Lyapunov Spectrum of Chaotic Maps with a Long-Range Coupling Mediated by a Diffusing Substance

Nonlinear Dyn (2017) 87:1589–1601 DOI 10.1007/s11071-016-3135-0

ORIGINAL PAPER

Lyapunov spectrum of chaotic maps with a long-range coupling mediated by a diffusing substance

R. L. Viana · A. M. Batista · C. A. S. Batista · K. C. Iarosz

Received: 1 June 2016 / Accepted: 6 October 2016 / Published online: 17 October 2016 © Springer Science+Business Media Dordrecht 2016

Abstract We investigate analytically and numeri- 1 Introduction cally coupled lattices of chaotic maps where the interaction is non-local, i.e., each site is coupled to all the Spatially extended dynamical systems with finite other sites but the interaction strength decreases expo- degrees of freedom have long been used as toy mod- nentially with the lattice distance. This kind of coupling els for the description of continuous systems like flu- models an assembly of pointlike chaotic oscillators in ids and plasmas [1]. A particular type of those sys- which the coupling is mediated by a rapidly diffusing tems are coupled map lattices, for which the space and chemical substance. We consider a case of a lattice of time are discrete variables, but allowing for a contin- Bernoulli maps, for which the Lyapunov spectrum can uous state variable [2]. Coupled map lattices are thus be analytically computed and also the completely syn- more complicated, in the dynamical sense, than cellular chronized state of chaotic Ulam maps, for which we automata due to their capacity of generating complex derive analytically the Lyapunov spectrum. spatio-temporal patterns [3]. Most investigations on coupled map lattices have Keywords Lyapunov exponents · Coupled map been focused on the so-called locally coupled lat- lattices · Long-range coupling tices, for which each lattice site interacts with its nearest neighbors [2]. This kind of coupling is physically related to diffusion, which is essentially a local phe- nomenon applicable to systems obeying Fick’s law. Indeed, a locally coupled map lattice can describe a R. L. Viana (B) reaction–diffusion system for which the reaction occurs Department of Physics, Federal University of Paraná, at discrete times. Curitiba, Paraná, Brazil On the other hand, many works have considered e-mail: viana@fisica.ufpr.br globally coupled lattices, in which each site interacts A. M. Batista with the mean field generated by all other sites, irre- Department of Mathematics, State University of Ponta spective of their relative position in the lattice [2]. This Grossa, Ponta Grossa, Paraná, Brazil kind of network has been first investigated by Win- C. A. S. Batista free in his pioneering works on the synchronization Center for Marine Studies, Federal University of Paraná, Pontal do Paraná, Paraná, Brazil of biological systems [4]. Moreover, the well-known Kuramoto model of coupled oscillators uses this kind K. C. Iarosz Institute of Physics, University of São Paulo, São Paulo, of coupling [5], with many physical applications such São Paulo, SP, Brazil 123 1590 R. L. Viana et al. as Josephson junction arrays [6], semiconductor laser pling strength for the transition to synchronization that arrays [7], Landau damping of plasmas [8], cortical is numerically observed. Recently, González-Avella oscillations [9], and bursting behavior of neurons [10]. and Anteneodo [21] computed the largest Lyapunov It turns out that a globally coupled lattice is non- exponent to characterize the chaoticity of delayed cou- local since it takes into account the interaction of each pled maps with adjustable range of interactions. There site with all its neighbors. However, the lattice distance are studies in which the maximal Lyapunov expo- between them does not play any role, what can be con- nent was considered as a diagnostic tool in a net- sidered a kind of approximation. In general, non-local work of Fermi-Pasta-Ulam model where the coupling couplings would need to consider the lattice distance decays as