Mode-locking in Coupled Map Lattices

R. Carretero-Gonz´alez∗ †, D.K. Arrowsmith and F. Vivaldi School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, U.K. (Physica D 103 (1997) 381–403)

Keywords: Coupled map lattices, travelling waves, limited range, with decreasing strength for distant neigh- mode-locking, symbolic dynamics. bours. Equation (2) defines the components of the global map F : We study propagation of pulses along one-way coupled map lattices, which originate from the transition between two F = ( . . . , F , F , F , . . . ) X = F (X ). superstable states of the local map. The velocity of the pulses −1 0 1 t+1 t exhibits a staircase-like behaviour as the coupling parameter A mapping of global states that will be of use in the is varied. For a piece-wise linear local map, we prove that the following is the shift mapping velocity of the wave has a Devil’s staircase dependence on the coupling parameter. A wave travelling with rational velocity G ( xt(i) ) = xt(i + 1) . (3) is found to be stable to parametric perturbations in a manner { } { } akin to rational mode-locking for circle maps. We provide It should be clear that F and G commute. evidence that mode-locking is also present for a broader range We consider homogeneous CML (same map F for all of maps and couplings. i cells) with linear nearest-neighbours interaction. The two most widely used models are 1. INTRODUCTION ε xt (i) = (1 ε)f(xt(i)) + (f(xt(i 1)) + f(xt(i + 1))) +1 − 2 − An interesting feature of coupled map lattices (CML) (4) [1,2] is the widespread occurrence of the so-called spatio- temporal chaos [3,4], which is irregular behaviour in time and as well as space. Equally interesting is the appearance xt (i) = (1 ε)f(xt(i)) + εf(xt(i 1)), (5) of coherent structures from an apparently decorrelated +1 − − medium [5]– [7]. For example, an interface separating two which are called diffusive and one-way CML respectively. different phases may travel along the lattice [8]– [11]. The The coupling strength ε (also called the diffusive param- movement of such a front depends on the strength of the eter) satisfies 0 ε 1 to ensure the conservation of coupling between lattice sites. This paper is concerned flux between neigh≤bouring≤ sites. with investigating some aspects of this phenomenon. In this paper we consider the propagation of localized We consider an infinite collection of cells with a local wavefronts along a linear one-way CML. Localized states dynamical variable x (i), characterizing the state of the t and step states (localized wavefronts) are introduced in i-th cell at time t. Both i and t are integer. The global the next section, where we establish some of their basic state of the lattice at time t is given by the state vector properties. In section 3 we introduce a one-parameter family of piece-wise linear local maps, for which wave- Xt = xt(i) = (. . . , xt(i 1), xt(i), xt(i + 1), . . .). (1) { } − fronts can propagate coherently in a simple way. In the The state of the i-th cell at time t + 1 given by following section we show that the dynamics of an im- portant class of wavefronts can be reduced to the study xt (i) = Fi(. . . , xt(i 1), xt(i), xt(i + 1), . . .), (2) of a discontinuous piecewise-linear circle map, depending +1 − on two parameters. We study this family with methods where the function Fi is determined by the local dynam- of symbolic dynamics. These results are then applied to ics in each cell as well as the interaction between cells. the pulse propagation in section 5. We show that for A physically meaningful interaction will have typically a each value of the parameters the velocity of the pulse is well-defined, and we use it to characterize the structure of the parameter space. The latter is found to be struc- turally similar to that of a subcritical circle map, with ∗e-mail: [email protected] velocity playing the role of rotation number. We estab- †New address: Center for Nonlinear Dynamics and its Appli- lish the occurrence of a critical transition from a stable cations (CNDA), Dept. Civil and Environmental Engineering, standing signal to a stable travelling one, as the coupling University College London, Gower Street, LONDON WC1E parameter is increased, and we prove parametric stability 6BT of pulses travelling with rational velocity. The velocity of the travelling pulse is found to have Devil’s staircase-like

1 dependence on the coupling parameter (see figure 1). In The mass of a step state is bounded from below the last section we indicate how to generalize our results ∗ ∗ ∗ M(Xt) ∆x = x x . (8) to a broader range of functions and couplings. ≥ | + − −| The simplest such state is a pure step state, that is, a step function between the two fixed points

2. LOCALIZED STATES AND PROPAGATING ∗ x− if i k FRONTS P (k) x (i) = ≤ (9) t ∗ ≡ ( x+ if i > k. We first define localized states, step states and their velocity of propagation and then establish some of their For this pure state µ(P (k)) = k and σ(P (k)) = 0. A step state has minimal mass or minimal variation, basic properties. In what follows, we deal with a linear ∗ i.e. M(xt) = ∆x (cf. (8)), if and only if its local states one-way CML, unless stated otherwise. ∗ ∗ are ordered, that is if xt(i) xt(i+1) when x− < x+ (or ∗ ≤ ∗ xt(i) xt(i+1) when x− > x+) for all i. It is plain that if the configuration≥ of the one-way CML has minimal mass 1. Localized states and step states at time t it is going to preserve its minimal mass at any time greater than t provided that f is a non-decreasing ∗ ∗ ∗ ∗ Let ∆xt(i) = xt(i + 1) xt(i) . The mass M(Xt) of function if x < x and non-increasing if x > x . | − | − + − + a global state Xt is defined as the variation of the local The global dynamics causes centre of mass and width states of a localized state to evolve. In this respect, two points ∞ need consideration. Firstly, the existence of the proba- Mt = M(Xt) = ∆xt(i). bility pt does not automatically imply that of pt+1, and i=−∞ secondly, the image of a localized state is not necessarily X localized. To achieve the above conditions, we impose We restrict our attention to states with positive finite some mild restrictions on the local map f as well as the mass (a zero-mass state is a uniform state). state Xt. The importance of the states defined above is For each lattice site i Z we define the probability ∈ that they are invariant under the dynamics of the CML — for the complete statements and proofs see appendix ∆xt(i) pt(i) = . (6) A. This gives the framework for the existence of steady Mt step states and travelling ones. The mean µ and the variance σ of the variable i with respect to the probability distribution (6) are given by 2. Travelling velocity and basic properties ∞ µt = µ(Xt) = i pt(i) i=−∞ From the previous definitions and results it is clear ∞ X that if the initial state X0 is a step state, then µ(Xt) and 2 2 2 σt = σ(Xt) = (i µt) pt(i). σ(X ) are defined for all t 0 (see appendix A for de- − t i=−∞ ≥ X tails). Our main interest is to determine the average ve- locity v of the centre of mass along a global orbit, namely The quantity µt is called centre of mass of the state Xt, and σt its width. The centre of mass and the width give, 1 v = v(ε, X0) = lim (µt µ0). respectively, the average position and spread of the lo- t→∞ t − calized state. A formal analysis on the stability of exponentially local- A state Xt with finite centre of mass and width is said to be localized. If, in addition, there exists a positive ized step states establishes the following quite general |i| property of the velocity of a step state in one-way CML: constant κ < 1 such that ∆xt(i) = O(κ ), then the state is exponentially localized. For an exponentially localized state, not only the mean and variance exist, but also all Theorem 1. central moments of the distribution (6). Let X0 be an exponentially localized step We are interested in localized states that can model state of a one-way CML. Let the local map f be bounded, wavefronts. So we define a step state to be a localized and let f be a contraction mapping in a neighbourhood of the fixed points (7). Then, for all sufficiently small ε, state for which the configuration xt(i) is asymptotic, for i , to two distinct fixed p{oints}of f [12]: v(ε) = 0 and v(1 ε) = 1, independently of X0. → §∞ − ∗ ∗ ∗ The proof of this theorem is given in appendix A. lim xt(i) = x§; f(x§) = x§. (7) i→§∞ The condition that f be a contraction mapping is a bit

