the notion of “forbidden words”, that is, symbol sequenceswhose appearance is restricted by the dynamical constraints, to systems with delays. Thebasic idea is to derive forbidden words for the delayed system from the properties of theundelayed map. We show how forbidden sequences are related to the time delays in thesystem and how this information provides useful information about the dynamics. We applythe theoretical findings to two important practical problems: Detecting synchrony in a largenetwork with multiple delays using measurements from only a few nodes, and determiningunknown values of the delays in the network. As might be expected from the “crudeness”introduced by symbolic dynamics, the method has a certain robustness against noise.
2. Symbolic dynamics for delayed maps
Let f : S → S be a map on a subset S of R n , and consider the dynamical system definedby the iteration rule x ( t + 1) = f ( x ( t )) , (1)where the iteration step t ∈ Z plays the role of discrete time. Let { S i : i = 1 , . . . , m } be apartition of S , i.e., a collection of nonempty and mutually disjoint subsets satisfying ∪ mi =1 S i = S . (We assume m > { . . . , s t − , s t , s t +1 , . . . } , where s t = i if x ( t ) ∈ S i . In theusual grammatical analogy, the symbols { , , . . . , m } form the alphabet , and finite symbolsequences are called words . We say the set S i avoids S j under f if f ( S i ) ∩ S j = ∅ . (2)Clearly, if S i avoids S j , so does any of its subsets. We also refer to a self-avoiding set if (2)holds with i = j . The significance of avoiding sets is that they yield forbidden words: If S i S j , then the symbolic dynamics for (1) cannot contain the symbol sequence ij . Thenotion is extended in a straightforward way to the k th iterate of f . Thus, if f k ( S i ) ∩ S j = ∅ ,then the symbol sequence for the dynamics cannot contain any subsequence of the form i ( ∗ · · · ∗ ) ( k − j , where ( ∗ · · · ∗ ) ( k − denotes k − k + 1 that starts with i cannot end with j . This constrains the symbolsequences that can be generated by a given map, and provides a robust method to distinguishbetween different systems by inspecting their symbolic dynamics. As examples of avoidingsets, we mention that, for the familiar unimodal maps of the interval [0 , x ∗ ,
1] and its subsets are self-avoiding, where x ∗ denotes the positivefixed point of f .More generally, partitions that contain avoiding sets can always be found. We give aconstructive proof. Suppose one starts with some partition { S , . . . , S m } of m sets for which(2) does not hold for any i, j ; that is, f ( S i ) ∩ S j = ∅ ∀ i, j. (3)Now fix some pair ( i, j ), i = j . Partition the set S i further into two disjoint sets as S i = S i ∪ S i , where S i := f − ( S j ) ∩ S i ,S i := S i \ S i . Thus, S i and S i contain those points of S i that are mapped to S j and those that are notmapped to S j , respectively, by the function f . Note that by definition f ( S i ) ∩ S j = ∅ , thatis, S i avoids S j under f . Furthermore, S i = ∅ by (3), and S i = ∅ because otherwise wewould have f ( S i ) ⊂ S j , which would imply f ( S i ) ∩ S i = ∅ (since S i and S j are disjoint setsby assumption), which would contradict (3). Hence, we can define a new partition of m + 1nonempty and mutually disjoint sets { S , . . . , S i − , S i , S i +1 , . . . , S m , S i } , (4)which is obtained from the original one by replacing S i by S i and adding S i , in which theset S i avoids S j . The same argument can be used to construct self-avoiding sets: Assume f ( S i ) S i (otherwise further partition S i to obtain a set which is not invariant under f , whichis possible except for the trivial case when f is the identity map.) Define S i = f − ( S i ) ∩ S i and S i = S i \ S i . Then S i is a self-avoiding set in the new partition (4). Hence, it is possibleto modify a given partition so that the sequence ij (or ii ) never occurs in the symbolicdynamics.The above arguments apply equally well to discrete-time inclusions x ( t + 1) ∈ F ( x ( t )) (5)where F is a set-valued function in R n . This case arises, e.g., when the actual function f is notprecisely known or is constructed from data, or when measurements are contaminated withnoise, so the value f ( x ) can only be determined up to some error bound. For instance, thepoint-value f ( x ) plus the “error disc” could be used to define the set-value F ( x ). We defineavoiding sets for set-valued functions F in the same way through (2), which similarly yieldforbidden sequences for (5). Hence, all results we present here remain valid when equalitiesare replaced by set inclusions. 