the distance [22] and in Hamiltonian systems between them. One physically interesting model for with long-range interactions [23]. In addition, Laffar- that was proposed by Kuramoto, who considered the gue et al. [24] presented large-scale fluctuations of the coupling among spatially localized oscillators medi- largest Lyapunov exponent in espatially extended sys- ated through a diffusing chemical substance [11–13]. tems that describe diffusive fluctuation hydrodynam- In this model, each oscillator secrets a chemical with ics. a rate proportional to the oscillator dynamics, and this This paper is organized as follows: in Sect. 2,we chemical diffuses in the medium, its local concentration outline the Kuramoto model for the diffusion-mediated affecting the dynamics of the oscillators themselves. coupling of maps. Section 3 considers the Lyapunov When the diffusion is much faster than the oscillator spectrum of a lattice of Bernoulli (piecewise linear) period, the local concentration of the mediator chemi- maps. In Sect. 4, we investigate the completely syn- cal relaxes to a stationary value, and the coupling inten- chronized state of a coupled logistic map (Ulam) lat- sity among oscillators decreases with their lattice dis- tice with diffusion-mediated coupling, the result being tance in an exponential way for one-dimensional lat- used for investigate the transition to synchronization as tices. This diffusion-mediated coupling model has been the coupling strength is varied through a critical value. used to describe bursting synchronization in neuronal The last section is devoted to our Conclusions. networks [14] and circadian rhythms in the suprachi- asmatic nucleus [15]. In spite of its importance in the description of a number of biologically relevant phenomena, there are 2 Diffusion-mediated coupling model relatively few studies on the dynamics of diffusion- mediated coupled map lattices. One of the basic quan- Since we deal with spatially extended systems, it is nec- tities that can be computed in such systems is the Lya- essary to introduce two different classes of vectors, rep- punov spectrum, from which other quantities of inter- resented with a different notation: (i) positions r in a d- est can be extracted like the Kolmogorov-Sinai entropy dimensional Euclidean space, to which the sites belong; = ( , ,... )T and the Lyapunov dimension [16]. More often, the (ii) state variables Xn x1n x2n xMn in a M- calculation of the Lyapunov exponents can be made dimensional phase space of the dynamical variables only numerically, but in some cases, it can be per- characterizing the state of each system at a given dis- formed analytically, what has been done for non-local crete time n. Hence, we have N pointlike sites located at discrete positions ri , where i = 1, 2, ···N,inthe coupled map lattices for which the coupling strength (i) decreases with the lattice distance as a power law d-dimensional Euclidean space; and Xn is the state [17–20]. Two cases can be considered for this task: variable for the ith site at time n, whose time evolution ( (i)) (i) when the map is piecewise linear (constant slope) is governed by the map M X (see Fig. 1). and (ii) when there is complete synchronization of all In this work, we will deal for simplicity with a one- dimensional system whose state variable is denoted sites. (i) The goal of this paper is to compute the Lyapunov simply by xn , and whose dynamics is governed by a → ( ) spectrum for these cases, namely lattices of chaotic map x f x . We suppose that the dynamics in each maps (Bernoulli) with diffusion-mediated coupling and site is affected by the local concentration of a chemical, (, ) complete synchronization in lattices of chaotic maps denoted as A r n , through a coupling function g: (logistic, Ulam-type maps). In the latter case, the Lya- (i) = ( (i)) + ( ( , )), punov spectrum can be used to predict the critical cou- xn+1 f xn g A ri n (1) 123 Lyapunov spectrum of chaotic maps with a long-range coupling 1591