2 ∗ stronger than the condition that x§ be attracting. We 1. The local mapping shall see that for an initial localized interface of the type (9), the critical behaviour is much richer than a discon- A simple one-parameter family of maps that fulfills the tinuous transition from v = 0 to v = 1, at some inter- above requirement is given by (see figure 2) mediate value of ε. Rather, the dependence of v on ε is characterized by infinitely many critical values, in corre- 1 if x a − ≤ − spondence to the boundaries of intervals in ε, where the 1 fa(x) =  a x if a < x < a 0 < a < 1, (12) velocity is a given rational number (see figure 1).  − A general feature of the velocity of the travelling wave  1 if x a ≥ is its symmetry. Let us define δ = 1 ε and consider a  − and by the step function between 1 and 1 centered at moving reference frame with a unit positive velocity: − ∗ the origin when a = 0. The fixed points x− = 1 and x∗ = 1 are superstable, with basins I = [ 1,−0) and yt(i) = xt(i) + − − ∗ I+ = (0, 1], respectively. The repeller is the origin x =  yt+1(i) = xt+1(i + 1) 0. The coupled map lattice now depends on the two . .  . . parameters ε and a, and the parameter space is the unit  . . yt+k(i) = xt+k(i + k). square. The one-way CML with local mapping fa and  coupling ε will be denoted by F .  ε,a The one-way CML in the moving frame now reads We partition the interval I = [ 1, 1] into the unstable − domain U = ( a, a) and the superstable domains S− = y (i 1) = δf(y (i)) + (1 δ)f(y (i 1)) − t+1 t t [ 1, a] and S+ = [a, 1]. Any site falling within S§ − − − − − ∗ maps, under fa, to x at the next iteration. = yt+1(i) = (1 δ)f(yt(i)) + δf(yt(i + 1)). § ⇒ − We begin considering states of minimal mass Xt whose (10) configurations xt(i) form a non-decreasing sequence. { } Because Xt is a localized state, only a finite number n of Equation (10) represents a one-way CML of the type (5) local states belong to U: but coupled in the opposite direction. Thus, taking into account the moving reference frame and the definition of i k + 1, . . . , k + n xt(i) U. (13) δ we can conclude that ∈ { } ⇐⇒ ∈ For i k (i > k + n), the local states belong to the ≤ 1 v(ε) = v(1 ε). (11) superstable domain S− (S+) and their image under fa is − − 1 (+1). Consequently, if we let X 0 be the state obtained − t Therefore the velocity of the travelling wave in a one- from Xt by replacing xt(i) with 1 for i k and with − ≤0 way CML of the form (5) is symmetric with respect to the +1 for i > k + n, then Fε,a(Xt) = Fε,a(Xt). The state 0 point (ε, v(ε)) = (1/2, 1/2) (see figure 1). This important Xt is the reduced state associated with Xt, and we write 0 symmetric property allows one to restrict the study of the Xt X . We will reduce all states after each iteration of ∼ t velocity to the ε-interval [0, 1/2]. the mapping Fε,a. So, without loss of generality, we shall restrict our at- tention to states of the type

3. A PIECE-WISE LINEAR LOCAL MAP Xt = ( . . . , 1, 1, xt(k + 1), . . . , xt(k + n), 1, 1, . . . ) − − with xt(i) < a, i = k + 1, . . . , k + n. In this section we deal with the coherent propagation ¯ ¯ size of a wavefront (without damping or dispersion) along the The number n¯ of lo¯ cal states in U is called the of lattice. We confine our attention to states with minimal Xt. The knowledge of the range (13) provides useful mass. In order to ensure that this property is preserved information about position and width of a step state. under iteration, we shall require the local map f to be Indeed the center of mass of Xt satisfies the bounds non-decreasing as stated before. In addition, we ask f to n(1 a) n(1 a) ∗ k + − < µ(Xt) < (k + n) − (14) be continuous and to have two stable fixed points x− and ∗ ∗ ∗ 2 − 2 x+ (with x− < x+). So f will have precisely one unstable ∗ ∗ ∗ ∗ and the width fixed point at x with x− < x < x+. The local dynamics ∗ ∗ takes place on the interval I = [x−, x+], where there are n(1 a) n ∗ ∗ ∗ ∗ − < σ(Xt) < . (15) two basins of attraction I− = [x−, x ) and I+ = (x , x+] 2 2 ∗ ∗ for the fixed points x− and x+, respectively. The image of Xt is

Xt = (. . . , 1, 1, f (xt(k + 1)), . . . , +1 − − − xt+1(k + n), f+(xt(k + n)), 1, . . . )

3 where 4. SYMBOLIC DYNAMICS OF MINIMAL STATES 1 ε f−(x) = −a x ε − (16) ε We develop a symbolic description of the dynamics of f+(x) = x + (1 ε). a − minimal states in the case of non-negative gap length.

Xt+1 has at most n + 1 sites in U. The reduced state Their dynamics can be reduced to the iteration of a one- 0 Xt+1 is one of the following: dimensional piecewise linear circle map — the auxiliary map. (a) (. . . , 1, 1, xt (k + 1), xt (k + 2), . . . , − − +1 +1 xt+1(k + n), 1, 1, . . . )

(b) (. . . , 1, 1, 1, xt (k + 2), . . . , 1. The auxiliary map − − − +1 (17) xt+1(k + n), xt+1(k + n + 1), 1, . . . ) Without loss of generality, we consider minimal states (c) (. . . , 1, 1, xt+1(k + 1), xt+1(k + 2), . . . , − − of the form xt+1(k + n), xt+1(k + n + 1), 1, . . . )

Xt = (. . . , 1, 1, xt(i), 1, 1, . . .), (20) Loosely speaking, as the time increases from t to t + 1, − − the state has not propagated in case (a), it has advanced where either x (i) a or x (i) = 1, for which we to the right by one site in case (b) and it has propagated t t introduce the |shorthand| ≤ notation − with dispersion in case (c). To decide among these possibilities, one needs to in- Xt = [x, i]t, vestigate the value of f− and f+ at the boundary sites i = k + 1 and i = k + n, respectively. Letting (the subscript t will often be dropped). Thus [ 1, i] = P (i) — cf. (9). The image of X is given by − a(ε a) a(a + ε 1) t γ− = − γ+ = − 1 ε ε Xt = (. . . , 1, 1, f (xt(i)), f (xt(i)), 1, . . .). (21) − +1 − − − + one verifies that the functions f satisfies the following § We define equations 1 a f ( a) = 1 f ( a) = 1 2ε εc = − (22) − − − + − − 2 f−(a) = 1 2ε f+(a) = 1 (18) − and note that εc is the area enclosed between the graph f (γ ) = a f (γ ) = a. of f and the diagonal, for x∗ < x < x∗ . The following − − − + + a + two results characterize completely the evolution of Xt gap We define the of the CML with local map fa to be in the parameter ranges ε εc and ε 1 εc. the open interval Γ = (γ+, γ−). Its length (with sign) is ≤ ≥ − given by Theorem 2. Let Xt be a minimal state of the form (20), and let a(1 a 2ε(1 ε)) γ(ε, a) = γ γ = − − − . (19) − − + ε(1 ε) εa a(1 ε) − x− = x+ = − . (23) 1 ε a a ε Let x U. Then f (x) U if x γ , and f (x) U − − − ∈ + ∈ ≤ + − ∈ if x γ . From this and the fact that x (i) is a non- (i) If ε εc, then for x(i) < x− (x(i) > x−) the state − t ≤ decreasing≥ function of i we obtain that the size of a step [x, i] reaches P (i) (P (i 1)) in a finite time. The state − state of minimal mass cannot increase, unless it is zero, [x−, i] is fixed and unstable. for a gap of non-negative size. (ii) If ε 1 εc, then for x(i) < x+ (x(i) > x+) the In particular, it is not difficult to prove that when the state [x, i]≥reac−hes P (i + k) (P (i + k 1)) in a finite time. gap size is non-negative, if the size is not greater than 1 − The state [x+, i] is fixed and unstable under G F , where at time 0, then it will remain so for all times. A minimal G is the shift mapping (3). ◦ state is a minimal mass step state of size 0 or 1. There- fore, if we start with a minimal state, the configuration The proof of the above results is given in appendix B. of the CML will remain a minimal state at any time. From Theorem 2 it follows that the velocity of the trav- elling interface satisfies