3o apply the above ideas to delayed dynamics, we first consider the following extensionof (1), x ( t + 1) = (1 − ε ) f ( x ( t )) + εf ( x ( t − τ )) , (6)where τ ∈ Z + is the time delay and ε ∈ [0 ,
1] is a parameter measuring the relative weightof the past in determining the next state. The significance of Eq. (6) is that it governsthe behavior of the synchronous solutions of coupled map networks with transmission delays[14, 15], which are studied in Section 3. The domain S of the map f is required to be a convexset in order for the iterations (6) to be meaningful. Clearly, for τ = 0 (6) reduces to (1).The symbolic dynamics is defined as before, but we define avoiding sets slightly differently.We say the set S i convexly avoids S j under f if conv( f ( S i )) ∩ S j = ∅ , where “conv” denotesthe convex hull of a set. Such sets give rise to forbidden sequences as follows: If S i convexlyavoids S j under f , then the symbolic sequence i ( ∗ · · · ∗ ) | {z } τ − i j (7)is not possible for the delayed system (6). This is a consequence of (6) and the observationthat if x ( t ) and x ( t − τ ) are both in S i , then any convex combination of f ( x ( t )) and f ( x ( t − τ ))belongs to the convex hull of f ( S i ) and so lies outside of S j . Similarly, if S i is convexly self-avoiding, then any sequence of τ + 1 symbols that begin and end with i cannot be followed byanother i . An important observation is that, although the dynamics of (6) can vary greatlywith ε [14], the forbidden sequences (7) are independent of the value of ε . Thus, (7) remainsa forbidden sequence for the symbolic dynamics of the time-dependent equation x ( t + 1) = (1 − ε ( t )) f ( x ( t )) + ε ( t ) f ( x ( t − τ )) , (8)where ε : Z → [0 ,
1] is allowed to be a function of time.Finally, we generalize to equations with multiple delays of the form x ( t + 1) = τ max X τ =0 ε τ f ( x ( t − τ )) , (9)where the coefficients ε τ are nonnegative and satisfy P τ max τ =0 ε τ = 1. Such equations governthe synchronous solutions of coupled maps with multiple delays, as will be shown in Section5. Note that the right hand side of (9) lies in the convex hull of the set { f ( x ( t − τ )) : τ =0 , . . . , τ max } . Therefore, if S i convexly avoids S j under f , then the sequence i i . . . i | {z } τ max +1 j (10)is forbidden for (9); that is, a sequence of consecutive i ’s of length τ max +1 cannot be followedby a j . Again, this result is independent of the values of the coefficients ε τ , so the latter canbe allowed to vary with time, subject to the constraint that they remain nonnegative andsum up to 1. Further restrictions are obtained if some ε m is identically zero, in which case(10) will be forbidden even when the symbol at position ( τ max + 1 − m ) is replaced by anarbitrary symbol in the alphabet.The additional condition of convexity of the sets in case of the delayed dynamics does notpresent an extra restriction in many practical situations. In fact, one often measures a single4omponent, say the first one, of the n -dimensional vector x = ( x , . . . , x n ). In this case, asimple partition of S given by the disjoint union S = S ∪ S , where S = { ( x , . . . , x n ) ∈ R n : x < x ∗ } and x ∗ is a scalar threshold value, which can be chosen to make both S and S nonempty.It is easy to see that both S and S defined in this way are convex whenever S is convex.Such partitions are almost surely non-generating, so the corresponding symbol sequences donot capture all features of the dynamics. (For a discussion of obtaining partitions in a simplesetting, see [13, Section VII].) Nevertheless, it will be seen that they still contain importantinformation that can be utilized to study some important aspects about the delayed dynamics.