On substituting Eq. (3) into Eq. (2) we obtain an equation expressing chemical coupling in the adiabatic approximation ⎛ ⎞ N (i) = ( (i)) + ⎝ σ( −) ( ( j))⎠ . xn+1 f xn g ri r j h xn (5) j=1

If g is a linear function of the state variable for each site, we write

N (i) = ( (i)) + σ( −) ( ( ( j))). xn+1 f xn ri r j g h xn (6) Fig. 1 Schematic figure of the Kuramoto’s model of chemical j=1 coupling among oscillators We will choose the following form for the coupling function: whereas the chemical concentration satisfies a diffusion equation of the form ( j) ( j) εf (xn ), if j = i, g(h(xn )) = ( ) (7) −εf (x i ), if j = i, N n ∂ A ( ) =−ηA + D∇2 A + h(x j )δ(r −r ), (2) ∂n n j j=1 which corresponds to the so-called future coupling, in that each coupled map remains in the same range as for the isolated map, and where ε>0 is a coupling where 1 is a small parameter representing the strength. Inserting (7)into(8) yields fact that diffusion occurs in a timescale faster than the intrinsic period of individual sites, η is a phenomeno- N logical damping parameter (representing the chemical ( ) ( ) ( ) x i = (1 − ε) f (x i ) + ε σ(r −r ) f (x j ). (8) degradation of the mediating substance), and D is a dif- n+1 n i j n j=1 fusion coefficient. The diffusion equation above has a source term h which depends on the state of individual Fourier-transforming Eq. (4), the interaction kernel sites located at r , since each site is supposed to release i is given by (in a d-dimensional physical space) a chemical with a rate depending on the current value of its own state variable. iq·(ri −r) Weassume,asinRef.[5], that the diffusion is so 1 d e σ (ri −r) = d q . (9) fast, compared with the period characteristic of each (2π)d η + D|q|2 site, that we may set A˙ = 0 such that the concentration relaxes to a stationary value that can be written in the If the coupling is isotropic (in the absence of a diffusion following form: flow, for example), the kernel is a function of rij = |ri −r j | and can be expressed as N ⎧ ( ) = σ( −) ( ( j)), ⎪ A ri ri r j h xn (3) ⎨⎪exp(−γ rij), if d = 1, j=1 σ( ) = (γ ), = , rij C ⎪K0 rij if d 2 (10) ⎪ (−γ r ) ⎩ exp ij , if d = 3, γ rij where σ(ri −r j ) is a Green function, which is the solution of where K0 is the modified Bessel function of the second kind and order 0, the constant γ is the inverse of the 2 (η − D∇ )σ(ri −r) = δ(ri ). (4) coupling length and is given by 123 1592 R. L. Viana et al.

η N γ = , ( ) ( ) ε −γ ( − ) (11) x i =( −ε) f x i + e s f x i s D n+1 1 n κ(γ) n s=1 + (i+s) , and the constant C is determined from the normaliza- f xn (18) tion condition where the normalization factor has been written in the form ddrσ(r) = 1. (12) N κ(γ) = −γ s, Let us adhere to the one-dimensional case, for which 2 e (19) = we work in a regular lattice parameterized by the spatial s 1 length z and with constant spacing , in such a way (i) where N = (N − 1)/2forN odd. that xn = x(z = i ), with i = 1, 2,...N. Hence, the non-locally coupled map lattice reads It is instructive to explore the limiting cases of this form of coupling. If γ goes to zero then κ(0) = 2N = N −1 and we have a global (all-to-all) type of coupling N ( ) ( ) −γ ( ) i = ( − ε) i + ε rij j , x + 1 f xn C e f xn n 1 N j=1 ( ) ( ) ε ( ) x i = (1 − ε) f x i + f x s . (13) n+1 n N − 1 n s=1,s=i (20) where the constant C is determined from the normalization condition (12), which now reads In the limit of large γ , the exponential factor decays very rapidly with the lattice distance , such that only N the term with = 1 contributes signiﬁcantly to the σ(|r −r|) = 1, (14) j summations, yielding κ(∞) = 2e−γ and the coupling j=1 term takes into account only the nearest neighbors of a given site such that (i) (i) ε (i−1) (i+1) ⎡ ⎤ x + =(1 − ε) f xn + f xn + f xn , −1 n 1 2 N −γ (21) C = ⎣2 e rij⎦ . (15) j=1 which is the well-known local (diffusive or Lapla- cian) coupling. Hence, depending on the value of the Without loss of generality, we will make = 1 from inverse coupling length, we can pass continuously from now on. a global to a local coupling type. We will use periodic boundary conditions:

(i±N) = (i),(= , ,... ), xn xn i 1 2 N (16) 3 Lyapunov exponents in such a way that the distance between two lattice sites The coupled map lattice with N sites is a dynamical is the less of the two possibilities: system in a N-dimensional phase space, which can be formally written as rij = min |i − j + N|. (17)

xn+1 = F(xn), (22) In this case, it is more convenient to change the sum- where the state vector can be written as x = mation index in the coupling term of (13)inamore n (1) (2) (N) T symmetric form: xn , xn ,...xn . Let the corresponding tangent 123 Lyapunov spectrum of chaotic maps with a long-range coupling 1593 T ξ = δ (1),δ (2),...δ (N) and, in terms of them, another matrix vector be written as n xn xn xn . On linearizing Eq. (22), we obtain the tangent map ε Bˆ (ik) = ( − ε)δ + B . 1 ik κ(γ) ik (31) ξ = ( )ξ , n+1 DF xn n (23) Using the latter, we can write the jacobian matrix in (29)as where DF(xn) ≡ Tn is the jacobian matrix of Eq. (22), with components ε T = BDˆ = ( − ε)1 + B D , n n 1 κ(γ) n (32) (i) ( ) ∂x + T ij = n 1 . n ( j) (24) ∂xn where the jacobian of uncoupled maps is a diagonal matrix: Starting from an initial displacement ξ , we iterate 0 (ik) = ( (i))δ . Eq. (23) n times and obtain Dn f xn ik (33)