0 if 0 ε εc v(ε) = ≤ ≤ (24) (1 if 1 εc ε 1. − ≤ ≤

4 As a consequence, we have that if ε εc then F (P (i)) 2. Symbolic dynamics ≤ ∼ P (i), and if ε 1 εc then F (P (i)) P (i + 1). In the ≥ − ∼ rest of this chapter we shall assume that εc < ε < 1 εc. − The following binary symbolic dynamics for the auxil- We partition U into three intervals, namely iary mapping Φε,a will play a decisive role in the rest of the paper. U+ = x : a x γ+ { − ≤ ≤ } We assign the symbols ‘0’ to U−, the symbol ‘1’ to U+, Γ = x : γ < x < γ and the symbol ‘10’ to Γ. This choice originates from the { + −} prescriptions (25), according to which the location of a U− = x : γ− x a . { ≤ ≤ } minimal state remains unchanged for x U , increases ∈ − Because we are assuming the gap length to be non- by one for x U+, and increases by one for every two it- ∈ negative, we are dealing with a minimal state and then erations for x Γ. To the orbit of the auxiliary mapping ∈ the three possibilities for the state (21) are: with initial condition x we associate the semi-infinite se- quence of symbols (a) xt U f (xt) U, f (xt) S ∈ − ⇒ − ∈ + ∈ + 0 S = S(x) = (s1, s2, s3, . . . ) sk 0, 1 (b ) xt Γ f−(xt) S−, f+(xt) S+ (25) ∈ { } ∈ ⇒ ∈ ∈ (b) xt U f (xt) S , f (xt) U. where we stipulate that the symbol ‘10’ generates two ∈ + ⇒ − ∈ − + ∈ binary symbols: ‘10’ ‘1’,‘0’. For this reason, our code corresponding to the reduced states — see (17) will not be unique. −→The space of binary sequences is denoted by Σ2, and is equipped with the usual topology. (a) Xt = (. . . , 1, 1, xt , 1, 1, . . .) +1 − − +1 Any orbit with an element in Γ will be called a gap 0 (b ) Xt+1 = (. . . , 1, 1, 1, 1, 1, . . .) orbit. If this orbit is periodic, then the period of its − − − symbolic representation is one greater than that of the (b) Xt = (. . . , 1, 1, 1, xt , 1, . . .) +1 − − − +1 orbit, because the gap Γ corresponds to two symbols. For the case (b0) we have the evolution The rotation number [13,14] of the orbit through x is defined as

[x, i]t [ 1, i]t+1 [1 2ε, i + 1]t+2. (26) T −→ − −→ − 1 ν = lim st (28) All possibilities are accounted for by defining the auxi- T →∞ T t=0 liary map Φε,a to be the 2-parameter map (see figure 3) X provided that the limit exists. If this is the case, it is f+(x) if x U+ plain that 0 ν 1. ∈ We will sho≤w ≤that the symbol sequence S(x) can be Φε,a(x) = 1 2ε if x Γ εc < ε < 1 εc. (27) − ∈ − characterized in terms the spectrum of ν, which is defined  f (x) if x U − ∈ − as the following sequence of integers  It is plain that Φε,a maps U onto itself and if, in addition, Spec(ν) = ν , 2ν , 3ν , . . . = a1, a2, a3, . . . (29) γ is positive, then U−, Γ and U+ have positive measure. b c b c b c This characterizes the domain and range of Φε,a. where is the floor function. Because 0 ν 1, The auxiliary map reduces the dimensionality of our consecutivb·c e elements of Spec(ν) differ by zero≤ or ≤one. system. Originally the CML is an infinite dimensional Thus defining system since it has an infinite number of sites. But in the case of minimal states the dynamics of the whole τ(a , a , . . . ) = (a a , a a , . . . ), 1 2 2 − 1 3 − 2 lattice is reduced to that of the auxiliary map Φε,a, as illustrated schematically in figure 4. Iterating the CML q we find that τ(Spec(ν)) is a binary sequence. times amounts to applying the auxiliary map q times on Using symbolic dynamics it is easy to follow the evolu- the interface site. During these q iterations, whenever the tion of the minimal state: every iteration corresponds to branch f− is used the interface is not shifted; whenever a new symbol in the sequence. If the symbol is ‘0’ then f+ is applied we shift the interface by one site to the the minimal state remains in the same place and if the right; and finally, when the interface site falls into Γ the symbol is ‘1’ the minimal state is shifted one site to the CML essentially undergoes two iterations — cf. (26) — right. We can assume the velocity of the travelling front but the final result is that the interface site value is 1 2ε to be constant, so that the spectrum of the velocity gives as well as being shifted one site to the right. − the position of the minimal state at every iteration, since the location of the minimal state can only take integer values. Thus τ applied on Spec(ν) gives a 0 when the position of the minimal state remains unchanged and a 1

5 when it jumps from on site to its right neighbour. There- the following way. Let us consider the orbit generated fore the symbolic binary sequence associated to a given from an arbitrary point x0 in Γ. If after q iterations xq rotation number ν can be computed as falls into Γ, the orbit is eventually periodic [15] of period q with rotation number p/q, p 1 being the number of − S = τ (Spec(ν)) . (30) times the orbit visits U+ (recall that there is an extra symbol ‘1’ coming from the gap). If ν is rational, then τ(Spec(ν)) is periodic, as easily Suppose now that we introduce a small perturbation verified. For example, the sequence S(3/7) is given by of the parameters and the initial condition: (x0, ε, a) 0 0 0 0 t 0 −→ (x0, ε , a ). The perturbed orbit xt = Φε0,a0 (x0) is ini- S(3/7) = τ (Spec(3/7)) 0 tially at a distance ∆x0 = x0 x0 from the unperturbed − 0 = τ(0, 0, 1, 1, 2, 2, 3, 3, . . .) one. Using the continuity of xt on the parameters ε, a = (0, 1, 0, 1, 0, 1, 0), and the initial condition, the distance ∆xt between the two orbits at time t can be made as small as we want by 0 0 0 where the bar denotes periodicity. making (x0, ε, a) sufficiently close to (x0, ε , a ). Because 0 The symbolic dynamics of Φε,a is that of a uniform Γ is an open interval, if xq belongs to Γ, then so does xq rotation with rotation number ν analogous to that of a for a sufficiently small perturbation. Therefore we have subcritical circle map, as we shall see later. The dy- established the crucial result: namics of the whole lattice is then reduced to the study of the one-dimensional map Φε,a whose rotation number Theorem 3. If the gap orbit of Φε,a is finite, then it is gives the velocity of propagation of the travelling front. stable under a sufficiently small perturbation of parame- Using arguments similar to that given in section 1.14 of ters and initial condition. [15], it is possible to prove that the rotation number is a well-defined, non-decreasing, continuous function of the Thus there is a region in the parameter space (ε, a), parameters (ε, a) and it is independent of the initial con- where the given rotation number is constant or mode- dition. The difference here is that the auxiliary map is locked. An example of mode-locking ε-regions for a fixed a non-decreasing function and not a strictly increasing value of a is given in figure 1. The mode-locked region, one as in [15], but the same line of proof may be fol- corresponding to a given rotation number ν = p/q, can be lowed. Therefore, when the gap size is non-negative, the computed by noting that in order that an orbit through velocity v(ε) of the travelling interface is a well defined, any point in Γ returns after q 1 iterations we must have non-decreasing, continuous function and does not depend that Φq−1(Γ) Γ. This condition− may be rewritten as ε,a ⊆ on the choice of the initial minimal state. Φq−2(1 2ε) Γ giving the inequalities ε,a − ∈ γ Φq−2(1 2ε) γ , (31) + ≤ ε,a − ≤ − 5. STRUCTURE OF PARAMETER SPACE since