3. Coupled map networks with time delay
We now move from single maps to networks of coupled maps, in the context of a modelwhich is sometimes referred to as the coupled map lattice [23]. We consider a general formallowing arbitrary coupling topology, directed and weighted connections, as well as time delayalong the connections: x i ( t + 1) = f ( x i ( t )) + εk i N X j =1 a ij [ f ( x j ( t − τ )) − f ( x i ( t ))] . (11)Here x i ( t ) is the state of the i th unit at time t , i = 1 , . . . , N , a ij ≥ j to i (zero if there is no link), ε ∈ [0 ,
1] is the coupling strength, and k i = P j a ij is the weighted in-degree of node i . (It is understood that the summation term is set to zeroin (11) for any unit for which k i is zero.) The delay τ is the time it takes for the informationfrom a unit to reach its neighbors and be processed. The system is said to synchronize if | x i ( t ) − x j ( t ) | → t → ∞ for all i, j and all initial conditions from some open set. In thiscase, the state of every node asymptotically approaches the same synchronous solution x ( t ),whose dynamics is governed by (1) and (6), respectively, depending on whether the delay τ is zero or nonzero. In the absence of delays, various aspects of the network have been studiedusing symbolic dynamics [13, 24]. Our focus here is on the delayed case.It is known that the network (11) can synchronize even under delays, where the unitsare unaware of the present states of their neighbors but still can act in unison [14]. Theimportant distinction from the undelayed case, however, is that the synchronous dynamics x ( t ) is no longer identical to the isolated dynamics (1) of the units, but is governed by thedelayed equation (6). A consequence is that the overall system (11) can exhibit a muchwider range of behavior than its constituent units through the coordination of their actions[15]. An important problem is to determine whether a large network is synchronized usinginformation from just a few nodes. As a first application, we study this problem in delayednetworks.Normally the symbol sequences observed from a node of a network can vary widely be-tween the nodes. However, in the synchronized state x i ( t ) = x ( t ) for all i , so that thesymbolic sequences observed from a node will be subject to the same constraints as that gen-erated by (6). The choice of the node is arbitrary so long as the network is capable of chaoticsynchronization (which is the case for the choice of parameters in our example systems).This gives a method of detecting synchronization of the network by choosing an arbitrary5 a) σ , ς (b) (c) (d) (e) ε σ , ς (f) ε (g) ε (h) ε Figure 1: Detecting network synchrony by comparing the transition probabilities from measurements at asingle node with those of the synchronous solution (6). Average deviations ς in transition probabilities(dashed line) and synchronization measure σ (solid line) are plotted against the coupling strength ε . Thesynchronous regime where σ = 0 coincides with the regions where ς = 0. The plots (a)–(d) are for a globallycoupled network of 20 nodes, and delay values (a) τ = 0, (b) τ = 1, (c) τ = 2, (d) τ = 3. (e) and (f) areplotted, respectively, for a random network of 100 nodes and a scale-free network of 200 nodes, where τ = 3for both. Subfigures (g) and (h) are for the same network and delay as in (d), but with additive Gaussiannoise of 2% and 5%, respectively, in the measurements. P ( j | ( i ∗ · · · ∗ i ) τ +1 ), that is, the conditional probabilitythat a sequence of length τ + 1 starting and ending with i is followed by j . Letting ς denotethe average squared difference between the observed transition probabilities of the networkand those of (6), synchronization is signaled when ς = 0.Fig. 1 illustrates the relation between synchronization and forbidden sequences, for thechaotic tent map f ( x ) = 1 − | x − | and the partition S = [0 , x ∗ ] , S = ( x ∗ , , (12)where x ∗ = 2 / f . Note that S is a (convexly) self-avoiding set under f .We evolve (11) starting from random initial conditions and estimate the transition probabili-ties using time series of length 1000 from a randomly selected node. (We note that the lengthof the time series used is independent of the network size.) Synchronization occurs when thevariance σ = D N − P i [ x i ( t ) − ¯ x ( t )] E t drops to zero, where ¯ x ( t ) = N P i x i ( t ) denotes theaverage over the nodes of the network and h . . . i t denotes an average over time. As seen fromFig. 