ξ = τ ξ , (25) n n 0 4 Lyapunov spectrum of Bernoulli maps where we have deﬁned the ordered product of jacobian Let x be a variable deﬁned in the unit interval [0, 1). matrices: The so-called Bernoulli map is

τ = ... . n Tn−1Tn−2 T1T0 (26) xn+1 = f (xn) = βx,(mod 1), (34)

Defining the matrix which, for β>1, is strongly chaotic (transitive). In fact, its Lyapunov exponent is λ = ln β>0. In the case 1/2n β = ˆ T 2, we have the well-known baker transformation, = lim τ τ n , (27) n→∞ n which is equivalent to a bilateral shift operator of a bi- infinite two-state symbolic sequence [25]. Moreover, { }N the baker transformation is topologically conjugate to with eigenvalues k k=1, the Lyapunov exponents of the coupled map lattice (22) are obtained as the horseshoe map [26]. Since f (x) = β there follows that λ = ,(= , ,... ). ˆ n ˆ n ˆ ˆ k ln k k 1 2 N (28) Dn = βI, Tn = βB, τ n = β B , = βB. N If {bk} = are the eigenvalues of B, then for the matrix If the Lyapunov exponents are ordered, the correspond- k 1 Bˆ the corresponding eigenvalues are given by {λ˜ }N ing spectrum is denoted by k k=1. ε In the non-locally coupled map lattice given by bˆ = ( − ε) + b ,(k = , ,...N), k 1 κ(γ) k 1 2 (35) Eq. (18), a standard calculation shows that the elements of the jacobian matrix are given by such that the eigenvalues of ˆ are = βbˆ .Dueto k k the periodic boundary conditions (16), the matrix B is T (ik) = ( − ε) f x(i) δ n 1 n ik circulant and its diagonal elements are identically zero ε (k) since such terms were ruled out when computing the + exp (−γ rik) f x (1 − δik), (29) κ(γ) n coupling part of the jacobian matrix. A standard matrix where the primes denote differentiation of the map calculation using Fourier transform yields function with respect to its argument. Let us define a N −γ 2πkm symmetric matrix B with elements b = 2 e m cos ,(k = 1, 2,...N) k N m=1 −γ rik Bik = e (1 − δik), (30) (36) 123 1594 R. L. Viana et al.

Fig. 2 Lyapunov spectrum 0.6 of a one-dimensional lattice 0.5 of N = 101 Bernoulli maps γ=0.2 with β = 1.9 and chemical γ=0.4 coupling with a γ = 1.0 0.4 γ=0.6 and various values of the γ=0.8 coupling constant; b 0 γ=1.0 ε = 0.5 and various values 0.2 of the coupling range for illustrative purposes 0 k ε=0.1