x Γ = x = Φε,a(x ) = 1 2ε. We now turn to the study of the parameter space of ∀ 0 ∈ ⇒ 1 0 − the coupled map lattice F , in the case where the gap ε,a We shall mention that the end points of (31) are included size γ(ε, a) is non-negative (cf. (19)). Then the gap orbit because an orbit arriving at γ after q 2 iterations is defined, and we denote by v = v(ε, a) the velocity of § will also reach 1 2ε after two more iterations− since the center of mass of the corresponding minimal state. − Φε,a (f§(γ§)) = Φε,a( a) = 1 2ε (cf. (18)) — these two orbits correspond §to the kneading− sequences of the local map [16,17]. The closed ε-interval region for which 1. Mode-Locking the velocity is mode-locked to a given rational is called a plateau. Using the symbolic coding we now prove that the gap Equation (31) ensures that the orbit starting at x1 = Γ forces the velocity of the travelling wave to be locked 1 2ε falls into the gap after q 2 iterations and there- − − to rational values if Φ has an eventually periodic or- fore repeats itself after q 1 iterations. The value of ε,a − bit containing the point 1 2ε. Mode-locking guarantees Φq−2(1 2ε) is determined by the coding sequence of the ε,a − stability of the propagating− signal with respect to para- associated rotation number. So if we want to compute metric perturbations. the mode-locked region for a given velocity ν = p/q we The symbolic coding of the orbit of the auxiliary map have to determine the (ε, a) region where (31) is satisfied Φε,a gives the velocity of the travelling interface via the in correspondence to the code S(p/q) given by (30). For rotation number (28). The presence of a non-negative example, to compute the mode-locked region for the ve- gap Γ induces a mode-locking of the rotation number in locity ν = 2/5 we first compute S(2/5) = (0, 1, 0, 1, 0), so

6 0 0 0 we know that the orbit visits the regions U−, U+, U− and v = m/n and v = m /n are two consecutive rationals of 3 F 0 Γ. Solving for Φε,a(1 2ε) = f− (f+ (f−(1 2ε))) gives the Farey series max(n,n ). Then, using the lattice repre- − − sentation, it is not difficult to see that if m/n < m0/n0 the γ f (f (f (1 2ε))) γ . 00 0 0 + ≤ − + − − ≤ − coding sequence of the velocity v = (m + m )/(n + n ) corresponding to their mediant is given by In figure 5 we show the mode-locking regions in the (ε, a)- m + m0 m m0 plane, computed via (31), for some rational velocities. S = S S , (32) These regions, represented as shaded areas, are the so- n + n0 n n0 µ ¶ µ ¶ µ ¶ called Arnold’s tongues [14,18]. It can be observed that where the right hand side denotes the concatenation of as we approach the zero-gap line (dashed line) — given two periodic sequences, defined as follows. If Sa = by γ = 0 in (19) — the tongue size decreases, as the (a1, . . . , ap) and Sb = (b1, . . . , bq) are periodic sequences latter is related to the gap size. of period p and q, respectively, their concatenation SaSb is the periodic sequence of period p + q defined as

2. Lattice representation SaSb = (a1, . . . , ap, b1, . . . , bq). (33) The result (32) derives from the fact that the parallelo- We now introduce a geometric representation of the gram (OPP 00P 0), with P = P (m/n), P 0 = P 0(m0/n0) and rational velocities of the wavefront. This representation P 00 = P 00(m00/n00) = P 00((m + m0)/(n + n0)), does not gives a straightforward method for computing the associ- contain any lattice point since m/n and m0/n0 are con- ated coding sequence as well as some properties related to secutive numbers of a Farey series [20]. the mediant of two given consecutive rational velocities An example of this construction is shown in figure 6 of a Farey sequence. where the velocities v = 1/3 and v0 = 1/2 give the medi- We represent the rational velocity v = m/n as the ant v00 = 2/5. The sequence for v = 1/3 is (010) (vertical point P (m/n) = (n m, m) on the integral lattice Z2. cross + lattice point) and the sequence for v0 = 1/2 is Consider the ray OP emanating− from the origin and pass- (10) (just a lattice point) and the resulting sequence for ing through P (m/n). It is obvious that distinct irre- the mediant (01010) is obtained by the concatenation ducible rational velocities correspond to distinct points (010)(10). and distinct rays. The ray OP will intersect the lattice grid infinitely many times. We drop the first intersec- tion in O and take all the remaining ones. Next, write 3. Unimodular transformations and envelopes a ‘0’ every time the ray OP crosses a vertical grid line, write a ‘1’ when it crosses a horizontal grid line and write The structure of the parameter space of the map- ‘10’ when its crosses through a lattice point (see example ping Fε,a at the boundary of a tongue can be described in figure 6). The resulting semi-infinite binary sequence analytically by means of envelopes. These are func- can be shown to be equal to the symbolic coding of the tions characterizing the structure of sequences of adja- velocity ν = m/n given by (30). cent tongues. The study of envelopes proceeds in two We sketch the proof. Recall that a ‘0’ in S(m/n) means stages. Firstly we derive upper and lower envelopes one iteration of the CML without advance of the interface for the so-called first order plateaus (i.e. v = 1/n and while a ‘1’ means that the interface has advanced one v = (n 1)/n, n = 1, 2, . . .). Then with the use of uni- site. Let us now consider the consequence of crossing modular−transformations we find envelopes for any family vertically and horizontally the grid lines of Z2. Crossing of plateaus. a vertical line of the grid means transition from the point The zeroth order plateaus are defined as the two (n m, m) to the point (n + 1 m, m), i.e. from P (m/n) plateaus given by (24) with velocities 0/1, 1/1 . In or- − − to P (m/(n+1)) — this corresponds to iterating the CML der to construct any order of plateaus,{ the k-th} order once without any advance of the interface (symbol ‘0’). plateaus, we take any two consecutive elements of the pre- Crossing a horizontal line means moving from the point vious order k 1 denoted by va = ma/na and vb = mb/nb (n m, m) to the point (n m, m + 1) = ((n + 1) (m + (v < v ). Then− the next order of plateaus is defined as − − − a b 1), m+1), i.e. from P (m/n) to P ((m+1)/(n+1)), which the following infinite increasing sequence of mediants corresponds to a shift of one site of the interface after 3m + m 2m + m m + m m + 2m one iteration (symbol ‘1’). And finally, crossing through , a b , a b , a b , a b , · · · 3n + n 2n + n n + n n + 2n · · · a lattice point is equivalent to a shift of one place of the ½ a b a b a b a b ¾ interface in two iterations (symbol ‘10’). (34) We recall that the Farey series [19,20] of order r, Fr, is defined to be the set of irreducible fractions, in ascending The first order plateaus are thus given by order, belonging to [0, 1] whose denominators are smaller 1 1 1 2 3 , , , , , , . than or equal to r. Suppose now that the velocities · · · 4 3 2 3 4 · · · ½ ¾

