Mean-field theory and critical behavior of coupled map lattices

Turbulence in globally coupled maps

“…The fine structure of this critical boundary is actually more complex; there are several narrow windows of turbulence and laminar gaps at smaller scales. Some of the complexity of the critical boundary between turbulent and laminar regimes has already been suggested by less detailed calculations on diffusively coupled onedimensional lattices [5,6]. For parameter values inside the critical boundary, the globally coupled map system exhibits sustained turbulence (0 < F ≤ 1).…”

“…In general, for CMLs with local interactions, the transition from a laminar regime to spatiotemporal intermittency (and viceversa, in the case of windows of turbulence) behaves in many aspects as a second order phase transition, characterized by the scaling relation F ∼ (ǫ − ǫ c ) β , where β is a critical exponent [4,5,8]. In contrast, the transition between laminar states and turbulence in globally coupled maps appears as a discontinuous jump in the quantity F , a feature associated to first order phase transitions.…”

“…In this respect, a CML can be viewed as a simplifying replacement for partial differential equations of hydrodynamics [9]. The onset of turbulence via spatiotemporal intermittency has mainly been investigated on CMLs with local, diffusive-type interactions on both low-dimensional Euclidean arrays [4,5,6,7] and on fractal lattices [8]. These studies have permitted the characterization of the transition to turbulence as a critical phenomenon in one and two dimensions and on several fractal dimensions with diverse local connectivities.…”

“…A very general scenario for the occurrence of turbulence is spatiotemporal intermittency, i.e., a sustained regime characterized by the coexistence of coherent-laminar and disordered-chaotic domains in space and time [2,3]. There are numerous studies of this phenomenon, some of the most extensive of which have been on model dynamical systems such as coupled map lattices (CML) [2,4,5,6,7,8]. The idea is that the ingredients of a CML -a discrete space, discrete time system of interacting elements whose states vary continuously according to specific functions -are sufficient to capture much of the phenomenology observed in complex spatiotemporal processes, in particular some relevant features of spatiotemporal intermittency and turbulence.…”

A method to discern complexity in two-dimensional patterns generated by coupled map lattices

“…Ergo the phenomenon called spatial-temporal intermittency can emerge [18,19,20,21]. This dynamical regime corresponds with a situation where each unit is weakly oscillating around a laminar state that is aperiodically and strongly perturbed for a traveling burst.…”