λ -0.5 ε=0.2 ε=0.3 ε=0.4 -0.2 ε=0.5 ε=0.6 -1 -0.4

-0.6 -1.5 (a) (b) 1 101 1 101 k k

in terms of them the Lyapunov exponents of the coupled whereas if k = N, the exponent is simply λN = ln β. Bernoulli map lattice read These results are in accordance with previous ﬁndings [27]. ε The γ →∞case brings about the locally coupled λ = β + ( − ε) + b . k ln ln 1 κ(γ) k (37) lattice, for which the spectrum (37) yields π Let us consider the two limiting cases of the above 2 k λk = ln β + ln 1 − 2ε sin . (41) expression. For γ = 0, global type of coupling, the N spectrum reads which also has been already found by Kaneko [28]. Now let us consider the intermediate case of γ = 1.0 N = 2ε 2πkm and a chain of N 101 coupled Bernoulli maps with λk = ln β + ln (1 − ε) + cos . β = . N − 1 N 1 9. In Fig. 2a, we plot the (ordered) Lyapunov m=1 spectrum λk versus k for coupling range γ = 1.0 and (38) different values of the coupling constant ε. The curves were obtained from Eq. (37), whereas the indicated Using Lagrange’s identity for the sum of cosines there points refer to numerically obtained values of the Lya- follows that punov exponent for selected sites. In the latter case, we computed the Lyapunov spectrum directly from the ε sin(kπ) deﬁnition, computing the product of jacobian matri- λ = β + ( − ε) + − + . k ln ln 1 1 π ces and its eigenvalues, using balancing, reduction to N − 1 sin k N the Hessenberg form and QR-algorithm [29]. We com- (39) puted only a few sites so as not to eclipse the agreement between the analytical and numerical expressions. If k = N, we have the same exponents, a (N − 1)-fold In Fig. 2a, it is clearly seen that the maximum Lya- symmetry: punov exponent is ln β = 0.6418, corresponding to the uncoupled maps (for this reason it is independent of ε N ). The spectrum of Lyapunov exponents is monotonic λk = ln β + ln 1 − ε , (40) N − 1 and its minimum decreases as ε increases. For ε 0.3 123 Lyapunov spectrum of chaotic maps with a long-range coupling 1595

Fig. 3 a Smallest Lyapunov exponent and b Density of KS-entropy as a function of the coupling strength and range, for a lattice of N = 501 chaotic Bernoulli maps with β = 1.9

(below the blue line in the figure), all the exponents are Eq. (40)], we deduce that all exponents will be positive positive, whereas part of them become negative as ε if increases further. The Lyapunov spectrum for fixed ε = 0.5 and vary- N − 1 1 N − 1 1 ε ≤ 1 − ,ε≥ 1 + , ing γ is shown in Fig. 2b. It is interesting that, as γ N β N β decreases, approaching the global coupling limit, the (43) spectrum becomes almost degenerate for around half of the sites, mirroring the full degeneracy observed when a condition that, for the same parameters used in Fig 3, γ = 0. namely β = 1.9 and N = 501, amounts to ε ≤ εc = The relative contribution of the positive Lyapunov 0.4727 and ε ≥ 1.5229 (the latter case is not depicted exponents to the entire spectrum can be quantified by in Fig. 3). Moreover, for the intervals (43), it turns out the density of Kolmogorov-Sinai (KS) entropy. For sys- that the density of KS-entropy is tems having a Sinai-Ruelle-Bowen (SRB) measure, the latter is given by [30,31] N − 1 N h = ln β + ln 1 − ε . (44) N N − 1 λ > j 0 = λ = 1 λ . h j ,λ > j (42) If ε is small enough, an approximate result is obtained j j 0 N j=1 as a power series ! " We used the analytical expression (37) in order to select 1 h ≈ ln β 1 − ε − ε2 + o(ε3) . (45) the positive ones and apply Eq. (42) to obtain the small- 2(N − 1) est Lyapunov exponent and the KS-entropy density shown in Fig. 3a, b, respectively, as a function of cou- By way of contrast, in the local case (γ large), pling strength and range. In this Figure, we consider the spectrum (41) indicates that, for small coupling N = 501 for illustrative purposes; however, in Sect. 6, strength, all exponents are positive, such that the we will show results with varying lattice size. entropy is as large as it is in the global case, a fact evi- In the global case (γ = 0), we have a clear-cut tran- dent in Fig. 3. The (non-ordered) Lyapunov spectrum sition to chaos, as the value of h increases suddenly for small values of ε have typically positive values, from zero to positive values, as ε decreases past a crit- starting from k = 1 and N, for which the exponents ical value εc ≈ 0.5. In fact, thanks to the quasi-total are ln β, and with the lowest value at k = N/2. The degeneracy of the Lyapunov spectrum for this case [see spectrum has a symmetry with respect to this value. 123 1596 R. L. Viana et al.