7 They form an unique family. On the other hand, between Replacing (40) in (31) we obtain the mode-locking region any two consecutive plateaus of a given order there is a for a given series of plateaus new family of plateaus of the next order and so on. This n ∗ ∗ f (γ ) αb(α (x ξ ) + ξ ) + βb f (γ ), construction is similar to that of the Farey series [19]. 10 + ≤ a 0 − a a ≤ 10 − In particular, any two consecutive plateaus p/q and p0/q0 (41) (p/q < p0/q0) of the same order and family are consecu- tive fractions of a Farey series (i.e. qp0 pq0 = 1). where we have taken into account the fact that the two − From the definition of the sequence of plateaus (34) final symbols ‘10’ of any sequence in Σa come from the and by applying repetitively the sequence concatenation gap Γ and therefore they have to be compensated by (32) it is possible to find the sequences associated to a replacing γ§ in (31) by f1(f0(γ§)) = f10(γ§) . given family of plateaus: As we mentioned earlier, the plateaus include their end points. These are obtained from solving the inequalities 3 2 2 3 . . . , SaSb, SaSb, SaSb, SaSb , SaSb , . . . , (41) for the extremes f10(γ§). Therefore, all the end points of a given family of plateaus are given by where w©e used the shorthand notation Sa = Sª(ma/na) n ∗ ∗ and Sb = S(mb/nb). It will be useful to distinguish the αb(α (x ξ ) + ξ ) + βb = f (γ ), (42) a 0 − a a 10 § left Σa and right Σb subfamilies of sequences defined by where the positive (negative) sign gives the left (right) 3 2 Σa = . . . , SaSb, SaSb, SaSb end points of the plateaus. As we vary n = 0, 1, . . . in (35) 2 3 (42) we browse all the extremes of the series of plateaus Σb = ©SaSb, SaSb , SaSb , . . . ª. going from the mediant of va and vb to va. Note that the sequence© SaSb, correspondingª to the medi- Solving (42) for n yields ant of v and v , appears in both families. The elements a b ∗ 1 f (γ ) βb αbξ of the subfamilies Σa (Σb) tend, to the left (right), to Sa 10 § a n§(ε, a) = ln − −∗ . (S ). Thus the subfamilies Σ (Σ ) correspond to veloc- ln αa αb(x0 ξa) b a b − ities going from the mediant of v and v to the plateau a b Finally, replacing the value of n in (37) by n (ε, a) gives v (v ). § a b the continuous functions From now on we consider only Σa. All the results generalize to Σb just by interchanging the subscripts a n§(ε, a) ma + mb v§(ε, a) = , (43) and b. The elements of Σa are of the form n§(ε, a) na + nb

n S Σa = S = S Sb n = 1, 2, . . . (36) that passes through all the left (v+) and right (v−) ∈ ⇒ a boundaries of the plateaus. Therefore, v§ gives the up- which correspond to the velocities per (+) and lower ( ) envelopes of the series of plateaus (37). − n n ma + mb v(Sa Sb) = n = 1, 2, . . . (37) In figure 7a we display the first order plateaus for a = n na + nb 0.4. We only plotted the left family (v 1/2) since the We recall that the transformations (37) are unimodular right family (v 1/2) is symmetrical≤from (11). An since we always deal with consecutive fractions of Farey example of a second≥ order family, for the same value of series. a, is shown in figure 7b where we display the lower and The sequences Sa and Sb correspond to a particular upper envelopes for the left and right families of second combination of f− and f+. Since f§ are linear, any such order plateaus between v = 1/2 and v = 1/3. combination will again give a linear function. We de- With the method described above, it is possible to find note by fSa (fSb ) the linear function resulting from the the envelopes of any sequence of high order plateaus via combination of f§ specified by the sequence Sa (Sb): unimodular transformations. The self-similarity struc- ture of the velocity function, easily observable in figures fSa (x) = αax + βa (38) 7a and 7b, is controlled by modular transformations (37) fSb (x) = αbx + βb. and envelopes. Since it is possible to find a periodic symbolic sequence for any given rational velocity arising Composing fSa with itself n times gives from an eventually periodic orbit of Φε,a through 1 2ε − n n ∗ ∗ then it is possible to find a non-empty ε interval where f (x) = fSn (x) = α (x ξ ) + ξ (39) Sa a a − a a the velocity is mode-locked (Theorem 3). Therefore the ∗ graph of the velocity v(ε, a) as a function of ε is in fact where ξa βa/(1 αa) is the fixed point of fSa . Com- bining (38)≡ and (39)− in (36) we obtain the iterates of a Devil’s staircase [18,21], i.e. a fractal staircase. This n mode-locking phenomenon is very similar to the one ob- x0 = 1 2ε under Φε,a for the coding sequences Sa Sb − served for a uniform rotation in a subcritical perturbed n n fSa Sb (x0) = fSb fSa (x0) . (40) circle map [14]. ¡ ¢ 8 ∗ If, on the other hand, the symbolic sequence is chosen a contraction mapping at x§. This is the case, for in- 00 ∗ ∗ 00 to be non periodic, the associated velocity ν is irrational. stance, if f (x) < 0 for x− < x < x and f (x) > 0 ∗ ∗ ∗ The continuity of the function v(ε) guarantees that there for x < x < x+, where x is the unstable fixed point in ∗ ∗ exists at least one value of ε such that v(ε) = ν. We between x− and x+. However, without superattractive- believe that it is possible to prove that there is only one ness, the travelling wave will be less stable to perturba- such ε. The idea is to take a sequence of rational approxi- tions. Then, if we are dealing with a medium involving mants of ν; the corresponding upper and lower envelopes random fluctuations we would like the map to be suffi- should approach each other, as the approximants tend to ciently contracting for a travelling interface to survive. ∗ ν, and coincide in the limit to give a unique ε such that On the other hand, the lack of superstability of x§ intro- v(ε) = ν. duces more sites in the interface since the convergence to ∗ x§ is slower. Moreover, in this case the number of sites ∗ ∗ contained in the open interval (x−, x+) is not necessar- 4. Convergence to minimal states ily finite (take the example of an exponentially localized state). For this reason the reduction of the dynamics in this case does not lead to any simplification. Neverthe- In this chapter we have described the structure of the less, if the shape of the front remains unchanged during parameter space associated with minimal states. Here we the evolution, it is possible to concentrate our attention generalize the notion of minimal state and briefly com- to a single site in the interface since the remaining sites plete our description of the parameter space. are determined by the shape of the front. We could, for A minimal state — a step state of size 0 or 1 — is stable example, consider the site that is closest to the unstable under iteration when the gap is non-negative. When the point x∗ and from it reconstruct a one-dimensional aux- gap is negative, the stable configuration of a step state iliary map that accounts for the whole dynamics, in the is found to have size assuming only two values, N and same way that Φ accounts for the whole dynamics of N 1. Such states will be called minimal N-states. ε,a minimal states. A more detailed study will be given in Numerical− experimentation reveals that the integer N [22]. does not depend on the initial condition. Any initial Secondly, the discontinuity of the derivative of the local configuration we have tried in the interface (increasing, map is not necessary for mode-locking. We have observed decreasing, oscillatory, random with small and large size, mode-locking for analytic maps as well. In figure 9 a win- etc . . . ) converged, after a transient (depending on the dow of the velocity as a function of ε is shown for some initial conditions) to a minimal N-state, with the value of polynomials maps (of odd degrees from 5 to 13). The N depending only on ε and a. Accordingly, the param- plateaus v = 1/2 and v = 1/3 are clearly visible. The eter space is partitioned into layers where the value of origin of mode-locking is not related to non-analyticity. N is constant, as depicted in figure 8. The boundary be- It ultimately lies with the asymptotic (t ) discon- tween the first layer (minimal 1-state) and the second one tinuity from the initial conditions associated→ ∞with the (minimal 2-state) is the zero-gap curve, given by γ = 0 repelling fixed point x∗ between the two stable points, in (19). Above that curve the gap is negative, and there which separates two dynamical behaviours. appears to be an infinite family of layers labeled by the In this respect the value of the derivative at the repeller value of N, separated by boundary curves which may not can be interpreted as a measure of this discontinuity. It be differentiable. is therefore expected that, for fixed value of the deriva- The dynamical analysis for such minimal N-states may tive at the attractors, the mode-locked plateaus become be carried out by reducing the dynamics of the whole more pronounced as the derivative at the repeller point lattice to a N-dimensional auxiliary map [22] analogous increases. In figure 9 the plateaus v = 1/2 and v = 1/3 to the one obtained previously for minimal 1-states. are almost imperceptible for the polynomial of degree 5 that has a derivative at the repelling point 1.9, mean- while they become more appreciable for the ∼polynomials 6. GENERALIZATIONS AND CONCLUSIONS of higher degree (7, 9, 11 and 13) where the derivative at the repelling point gets larger ( 2.2, 2.5, 2.7 and 2.9 respectively). ∼ We have described a mechanism that allows travelling Finally, the travelling wave property arises from one- interfaces in a one-way CML (5) for a special family of way coupling in the CML and the direction of the wave is piece-wise linear maps (12). We now give some ideas on given by that of the coupling. By comparison, a front on the generalization of these results to a broader range of a diffusive CML (4) with a symmetric map at each site is functions and couplings. going to remain stationary in the long term. For the front First of all, the superattractiveness of the points x∗ − to advance one needs asymmetry, either on the coupling and x∗ is not necessary. It suffices that the map f be + or in the local map (with respect to the repeller point x∗) which generates a bias between competing attractors.

