ELSEVIER Physica D 78 (1994) 141-154 PHY$1GA Long-range order with local chaos in lattices of diffusively coupled ODEs Leonardo Brunnet a,t,, Hugues Chat6 a,e, Paul Manneville a,¢ a Service de Physique de l'Etat Condense, CEA - Centre d'Etudes de Saclay, 91191 Gif-sur-Yvette, France b lnstituto de Ffsica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 91501-970, Porto Alegre, Brazil e LadHyX- Laboratoire d'Hydrodynamique, Ecole Polytechnique, 91128 Polaiseau, France Received 8 April 1994; accepted 6 May 1994 Communicatedby Y. Kuramoto Abstract The existence of non-trivial collective behavior in lattices of diffusively coupled differential equations is investigated. For a two-dimensional square lattice of Rrssler systems, a rotating long-range order is observed. This case is best described in terms of a complex Ginzburg-Landau (CGL) equation submitted to the local noise produced by the chaotic Rrssler units. The parameters of this CGL equation are estimated to be in the so-called "Benjamin-Feir stable" region. The collective oscillation regime thus corresponds to the linearly-stable, spatially-homogeneous solution of the equivalent CGL equation. The possibility of more complex collective behavior in similar systems is discussed. 1. Motivation Recently, evidence was given that spatially ex- tended dynamical systems may exhibit "non-trivial collective behavior" characterized by the non-steady, usually regular, evolution of space-averaged quanti- fies emerging out of homogeneously distributed local chaos [ 1,2]. For example, it was shown that a five- dimensional hypercubic lattice of logistic maps cou- pled diffusively to their nearest neighbors can show a quasiperiodic evolution of the spatial average of the site values though the sites are not synchronized and evolve in a chaotic manner. Extensive numerical investigations [ 1] revealed that these types of behavior occur more frequently in high-dimensional systems and that they are neither transient nor due to finite-size effects. Synchronous updating (implicitly considered here, since we deal with dynamical systems), a high connectivity (prefer- ably due to a large space dimension), and a local dynamics allowing the possibility of local chaos were shown to be essential ingredients for observing non-trivial collective behavior. In spite of a number of attempts [ 3 ], it is our belief that no convincing explanation of even a single case has been given until now. Consequently, these types of behavior remain mysterious and somewhat con- troversial [4]; in particular, because they go against any intuition based on equilibrium statistical mechan- ics - often invoked in this matter even though the systems considered are essentially out-of-equilibrium ones. This unsatisfactory situation has led us to push further our numerical exploration. So far, non-trivial collective behavior has been reported for discrete-time lattice systems (coupled map lattices and cellular au- tomata), so that the question of the role played by the discreteness of time (and space) arises. As a first step, we report here on our investigations of discrete-space, continuous-time systems, i.e. lattices of coupled ordi- nary differential equations (ODEs). An interesting collective behavior was recently found in collections of globally coupled oscilla- 0167-2789/94/$07.00 (~) 1994 Elsevier Science B.V. All fights reserved SSD10167-2789(94)00131-9