The value of λN/2 is positive provided either of the tive Lyapunov exponents, the difference being the nor- inequalities below are fulﬁlled malization used in each case. In particular, for weakly coupled maps, the system attractor has practically the 1 1 ε< 1 − = 0.23684, same dimension as the phase space itself (N) due to the 2 β many positive exponents. 1 1 ε> 1 + = 0.76315. (46) As the coupling strength grows, however, a number 2 β of these exponents turns negative producing synchro- As ε grows further, this exponent becomes negative, as nization and, as we will see in the next Section, a drastic well as a number of other ones with different values reduction in the attractor dimensionality. In the extreme of β. Nevertheless, even if ε if large enough, a certain case in which all maps are synchronized the dimension number of exponent remain positive. reduces to the unity (which is the dimension of the Another quantity of interest which we can compute uncoupled systems). from the Lyapunov spectrum is the Lyapunov dimension D of the attractor for the coupled map lattice in its N-dimensional phase space. Let p be the largest inte- # 5 Lyapunov spectrum of the completely ger for which p λ ≥ 0. Then D is deﬁned by one j=1 j synchronized state of the following relations [32]

⎧ More than 25years ago, it was discovered that two or ⎪0 if there is no such p more chaotic systems, despite having the characteris- ⎨ # 1 p tic sensitive dependence to initial conditions, can syn- D = p + = λ j if p < N . ⎪ |λp+1| j 1 ⎩ chronize their chaotic evolutions [33,34]. Since then N if p = N this subject has developed into a subfield of nonlin- (47) ear dynamics with various applications in physical and biological systems [35]. It has been conjectured that the Lyapunov dimension We define a completely synchronized state of the provides an estimate for the information dimension of coupled map lattice in the following way the attractor, which is in general less than its capacity ( ) ( ) ( ) ∗ dimension. x 1 = x 2 = ...= x N = x , (48) Figure 4 shows the values of D as a function of the n n n n coupling parameters. A cursory look at both Figs. 3 and 4 shows a resemblance between them, which comes for all times. If we substitute this expression into (18), from the similar definitions in the case of many posi- we find an identity, showing that a completely synchronized state is a valid solution as long as the maps be identical. Geometrically, Eq. (48) defines a one-dimensional 600 synchronization manifold S in the N-dimensional phase space of the coupled lattice. The remaining

400 (N −1) directions are transversal to S. We shall denote the Lyapunov exponents of the completely synchro- D N nized state as {λ˜ ∗} . From the deﬁnition (48), we can 200 k i=1 derive the following quantities for the completely syn-

1 chronized state of one-dimensional maps 0 0 0.8 0.2 0.4 0.6 ∗ ∗ 0.4 = ( ) , = ( ) ˆ , 0.6 Dn f xn I Tn f xn B (49) ε 0.8 0.2 γ ⎡ ⎤ 1 0 1/n n$−1 n$−1 ∗ ˆ ˆ ⎣ ∗ ⎦ ˆ τ n = f (x )B, = lim f (x ) B. Fig. 4 Lyapunov dimension as a function of the coupling j n→∞ j strength and range, for a lattice of N = 501 chaotic Bernoulli j=0 j=0 maps with β = 1.9 (50) 123 Lyapunov spectrum of chaotic maps with a long-range coupling 1597