9 In order to illustrate this phenomenon let us take the A similar inequality holds for i . From the above → −∞ logistic map fµ(x) = µx(1 x). The logistic map itself and the fact that any finite sum of ∆xt (i) is bounded − +1 does not fulfill the requirements (two attractive points (because f is bounded) it follows that Mt+1 is finite (and with a repeller in between) but consider its second iter- non-zero, by hypothesis), so that the probability pt+1 2 ate fµ. At the parameter value µ = 1 + √5 the map- exists. Finally, multiplying both sides of the inequality 2 ∗ 2 ping f has two superattractive fixed points x− = 1/2 (A.1) by i and by i , respectively, and summing over ∗ i > N shows that µt+1 and σt+1 are finite, whence Xt+1 and x+ = 1/2 + √2 3 √5/4) (corresponding to a superattractive 2-cycle for− f) separated by a repeller is localized. p If X is exponentially localized, then there exist posi- x∗ = 1 √2 3 √5/4. Consider iterating the diffu- t tive constants c and κ < 1 for which sive CML− with the− map f 2 at each site. For an initial p µ i step state we obtain a travelling interface where the ve- ∆xt(i) < cκ . (A.2) locity again behaves as in the examples shown above — it has a transition from v(ε) = 0 to v(ε) > 0 at εc 0.198 Combining the above with (A.1) we obtain ' —, see figure 10. In this case the mode-locking is not i ε ∆xt (i) < κ cρ (1 ε) + , i > N. (A.3) apparent at first glance. However, on amplification, the +1 − κ plateaus are shown to occur. For example, the plateaus ³ ´ A similar estimate holds for negative i, showing that corresponding to v = 1/7 (right) and v = 1/8 (left) are X is exponentially localized (possibly with a larger shown in the enlargements in figure 10. t+1 constant c). 2 In conclusion, the mode-locking of the velocity of the travelling interface, appears to be an universal phe- From Lemma A.1 and the fact that the mass of a step nomenon in coupled map lattices. It implies that waves state is bounded from (cf. (8)) we obtain immediately travelling with rational velocities are stable under both spatial and parametric perturbations. Corollary A.2. Let Xt be a step state of a one-way ∗ CML, and let f be continuous at x§ and bounded. Then Appendix A Xt+1 is a step state. If, in addition, Xt is exponentially localized, so is Xt+1. In this appendix we provide various statements and proofs concerning the image of localized states. We have then establish the invariance of the localized Lemma A.1. Let Xt be a localized state of a one-way and exponentially localized states under the dynamics CML, let x§ = limi→§∞ xt(i), and let M(Xt+1) = 0. of the CML and we are ready now to give a proof for 6 Then, if f is bounded and continuous at x§, Xt+1 is a Theorem 1. localized state. If, in addition, X is exponentially local- t Theorem 1. Let X be an exponentially localized step ized, so is X . 0 t+1 state of a one-way CML. Let the local map f be bounded, Proof. We begin considering the case i . Because Xt and let f be a contraction mapping in a neighbourhood → ∞ is localized, x+ is finite. Moreover, since f is continuous of the fixed points (7). Then, for all sufficiently small ε, at x , then there exists a constant ρ > 0 and an integer N v(ε) = 0 and v(1 ε) = 1, independently of X . + − 0 such that, for all i N we have f(xt(i+1)) f(xt(i)) < ≥ | − | Proof. We deal with the case of small coupling first, ρ xt(i + 1) xt(i) , whence | − | and positive i. From Corollary A.2 we know that Xt is an exponentially localized step state for all t 0. By ∆xt (i) = (1 ε) f(xt(i + 1)) + ε f(xt(i)) +1 | − assumption f is a contraction mapping in some≥domain (1 ε) f(xt(i)) ε f(xt(i 1)) ∗ 0 0 x x+ < r. Let N be such that for all i N we have − − − − | | − | ∗ ≥ 0 (1 ε) f(xt(i + 1)) f(xt(i)) xt(i) x+ < r/2, so that ∆xt(i) < r. Then, for i N , ≤ − | − | the| constan− | t ρ appearing in (A.1) can be chosen ≥to be + ε f(xt(i)) f(xt(i 1)) | − − | smaller than 1. 00 00 < ρ (1 ε) xt(i + 1) xt(i) Let N be such that ∆xt(i) < 1 for all i N , and − | − | 0 00 ≥ + ρε xt(i) xt(i 1) let N = max(N , N ). Then since Xt is exponentially | − − | localized, we can choose κ so that the bound (A.2) holds = ρ (1 ε) ∆xt(i) + ρε∆xt(i 1), i > N. − − for i N with c = 1. Letting c = 1 in (A.3) it follows ≥ (A.1) that i Then, from absolute convergence ∆xt+1(i) < κ i > N (A.4)