1, the region for synchronization exactly coincides with the region where the transitionprobabilities for the network are identical to those of Eq. (6). Hence, regardless of networktopology and size, both synchronized and unsynchronized behavior can be detected over thewhole range of coupling strengths using only measurements from an arbitrarily selected node.Moreover, Figure 1(g-h) show that the “crudeness” introduced by using symbolic sequencesalso provides some robustness against noise.As a second application, we consider the reverse problem of determining the value of thedelay τ in (6) from observed symbolic dynamics. For this purpose, we check the presenceof subsequences of the form (7) of various lengths, knowing that such a sequence of length τ + 2 would be forbidden. Plotting the occurrence frequencies of (7) against τ , the actualvalue of the delay is found at the point where the frequency drops to zero (or attains itsminimum, in the presence of small noise). Similarly, the value of the delay in the network(11) can be found from a knowledge of its synchrony. A practical situation is when the valueof τ is unknown but the network is known to be synchronized or can be made to synchronizeby the adjustment of control parameters. The value of τ can then be obtained by using themeasurements from a node and checking the presence of the forbidden sequences (7). Fig. 2gives an illustration for the tent map and the partition (12), by plotting the probability P (2 | (2 ∗ · · · ∗ τ +1 ) versus τ , that is, the conditional probability that a sequence of length τ + 1 starting and ending with 2 is followed by another 2. By the arguments above, such asequence cannot occur for the synchronized dynamics (6) since S is convexly self-avoiding.The true value of the delay is thus found at the point where the occurrence probability ofthe sequence drops to zero. Fig. 2 shows that the method works well also under noise.
4. Time-varying delay and connection structure
The arguments above remain valid also when the connection topology is changing withtime, as in the network x i ( t + 1) = f ( x i ( t )) + ε ( t ) k i ( t ) N X j =1 a ij ( t ) [ f ( x j ( t − τ )) − f ( x i ( t ))] , a) τ P ( | ( ∗ ... ∗ ) τ + ) (b) τ Figure 2: Finding the unknown value of delay in a synchronized network using measurements from anarbitrarily selected node. The observed probability P (2 | (2 ∗ · · · ∗ τ +1 ) is plotted against τ . The true valueof τ corresponds to the point where the probability drops to zero. The network has 20 nodes that are globallycoupled with ε = 0 .
75, where the true value of the delay is (a) 5 and (b) 6. The dotted lines are in thepresence of 10% noise. where ε ( t ) ∈ [0 ,
1] and k i ( t ) = P j a ij ( t ) for all t ∈ Z . This is a consequence of the observationthat the synchronized solution does not depend on the network topology and its forbiddensequences (7) are independent of ε . The conditions for synchronization, of course, dependon the connection structure. In the undelayed case, synchronization conditions involve theexistence of spanning trees of the union graphs and can be quantified in terms of the Hajnaldiameter of infinite sequences of connection matrices [25, 26]. On the other hand, the preciseconditions for synchronization of delayed time-varying networks is a more involved problem.A further generalization is to allow delays that change with time; τ = τ ( t ). In this case,sequences such as (7) will be forbidden at some time point t for the corresponding value of τ ( t ). For instance, if the partition set S i convexly avoids S j under f , then s t +1 = j whenever s t = s t − τ ( t ) = i . Even the precise time dependence τ ( t ) is not known, one can still obtainuseful information by studying such subsequences. Thus, if τ ( t ) often assumes some value k ,then the ( k + 1)-symbol block i ( ∗ · · · ∗ ) | {z } k − i j (13)will correspondingly appear more seldom; hence, rather than being forbidden within thewhole symbolic history, it will have reduced frequency of occurrence. This observation helpsdetermine the unknown values of the time-varying delay. To illustrate, we return to ourexample of coupled tent maps used for Figs. 1 and 2, this time considering a time-varyingconnection delay τ whose value at each time step is chosen randomly from the set { , } .In Fig. 3 we plot the occurrence frequencies of the sequences (7) for various values of τ .In contrast to Fig. 2, the frequency does not drop to zero but displays two marked dips atthe values τ = 5 and τ = 7, agreeing with the fact that the delay was randomly switchingbetween 5 and 7. 8 τ P ( | ( ..... *2 ) ( τ + ) ) Figure 3: Finding the unknown values of time-varying delay in a synchronized network using measurementsfrom an arbitrarily selected node. The delay randomly takes one of the values { , } with equal probabilityat each time step. The observed probability P (2 | (2 ∗ · · · ∗ τ +1 ) is plotted against τ , displaying marked dipsat the true values of the delay. The network consists of 20 all-to-all coupled nodes, with coupling strengthsof ε = 0 .