As in the previous case, the eigenvalues of the matrix which is the logistic map for a = 4. The initial condi- ˆ (i) = , ,... B are given by (35), in such a way that the eigenvalues tions x0 , i 1 2 N are randomly chosen in the of the matrix ˆ read interval [0, 1). Due to its topological conjugacy to the ⎛ ⎞ baker transformation, in this case the invariant density 1/n n$−1 is known to be [25] ⎝ ∗ ⎠ ˆ k = lim f (x ) bk, (51) 1 n→∞ j ρ(x) = √ , (56) j=0 π x(1 − x) and, taking the natural logarithm, we obtain the Lya- such that its Lyapunov exponent is, according (53), punov exponents corresponding to the completely syn- 1 | ( − x∗)| λ = 1 ln√4 1 2 ∗ = . chronized state U ∗ ∗ dx ln 2 (57) π 0 x (1 − x ) n−1 ∗ 1 ∗ ε λ = lim ln | f (x )|+ln (1 − ε) + b , k n→∞ n j κ(γ) k j=0 6 Transversal stability of the completely (52) synchronized state where b are given from (36). k The completely synchronized state can only be The uncoupled maps are supposed to be strongly observed in numerical computation if the synchroniza- chaotic, and the ergodic property is shown to hold [25]. tion manifold S is stable with respect to all the transver- However, the ergodicity of a coupled chaotic map lat- sal directions. The dynamics on S is the same as the tice, although not rigorously proved, can be assumed to uncoupled maps. If the latter are chaotic, the maximal hold for weak enough coupling. As a matter of fact, the exponent is positive (λ˜ ∗ > 0, for λ˜ ∗ stands for the existence of a SRB measure has been proved for a few 1 k ordered spectrum). The remaining (N − 1) exponents coupled map lattices only [36]. Assuming ergodicity, are related to transverse directions with respect to S. we can replace the time average in (54)byanaverage Hence, the completely synchronized state is trans- over the attractor of uncoupled maps: λ˜ ∗ ≤ S versely stable if 2 0, i.e., the manifold loses n−1 1 ∗ transversal stability when the largest transversal expo- lim ln | f (x )| n→∞ n j nent is zero. Some authors call this transition a blowout j=0 λ˜ ∗ > bifurcation [37]. If 2 0, the synchronized state b is transversely unstable and, in practice, cannot be = ∗ρ( ∗) | ( ∗)|=λ , dx x ln f x j U (53) a achieved for typical initial conditions, since any initial condition off-S will generate a trajectory which does where λU > 0 is the Lyapunov exponent of an uncou- ∗ not converge to the synchronized state. pled map, and ρ(x ) is the invariant density of iterations Note that in the Lyapunov spectrum (54) λ˜ ∗ corre- on the interval [a, b] for the chaotic map f . Substituting 2 sponds to k = 1(ork = N − 1 due to the two-fold (53) and (36)into(54) there results that degeneracy) and k = N (or k = N + 1forthesame reason). Hence, ε λ∗ = λ + ( − ε) + 2 k U ln 1 κ(γ) ∗ ε λ = λU + ln (1 − ε) + b1,N , (58) 2 κ(γ) N π −γ m 2 km e cos ,(k = 1, 2,...N). N m=1 where

(54) N −γ 2πm b = 2 e m cos , (59) 1 N In the following, we shall consider the Ulam map, m=1 ∈[ , ) deﬁned in the interval x 0 1 N π( − ) −γ m N 1 m bN = 2 e cos . (60) = N xn+1 = f (xn) = 4xn(1 − xn), (55) m 1 123 1598 R. L. Viana et al.

Fig. 5 Critical lines for (a) (b) completely synchronization 1.6 1.6 of chaotic maps in the parameter plane of coupling strength ε versus coupling range γ , for different lattice sizes. The curves were obtained using the analytical result given by 1.2 1.2 Eq. (64), whereas the symbols stand for numerical

results using the order ε parameter computed from Eq. (70). The region of transversely stable states 0.8 N=11 0.8 N=1001 shrinks with increasing γ N=101 N=5001 N=501 N=10001

0.4 0.4 0 0.05 0.1 0.15 0.2 0 0.002 0.004 0.006 0.008 0.01 γ γ

Using (58), the condition for transversal stability of γ for a coupled Ulam map lattice with various values of the completely synchronized state is that the coupling the lattice size N, using the analytical formulae above. γ ε ε constant lies in the interval (for given values of and The region in-between the curves for c and c corre- N): sponds to a transversely stable synchronized state. We observe a strong dependence on the lattice size ε ≤ ε ≤ ε , ε c c (61) N:ForlargeN, the upper value of c is almost constant at ∼ 1.5, whereas the lower εc increases monotonically where with γ until it crosses the upper curve. For γ higher than this crossing value, there is no region for transversely −1 −λ b1 stable synchronized state, i.e., it ceases to be observed ε = 1 − e U 1 − , (62) c κ(γ) numerically. Practically, we thus say that the lattice −1 cannot be synchronized at all. This crossing value is −λ b ε = + e U − N . c 1 1 κ(γ) (63) observed to decrease with increasing N; hence, it is worthwhile to consider the thermodynamical limit of →∞ Let us now consider the Ulam map. In this case, the the lattice (N ). critical values of the coupling constant given by (62) This analysis is easier to be performed in the globally (γ = ) κ = − and (63) are, respectively coupled case 0 ,forwhich N 1 and

b1 =−1, (66) 1 3 ε = ,ε= , π(N−1) c c (64) sin 2 2 =− + 2 , bN 1 π( − ) (67) sin N 1 where 2N

such that (64) and (65) yield = − b1 , = − bN . 1 κ(γ) 1 κ(γ) (65) N − 1 εc = , (68) In Fig. 5a, b we plot the critical curves for the transi- 2N tion to a completely synchronized state in the parameter 3(N − 1) ε = . (69) plane of coupling strength ε versus the coupling range c 2N 123 Lyapunov spectrum of chaotic maps with a long-range coupling 1599