∞ ∞ provided that ∆xt (i) < ρ(1 ε) ∆xt(i) i=N+1 +1 − i=N+1 ∞ κ 1 ρ P + ρε i=N+1 ∆xt(i P1) < . ε < − − ∞ ρ 1 κ P − 10 which is satisfied for sufficiently small ε, since the right (P (i 1)). For the marginal case xt(i) = x− the state of − k hand side is positive. the lattice Xt+k = [f−(x−), i] = [f−(x−), i] is fixed and It remains to consider the case i = N. Because f is unstable since x− is an unstable fixed point of f−. bounded, the quantity (ii) Using a similar reasoning, if ε 1 εc, x is an ≥ − + unstable fixed point of f+ and the state of the lattice at ∆f = sup f(x) infxf(x) k x − time t + k is Xt+k = [f+(xt), i + k]. Therefore the lattice will reach the state P (i + k) (P (i + k 1)) for xt(i) < x+ is finite. We have, in place of (A.1) − (xt(i) > x+). In the marginal case xt(i) = x+, the k k state of the lattice is Xt+k = G [f+(x+), i + k] = ∆xt+1(N) < ρ (1 ε) ∆xt(N) + ε ∆f k − G ([f+(x+), i]), which is fixed and unstable under G F . 2 ¡ ¢◦ giving, for all t 0 ≥ 1 αt+1 Acknowledgments ∆x (N) < αt+1κN + ε ∆f − , t+1 1 α RCG would like to thanks DGAPA for financial sup- − port during the preparation of this paper and Carmen where α = ρ(1 ε) < 1. The right-hand side of the Cisneros and Alvaro Salas Brito for all their inconditional − N above inequality can be made smaller than κ for all t, support. provided that ε is sufficiently small. Thus the inequality (A.4) holds also for i = N. −i Let i be negative. Assuming that ∆xt(i) < κ for all sufficiently large (negative) i, and proceeding as above, −i we find that the bound ∆xt(i) < κ for i N implies that ≤ − [1] K. Kaneko, editor. Theory and applications of coupled −i −i ∆xt+1(N) < κ ρ (1 ε + εκ) < κ ; i < N map lattices. John Wiley & Sons, 1993. − − [2] E. Atlee Jackson. Perspective of nonlinear dynamics, vol- for all ε (all constants have been redefined). The case ume 2. Cambridge Univ. Press, 1991. Chap. 10. i = N gives the recursive inequality [3] K. Kaneko. Spatiotemporal chaos in one- and two- − dimensional coupled map lattices. Physica D, 37:60–82, −N+1 ∆xt+1( N) < (1 ε) ∆f + ρεκ 1989. − − [4] D. Keeler and J.D. Farmer. Robust space-time intermi- −N which, as before, yields ∆xt+1( N) < κ for suffi- tency and 1/f noise. Physica D, 23:413–35, 1986. ciently small ε. − [5] L.A. Bunimovich, A. Lambert and R. Lima. The emer- The above induction establishes the time-independent gence of coherent structures in coupled map lattices. J. bound Stat. Phys., 61:253, 1990. [6] L. Ying-Cheng and C. Grebogi. Synchronization of spa- |i| tiotemporal chaotic systems by feedback control. Phys. ∆xt(i) < κ , i N, t 0. | | ≥ ≥ Rev. E, 50(3):1894–9, 1994. for a suitable choice of κ and N, and for all sufficiently [7] R. Kapral and G.-L. Oppo. Competition between sta- small ε. In this ε-range, from the boundedness of f we ble states in spatially-distributed systems. Physica D, 23:455–63, 1986. conclude that µt and σt are bounded from above for all [8] R. Kapral, R. Livi, G.-L. Oppo and A. Politi. Dynamics times. Thus v|(ε)|= 0, and X remains a step state also t of complex interfaces. Phys. Rev. E, 49(3):2009–49, 1994. in the limit t . 2 → ∞ [9] K. Kaneko. Chaotic travelling waves in a coupled map lattice. Physica D, 68:299–317, 1993. Appendix B [10] R.E. Amritkar, P.M. Gade and A.D. Gangal. Stability Proof of Theorem 2. of periodic orbits of coupled map lattices. Phys. Rev. A, 44(6):3407–10, 1991. (i) The dynamics of the i-th site is given by xt+k(i) = k [11] P.M. Gade and R.E. Amritkar. Spatially periodic or- f−(xt(i)) since it is coupled to the (i 1)-th site whose bits in coupled map lattices. Phys. Rev. E, 47(1):143–53, value is 1. For the dynamics of the− (i + 1)-th site, − 1993. since ε εc and a f (x) for any x U, we have ≤ ≤ + ∈ [12] B. Fernandez. Existence and stability of steady fronts in that xt+k(i + 1) S+. Therefore at time t + k the state bistable CML. Preprint, 1995. ∈ k of the lattice will be equivalent to Xt+k = [f−(xt), i] [13] J. Stark. Smooth conjugacy and renormalization for which depends exclusively on the initial condition xt(i). diffeomorphisms of the circle. Nonlinearity, 1:541–575, 0 If ε εc the derivative of the map f−(x) = (1 ε)/a 1 1988. ≤ − k ≥ [14] G.L. Baker and J.P. Gollub. Chaotic Dynamics, an in- and then f− has a repeller at x−. Therefore f−(xt(i)) troduction. Cambridge Univ. Press, 1990. Chap. 4. will decrease (increase), for xt(i) < x− (xt(i) > x−) until it reaches S− (S+) when the state of the lattice is P (i)

11 [15] R.L. Devaney. An introduction to dynamical systems. Addison-Wesley, second edition, 1989. [16] J.H. Hubbard and C.T. Sparrow. The classification of topologically expansive Lorenz maps. Comm. on Pure and Appl. Maths., 43:431–443, 1990. [17] P.A. Glendinning and C.T. Sparrow. Prime and renor- malisable kneading invariants and the dynamics of ex- panding Lorenz maps. Physica D, 62:22–50, 1993. [18] P. Bak. The Devil’s staircase. Physics Today, December:38–45, 1986. [19] R.L. Graham, D.E. Knuth and O. Patashnik. Con- crete mathematics, a foundation for computer science. Addison-Wesley, 1989. Chap. 4.5. [20] G.H. Hardy and E.M. Wright. An introduction to the the- th ory of numbers. Oxford Univ. Press, 4 edition, 1960. Chap. 3. [21] M. Schroeder. Fractals, Chaos, Power Laws. W.H. Free- man and Company, 1991. Chap. 7. [22] R. Carretero-Gonz´alez, D.K. Arrowsmith and F. Vivaldi. Reduction dynamics for travelling fronts in Coupled Map Lattices. In preparation, 1996.

12

)#02-%!

‹ 

^

jˆj1j>jˆj1j>jj>j>jj>j1j

‰ ‰

1.0

‰ j

0.9 ’

Š

^

’

jˆj>j1jˆj

h

Š

’ Ž

0.8 Œ

e

( +0•?–:^7=—BDS #0 

„O ¤†

’

’ 

0.7 ‹

’”“

#0?#0 ";

^

 j>jj>j1j

0.6

Š

j

‰8Š ‰

) 

ε 0.5 ‰

v(

^

jˆj>j1jˆj1j>jˆj1j>jˆj1j>j

h Ž

0.4 Œ

;=/"L-¤#: ?-o‘.;=#0; b

0.3

¢¡¤£¦¥¨§?©q˜™ ›š:IK)C¦!#0 q ?-:#0/.K)H 0;4 ?-:-¤0K) Dœ- NDG"D.?

t:u@v U¦:/‚ 0¦?$G)žV#070K)DG";= ¦?

0.2 ;0#?#=;‚ -A‡ 7%9:$

t:u@v ;N=K)CV!K,#= .,^70K)DG";= ¦?o;=$M;=#0D,UJA/‚ "$@K)=

0.1 ?-O-¤=KM DŸ;‚#=/.?C[ -N#0/.M#=%-% !S;=#03 #=;[K)$M¦?DGE!;F;

e

Y¦<‡.Y)$[#=/‚?C)ƒ"E 0$4DR U 6S„O ¤† 0.0 a=1/2 a=2/5 a=2/7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ε

1.0 ¢¡¤£¦¥¨§ © ¦ ¨  "!#%$'&)(%*+, -.#0/1)#02-%!34-¤ 5

0.9

!8729:<;=><?@ "729A$B! C 5D. E #0#=E!1?;,>-¤C¦!#0 

6 6 6

-F#0/5! C " ?8 DGH#=G*I3J! D C#=IK>)C"DG=E!H $,-¤ .K)L-E7

6 6 6

0.8

-¤=M#N C";O -P#=/"4 "!?QDR  8 DS#=NTVUJ:/"W "!#%$./¦?;

6.6

J;0#?8! ;07%X OYV/¦) CZ;=$MDGDG#0=E!Z9:#=/[=; !#O#0 4#0/ )# 6 6 __1 __4 0.7

5 5 (2*\&M(%*+=+Z]B(^_`\^_ `+HU

__2 __7 0.6 9 9 __2 __5 7 __2 __3 7 5 5

a 0.5

d e

( +

c

g

a

c m k 1 3 g __ __

^ 0.4 __0 4 4 __1

j f g 1 1

f __1 __2 g 3 3

f 0.3

g __1

f 2 g

f 0.2

g

f

g T

^ 0.1

f

h h

g

bJc

f

Ti^

c k

g 0.0

f 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

g

;:#= C;OR#=/[(2*I\T)+¤7 E "N! ==H; ¦K)"