85, 0 .
86 and 0 .
9, shown by the three curves.
5. Multiple delays
The foregoing ideas can be extended to systems with multiple delays, e.g., to the coupledmap network x i ( t + 1) = f ( x i ( t )) + εk i N X j =1 a ij [ f ( x j ( t − τ ij )) − f ( x i ( t ))] , (14)where τ ij denotes the transmission delay from j to i . Whereas the network (11) with aconstant delay always admits a synchronous solution, one needs additional conditions inthe case (14) of multiple delays. Namely, Eq. (14) has non-constant synchronized solutionsprovided that the fraction of weighted incoming connections having a given value of delay isthe same for every vertex. To see this, suppose x i ( t ) = x ( t ) for all i . Substituting into (14)and rearranging, we have x ( t + 1) − (1 − ε ) f ( x ( t ) = ε N X j =1 a ij k i f ( x ( t − τ ij )) , (15)where we have used the fact that k i = P j a ij . One can decompose the summation furtherover the delay values since the delays τ ij are integers, thus obtaining x ( t + 1) − (1 − ε ) f ( x ( t )) = ε τ max X τ =0 f ( x ( t − τ )) X j ∈ J i ( τ ) a ij k i , (16)where τ max = max i,j { τ ij } is the maximum delay in the network and J i ( τ ) = { j : τ ij = τ } is the index set of the connections to i that are subject to a delay of precisely τ . Now if f is constant over the synchronous trajectory X := { x ( t ) : t ∈ Z } , then the term f ( x ( t − τ ))can be taken outside the summation and the double summation adds up to 1, reducing theequation to (6). This case happens, in particular, when the synchronous solution is constant.9n general, however, for non-constant synchronous solutions, f will not be constant over X .In this case, since the left hand side of (16) is independent of i , we require that the quantity X j ∈ J i ( τ ) a ij k i be also independent of i . In other words, for any given value of delay, the weighted fractionof incoming links having that delay value should be the same for each node. We let p τ = P j ∈ J i ( τ ) a ij /k i denote this common fraction, and define ε τ = (cid:26) εp τ , if τ ≥ − ε ) + εp , if τ = 0 . Note that P τ p τ = 1; therefore, P τ ε τ = 1.Thus (16) becomes x ( t + 1) = τ max X τ =0 ε τ f ( x ( t − τ )) , which is the same as Eq. (9) considered in Section 2. Thus the synchronous solution x ( t ) ofthe system (15) with multiple delays obeys (9), and by the results of Section 2, symbolic se-quences of the form (10) are forbidden for the synchronous dynamics. Such symbol sequencescan thus be used to determine whether a delay value of m is present in the network (14),yielding a systematic way of finding the values of all delays from a knowledge of synchrony.Conversely, synchronization can be detected by comparing the transition probabilities ofsymbolic sequences from an arbitrary node to those of (9) if the delays are known.As an example of networks with multiple delays, we consider a network of four nodesarranged on a circle, where each node is coupled to its nearest neighbors on its left and rightwith delay equal to 1 and to its far neighbor on the opposite side with delay equal to 2.The local map is the chaotic shift map f ( x ) = 2 x (mod 1) on the unit interval, which wepartition as S = [0 , . S = [0 . , . S = [0 . , . S = [0 . , S and S are convexly self-avoiding under f . Hence, from (10), symbolsequences containing τ max + 2 consecutive 2’s are forbidden. One can then check the lengthsof uninterrupted subsequences of 2’s (or 3’s) from a node of the synchronized network anddetermine the largest value of the delay. Fig. 4 shows that subsequences of four consecutive2’s are not observed, implying that τ max is indeed equal to 2. Furthermore, the result isindependent of the value of the coupling strength.
