Global synchronisation in translation invariant coupled map lattices

A sufficient condition for global synchronisation in coupled map lattices (CML) with translation invariant coupling and arbitrary individual map is proved. As in [Jost & Joy, 2001] where CML with reflection invariant couplings are considered, the condition only involves the linearised dynamics in the diagonal, namely for all points in the diagonal, the derivative must be contractive in all transverse directions. In addition to this result, a (weaker) condition that ensures the CML attractor to be composed of either 2-periodic or constant configurations, is also obtained. Synchronisation is probably the most commonly observed dynamical phenomenon in interacting (coupled) nonlinear systems. Generally speaking, it is said to occur in a multidimensional dynamical system when the attractor lies inside a one-dimensional subset of the phase space so that the knowledge of the trajectory of a single coordinate allows to determine all coordinate trajectories [Boccaletti et al., 2006; Pikovsky et al., 2001]. In other words, all components asymptotically evolve ”in phase”, possibly in a chaotic motion [Pecora & Caroll, 1990]. In practice, the phenomenon takes various forms upon the system under consideration. For instance, phase synchronisation takes place in systems of coupled oscillators [Fujisaka & Yamada, 1983] and master-slave, or generalized synchronisation are possible phenomena in unidirectionally coupled systems [Hunt et al., 1997; Rulkov et al., 2001; Tresser et al., 1995]. Another specific form of synchronisation happens in dynamical systems with symmetry where it consists in the convergence to the symmetry fixed point set (which is invariant under dynamics, see e.g. [Ashwin et al., 1996]). Challenging problems in this framework then concern conditions on parameters (or on components) for synchronisation. These conditions often involve the notion of transverse Lyapunov exponent in order to specify the basin of attraction of the symmetry fixed point set. In particular, a seminal result [Alexander et al., 1992] states that this basin has positive Lebesgue measure (in phase space) when all transverse Lyapunov exponents are negative, for Lebesgue almost every point in the invariant set. Moreover, it is likely to have a fractal ”riddled structure” (see the chapter by P. Ashwin in [Chazottes & Fernandez, 2005] for a mathematical definition) and has actually such a structure in various examples. However, [Ashwin et al., 1996] showed that this basin is indeed a neighbourhood of the invariant set when the supremum of all transverse Lyapunov exponents with respect to all ergodic measures supported in the invariant set, is negative. Naturally, and as observed in [Ashwin et al., 1996], without any further specification of the dynamics, these results are optimal and one cannot expect to control the global behaviour. In some cases however, it is possible to determine the fate of every orbit in phase space from properties of the linearised dynamics in the invariant set. CML have been introduced at the beginning of the eighties as simple discrete-time models of reactiondiffusion systems [Chazottes & Fernandez, 2005; Kaneko, 1993]. Their specificity resides in the definition of their mapping which is the composition of an individual map and of a linear coupling. This presents the 1 ha l-0 02 61 30 6, v er si on 1 6 M ar 2 00 8 double advantage of being well-adapted to numerical simulations and to mathematical analysis. Recently, [Lu & Chen, 2004] has established and analyzed synchronisation conditions in CML with arbitrary linear coupling operators, mostly in absence of symmetry but yet with invariant diagonal. For reflection invariant coupled map lattices (CML), [Jost & Joy, 2001] proved that if all transverse eigenvalues of the jacobian matrix are contractive for all points in the diagonal (the invariant set in this case), then all points in phase space asymptotically approach the diagonal, a property called global synchronisation. Although slightly stronger than the previous one, this condition is much simpler to check in practice. In this Letter, we focus on translation invariant (but not necessarily reflection invariant) CML on periodic lattices. Translation invariance is usually assumed in most studies [Chazottes & Fernandez, 2005; Kaneko, 1993] as it reflects the simplifying assumptions that the individual systems are all identical and that the coupling is of diffusive type. As in [Jost & Joy, 2001], we show that global synchronisation holds provided that all transverse directions are contractive for all points in the diagonal. Our condition is actually a bit more general than the one in [Jost & Joy, 2001] and the technique is different. The results in particular complete a previous result in [Lin et al., 1999] on global synchronisation for lattices of 2,3 and 4 coupled logistic maps. A translation invariant CML on the periodic lattice ZL := {s ∈ Z modL} (L > 1) is the dynamical system generated by the following induction relation in R x s = ∑ n∈ZL cnf(xs−n), s ∈ ZL. (1) Here the coefficients cn are non-negative (cn > 0) and normalized ( ∑ n∈ZL cn = 1). The most frequent examples are respectively, the asymmetric nearest neighbour coupling, the symmetric one (L > 2) and the global coupling for which the coefficients are respectively given by ( ∈ [0, 1]) cn =  1− if n = 0 if n = 1 0 otherwise , cn =  1− if n = 0 2 if n = ±1 0 otherwise and cn = { 1− if n = 0 L otherwise More generally, a coupling on the periodic lattice with L sites can be defined from any normalised sequence {γn}n∈Z of non-negative coefficients by summing over the periods, namely cn = ∑