¢¡¤£¦¥¨§?©@¡o O¢O=" EK”£ ε

6 6

f

g

#0 W8 #0 ¦? H !#0;Z -o#0/:#0OM#=%-% !8:-¤ ¨#=/"O "729A$

f

g

e

! C K.D. E?#=#0E!F¤ U

6 6 ¤†

f

g

f

^

g

h

j

«_ « «_?¬ « _­ «_ ® «_¯

c¦k l

g

d?e

¢¡¤£¦¥¨§ ©'no Bp¨!7%9:;0q"?. !?ND. -¤ .#=/"r 7%9As$

6

t:u@v

U

´¸·

±°

`_ ` `_« ` _?¬ ² `_ ­ `_®

¥

g

T

¨

¥

j

w

w

¨

g

¥

¨

w

d

m

¥

¨

g

w

¨

¥

¨

g

w

¥

¨

°

w

² ¥

¨

g

^ ` *

´¶µ ´

¨

h

j j

¥

yz {| x

¨

g

x

¥

±° ±° ±

¨

² ² ² ¥

§©¨

¦ ^_^ ^_` ^s_« ^_ ¬ ^_­

x

¦

¥

§

¦

x

¥

¦

°

²

§ ¦ ¥

x

d

¦ °

l

¥

²

§

¦

x

¥

¦

§ ¦

¥

}

x

¦

¥

§ ¦ € b

x

¥

¦

³

²

ª_"^ ª _` ª_ « ª_?¬

x

x

¢¡¤£¦¥¨§?©ˆ¹o Gº%)#0?¢E?#=#0E!‡0 0;=M#?#= ˆ ?-O#=/"G?#0 V Z 87

6

x

 "!#=H;U »¨ 0$G8 #0 V " < !#%$G¼G_½@! =0; ¦KM;O#= ‚#=/ M#

6 6

~ ~

m 

x

l

¾‚(2¼S_ ½o+Z]'(2½ ¼›\H¼S+¢ .#=/NE?#=#0E!U¿º%#=;A;=$MD‡Y¦ E! ;0À C"¦!‚;

T

h

h

j

#0/@! ;0#=0C¦!#=IKÁYM$‡#0/F!= ;=;0;: ?-ZÂQ¾B9:<#=/G#0/N 0EK‡";–¨ T T

m

l

} }

h

à Ã

ª"£ ! 0=; VK);¢#0 N!= ;=;0FO=#0E!?)"3  ^£sO/" 0Ä M#?

6

à t

?¦K[ ^ª"£ ! =0; VK);:#0 W!8= ;=;=<‚ÅÆ #0#=E! M#U V!?#=¦?#%7

6 6

e

¢¡¤£¦¥¨§ ©‚o ¢:/"O?C)ƒ<E 0$ND. 9:/E!0/ 0 MEK);Z#=/"OK)$M¦?D‡7

64„O ¤† 6

"G;=ÀCV!;N- O#%9¢ SK)LÇo0)#N !8#=;F¼G_½5 VK.¼‡ÈE_½¦È ! %7

E!;: ?-¨#=/FM#=%-% !U

È È

0; ¦KM;W#0 .#=/"4!#= FKK)#0 . -AÂɾ4(%¼G_½o+O VKrÂɾ‚(2¼ _½ +3

6

?¦KG#0/F=;0C#0‡ "!#2$[;:#0/NDGK)Æ M#O(2¼'ʼ +=_M(2½‚ʈ½ +HU

È È

13 0.50 0.34

0.45 0.33

0.32 0.40

0.31 0.35

0.30 0.30 ε)

) 0.29

ε P5 (x) v( P 7(x)

v( 0.25 P9 (x) 0.28 P 11(x)

0.20 P 13(x) 0.27

0.15 0.26

0.10 0.25

0.05 0.24

0.23 0.00 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.29 0.31 0.33 0.35 0.37 0.39 0.41 0.43 0.45 0.47 0.49 0.51 0.53

ε ε _ &0\C -A,uBX Sv "(i)/+w%5# -A0X )CUx(o yvR`_ba

0.50 ¢¡¤£¦¥¨§"©tsQ !X )/+z',W B 5)/+r¦  U&- 5WX{ T -A#"|W"'l}~p=€k+E¦  U&- 5WYX1  '

0.48 }¨q0}ƒ‚„q0}ƒ "q0}I†0†"¦Y}V†C‡ˆ+2%5]B#S5/##'*‰-Š‹ŠŒ-Š¦JJˆ"¦J2O|/#',2A,\ )/X %5#^U&=Žk+T-1 )/2"3'D*7KJ2L5O*"¦jD*7KJ2L <*>¦2A0 W`X ¦A0,„ '/X S U

0.46 %BX ',X >- !X ),+),+-&#S,#f "(*),+ 5 UB WX1" '*A0 5/,#'/¦ ¦&X -S)/ X A0,2"',X -S),+-*&#,X^%5"),X^%5]")),+-* 5/X S5X =

0.44

0.42 ) ε

v( 0.15

0.40 0.14

0.13 0.38 0.12

0.36 0.11

0.10 0.34 0.09

0.32 0.08

0.37 0.39 0.41 0.43 0.45 0.47 0.49 0.51 0.53 ε)

ε v( 0.07

¢¡¤£¦¥¨§ © ¦ ¦  "!¢#$#&%# ¦#' "(*),+-.¦,'/) 0& 0.14305

0.06 0.12510 <¦=>¦$¦#? @ "!¢#$#&%# ¦#'$ (*)/+ 1"),2"3'4(¤ 5$687:9; 0.14295 0.05 ',2A0 ¦ 0B#1")/2 3-'>¦#)C!¢##EDBFHG0I7KJ2L5M4 D&FCG0I7KJNLO-= 0.12505 __1 0.14285 7 0.04

0.12500 __1 0.14275 0.03 8 0.14265 1.00 0.02 0.12495 0.14255 0.9070 0.9075 0.9080 0.9085 0.9090 0.9095 0.9100 0.01 0.12490 0.95 0.7208 0.7209 0.7210 0.7211 0.7212 0.7213 0.7214 0.00 minimal 4-state 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.90 ε

0.85 minimal 3-state ¢¡¤£¦¥¨§"©‘’Q ]_ B0\H^ A,uBX S$ (I),+-|%# -A0X )CU (I)/+*),02%#  X -SjX &\ )/#H(CA0TX ]BX”“Q3'/X %5TR`_faV="k+^ A#"W"*X 'V)/+'/2A0 ¦jX )/#0"),

0.80

‰ŠQ"¦ "(¢)/+j^ 5SX ',)/X1ATW"f•"–¦FH—¦`7™˜Ž—QFZJh—¦T!X )/+f˜@7šJ¢›8œ

0.75 zero-gap )/+eX^&),H(CA0eX 'T'/3 5/),2>&UX^),'])H! j'/3¦#/',)Z >- j¦ X &)/'2=]k+ %# AZX )HUY',##W'`), j+2%5*4',W B 5)/+S0"+ŽŠ&>-3)T),+-]W B0\C A,uBX S

a \

0.70 { )/2 3-'eAZ 5/,#'/¦ 5&X Sf)/ D7gJ2L5‹? ¦D7žJNLŸ>¦2A0 W4#%&X &&)T3&#TW"SX”.QA )/X 5=

0.65 minimal 2-state

0.60

0.55

0.50 minimal 1-state

0.45 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

ε

¢¡¤£¦¥¨§ ©$PQ R¢ &%#/S5#A0T12U5,'V(¤ VWYX X WY "',)Z )/#'X $),+[X 2A00\

!X ',] X^#R`_baI=cKd\CWX X WY2',)Z )/]X 'Te'/)0"),]!X )/+fdg 5[dih

J',X ),'j 8),+-X &)/#H(CA0=8k+lA03,%'/#¦"0"),X -S@)/+12U#,'fJ0\

WX X WY" '/)0"),]"¦$MN\CWX -X W2',)Z )/X '¢A   24),+-m##, "\CSB YA03,% FonpFHG2q06BV7r9#=

14