6. Discussion and conclusion
In this paper we have used symbolic dynamics to study discrete-time systems and net-works with time delays. We have derived forbidden symbol sequences for the delayed systemfrom the properties of the undelayed map. Although the partitions used are usually non-generating, the forbidden sequences are related to certain characteristics of the dynamics,and in particular to delays. Consequently, the value of the delay in the system can be deter-mined by the presence and absence of such sequences. Conversely, a knowledge of the delaysenables one to detect synchronization (or phase locking) in the network using measurementsfrom a single node. The method has the advantage of being based on a phase-space partition10 τ P(2|(2*.....*2) ( τ + ) ) ε = 0.36, ε = 0.24 ε = 0.32, ε =0.26 ε = 0.40, ε =0.30 Figure 4: The value of the largest delay in networks with multiple delays is obtained from the non-occurrenceof ( τ max + 1) consecutive symbols corresponding to a self-avoiding set. that is much easier to obtain than a generating partition. Furthermore, it can utilize rathershort measurements from a single node of the network. The computations are therefore fastand independent of the network size, and do not require knowledge of the connection struc-ture. As such, they can complement or be an alternative to existing techniques for detectingsynchronization based on phase-space reconstruction [27, 28, 29] and methods for estimatingdelay times [16, 17, 18, 19, 20, 21, 22].Although we have restricted our discussion to complete synchronization, the ideas applyalso to some other types of collective behavior, for instance to phase-locked solutions and traveling waves . In such collective states, the symbol sequences of all nodes in the networkare identical except for a time shift (that depends on the particular node), which does notchange symbol statistics. Hence, the forbidden sequences can be derived as before from theproperties of the local map, with the same implications as in the applications presented here.Beyond the coupled map lattice model (11), there is also a growing interest in moregeneral coupling schemes such as x i ( t + 1) = f ( x i ( t )) + εk i N X j =1 a ij g ( x i ( t ) , x j ( t − τ )) . (17)Synchronized solutions s ( t ) of (17) satisfy s ( t + 1) = h ( s ( t ) , s ( t − τ )) (18)where the function h is defined by h ( x, y ) := f ( x ) + εg ( x, y ) . (19)Thus, for instance (6) becomes a special case with h ( x, y ) = (1 − ε ) f ( x ) + εf ( y ). The notionof avoiding sets can be extended to the more general case: We say that the set S i avoids S j under h if h ( S i , S i ) ∩ S j = ∅ . (20)11f (20) holds, then the symbol sequence (7) is a forbidden sequence for the dynamics (18).More generally, if h ( S i , S k ) ∩ S j = ∅ for some sets in the partition, then the symbol sequence k ( ∗ · · · ∗ ) | {z } τ − i j will be forbidden. With a knowledge of forbidden sequences, one can pursue the line ofreasoning of the previous sections to derive results about the coupled system (17). Theadditional challenge now is to relate the avoiding sets to the properties of the functions f and g and the coupling coefficient ε . The difficulty is of course not unique to the symbolic-dynamics approach, since the dynamics of the system (17), with more parameters in itsstructure, is not easy to characterize in its full generality, although there has been somerecent progress in this direction. For example, for the undelayed case, conditions for thestability of the synchronous state have been given in [30], and