1 In the limit, N →∞we have that εc = 1/2 and ε = / c 3 2, in accordance with Eq. (62). 0.8 In order to check these analytical results, we resort power to a numerical diagnostic of phase synchronization, exponential which is the complex order parameter introduced by 0.6 ε=0.7

Kuramoto [11–13] R 0.4 N ϕ( ) 1 θ ( ) z(t) = R(t)ei t = ei j t . (70) N 0.2 j=1

ϕ ∈[, π) 0 The quantities R and 0 2 are, respectively, 0 0.2 0.4 0.6 0.8 1 the amplitude and angle of a gyrating vector which is α , γ (black circles) (red squares) equal to the vector sum of phasors for each oscillator in a one-dimensional lattice with periodic bound- Fig. 6 Average order parameter for the power-law (black circles) ary conditions. In order to see the meaning of z(t),let and exponential (red squares) coupling models, for ε = 0.7. us analyze two limits cases. Firstly, for a completely (Color figure online) phase-synchronized state, the order parameter magnitude is R(t) = 1 for all times. In the second place, let where the normalization factor is now us consider a completely non-synchronized pattern, for which the phases are so spatially uncorrelated that they N −α can be considered as randomly distributed over [0, 2π). η(α) = 2 s , (72) The order parameter in this case has zero magnitude, s=1 since it is a space-average over randomly distributed = ( − )/ variables. with N N 1 2, as before. In general, the order parameter magnitude fluctu- In order to compare the power-law coupling pre- ates in time, so we usually compute its time average, scription with the exponential model here considered, after transients have died out. In our simulations, we in Fig. 6, we show the variation of the time-averaged have discarded 5000 iterations to eliminate transient order parameter for the power-law (black circles) and behavior. The symbols in Fig. 5 stand for numerically exponential (red squares) coupling models as a func- α γ ε = . obtained values of the critical values of ε and γ for tion of and , respectively, by keeping 0 7. For α = γ = which there is a transition to a completely synchro- 0, both prescriptions stand for a global (all- nized state, i.e., the symbols mark the parameter values to-all) coupling, and thus we expect that the lattice is ε for which the (time-averaged) order parameter magni- completely synchronized for large enough. However, α κ tude R either increases from zero or decreases from the as both and decrease, the decay of R is faster for unity. The numerical results using the order parameter the exponential coupling, with respect to the power-law κ = . agree with the analytical predictions based on the Lya- coupling. For example, if 0 4, the power-law cou- punov spectrum for all parameter values considered. pling leads to an almost synchronized lattice, whereas α = . A similar coupling prescription uses a power-law 0 4 already results in a non-synchronized state. spatial dependence ∼ r −α, instead of an exponential As both parameters increase further, both prescriptions decay as here, and it was used to discuss many dynam- tend to exhibit similar behavior. ical aspects of non-local coupled maps, like shadowing breakdown [38], spatial correlations [39], and unsta- 7 Conclusions ble dimension variability [40]. The coupled map lattice with that prescription would read A non-local type of coupling between dynamical sys- (i) = ( − ε) (i) xn+1 1 f xn tems in a spatial lattice can be obtained by considering the interaction of the system units being mediated by ε N + s−α f x(i−s) + f x(i+s) , a chemical which is both secreted and absorbed by the η(α) n n (71) s=1 systems, the corresponding rates being affected by the 123 1600 R. L. Viana et al. system local dynamics. The mediating chemical dif- form the analytical computations in an exact form. In fuses along the space in which the systems are embed- this way, we believe that this work can be useful for ded. If the diffusion timescale is fast compared with the other maps belonging to the same class. Considering typical period of the local dynamics, we can neglect coupled Ulam maps with a long-range coupling, it is the time dependence and consider the stationary limit possible to find analytically the Lyapunov spectrum. of the chemical concentration. In one spatial dimen- We used coupled Ulam maps to illustrate the synchro- sion, this leads to a coupling between systems whose nization of chaos because they present strong chaos. intensity decreases exponentially with the lattice dis- As a future work, this method can be applied to other tance. 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