Ref. [31] has shown the rangeof rich dynamics such systems can exhibit at synchrony. For the delayed case, however,considerably less is known at present.Finally, we note that our treatment is based on systems with known dynamics, such asEq. (1), or models of approximately known dynamics, such as the discrete-time inclusion (5).On the other hand, in certain important applications only a time series of measurements isavailable without any detailed knowledge of the dynamical process generating it. In the ab-sence of a priori information about the forbidden sequences, the applicability of the methodsof this paper is restricted. Nevertheless, it may still be possible to exploit similar ideas incombination with the methods of time series analysis. One possibility is to use a posteriori statistics of subsequences from the time series. In fact, here one need not confine himself toforbidden sequences but instead can use statistical information of symbol sequences to com-pare the network’s behavior with that of individual units; see e.g. [13] for an example in theundelayed case. Another alternative is to first build a mathematical model of the dynamicalprocess from time series using several well-established methods [28]. Once a model is fit todata, the analysis presented here can be carried out as before. References [1] S. Smale, Bull. AMS (1967) 747–808.[2] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding , CambridgeUniversity Press, Cambridge, 1995.[3] B.-L. Hao and W.-M. Zheng,
Applied Symbolic Dynamics and Chaos , World Scientific,Singapore, 1998.[4] D. J. Rudolph,
Fundamentals of Measurable Dynamics, Ergodic theory on Lebesguespaces , Clarendon Press, Oxford, 1990.[5] J. P. Crutchfield and N. H. Packard, Physica (1983) 201–223 .[6] F. Giovannini and A. Politi, Phys. Lett.
161 A (1992) 332.[7] L. Jaeger and H. Kantz, J. Phys. A (1997) L567.128] R.L.Davidchack, Y.-C. Lai, E.M. Bollt, and M. Dhamala, Phys. Rev. E (2000) 1353–1356.[9] M. B. Kennel and M. Buhl, Phys. Rev. Lett. (2003) 084102.[10] M. Buhl and M. B. Kennel, Phys. Rev. E (2005) 046213.[11] E. M. Bollt, T. Stanford, Y.-C. Lai and K. ´Zyczkowski, Physica D (2001) 259–286.[12] F. M. Atay, S. Jalan and J. Jost, Complexity (2009) 29–35.[13] S. Jalan, J. Jost, and F. M. Atay, Chaos , (2006) 033124.[14] F. M. Atay, J. Jost, and A. Wende, Phys. Rev. Lett. (2004) 144101.[15] F. M. Atay and J. Jost, Complexity (2004) 17–22.[16] S. Lepri, G. Giacomelli, A. Politi, and F. T. Arecchi, Physica D (1993) 235–249.[17] M. J. B¨unner, A. Kittel, J. Parisi, I. Fischer, and W. Els¨aßer, Europhys. Lett. (1998)353-358.[18] H. C. So, Signal Processing (2002) 1471–1475.[19] A. B. Rad, W. L. Lo, and K. M. Tsang, IEEE Trans. Control Systems Technology, (2003) 957–959.[20] S. V. Drakunov, W. Perruquetti, J. P. Richard, and L. Belkoura, Annual Reviews inControl (2006) 143–158.[21] D. Yu, M. Frasca, and F. Liu, Phys. Rev. E (2008) 046209.[22] D. Yu and S. Boccaletti, Phys. Rev. E (2009) 036203.[23] K. Kaneko, Physica D (1989) 1–41; Phys. Rev. Lett. (1990) 1391–1394; PhysicaD (1990) 137–172.[24] S. D. Pethel, N. J. Corron, E. Bollt, Phys. Rev. Lett. (2006) 034105.[25] W. Lu, F. M. Atay, and J. Jost, SIAM J. Math. Anal. 39 (2007) 1231–1259;[26] W. Lu, F. M. Atay, and J. Jost, Eur. Phys. J. B 63 (2008) 399–406.[27] J. D. Farmer and J. J. Sidorowich, Phys. Rev. Lett. (1987) 845–848.[28] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis , Cambridge UniversityPress, Cambridge, 1997.[29] M.J. B¨unner, M. Ciofini, A. Giaquinta, R. Hegger, H. Kantz, R. Meucci, and A. Politi,Eur. Phys. J. D10