Conversion of local transient chaos into global laminar states in coupled map lattices with long-range interactions (original) (raw)

Collective behavior in coupled chaotic map lattices with random perturbations

Physica A: Statistical Mechanics and its Applications, 2008

Numerical simulations of coupled map lattices with non-local interactions (i.e., the coupling of a given map occurs with all lattice sites) often involve a large computer time if the lattice size is too large. In order to study dynamical effects which depend on the lattice size we considered the use of small truncated lattices with random inputs at their boundaries chosen from a uniform probability distribution. This emulates a "thermal bath", where deterministic degrees of freedom exhibiting chaotic behavior are replaced by random perturbations of finite amplitude. We demonstrate the usefulness of this idea to investigate the occurrence of completely synchronized chaotic states as the coupling parameters are varied. We considered one-dimensional lattices of chaotic logistic maps at outer crisis x → 4x(1 − x).

Spatiotemporal chaos in one- and two-dimensional coupled map lattices

Physica D: Nonlinear Phenomena, 1989

Coupled map lattices are investigated as a model for spatiotemporal chaos. Pattern dynamics in diffusivel~y coupled logistic lattice is briefly reviewed with the use of power spectra, domain distribution, a~d Lyapunov spectra. Mechanism of pattern selection with the suppression of chaos is discussed. Pattern dynamics on a 2-dimensional lattice is shown, in a weak coupling regime, a similarity with the one-dimensional case is found; frozen random pattern, pattern selection, Brownian motion of a chaotic string, and intermittent collapse of the pattern with selective flicker noise. In a strong coupling regime, frozen pattern is found to be unstable by the surface tension, which is in contrast with the one-dimensional case. Convective coupling model is introduced in connection with the fluid turbulence of Navier-Stokes type. Soliton turbulence and vortex turbulence in the model are reported. Physical implications of coupled map lattices are discussed.

Stochastic analog to phase transitions in chaotic coupled map lattices

Physical Review E, 2001

Stochastic dynamical systems are shown to exhibit the same order-disorder phase transitions that have been found in chaotic map lattices. Phase diagrams are obtained for diffusively coupled two-dimensional ͑2D͒ lattices, using two stochastic maps and a chaotic one, for both square and triangular geometries, with simultaneous updating. We show how the use of triangular geometry reduces ͑or even eliminates͒ the reentrant behavior found in the phase diagrams for the square geometry. This is attributed to the elimination ͑via frustration͒ of the antiferromagnetic clusters common to simultaneous updating of square lattices. We also evaluate the critical exponents for the stochastic maps in the triangular lattices. The strong similarities in the phase diagrams and the consistency between the critical exponents of one stochastic map and the chaotic one, evaluated in an early work by Marcq et al. ͓Phys. Rev. Lett. 77, 4003 ͑1996͒; Phys. Rev. E 55, 2606 ͑1997͔͒ suggest that the ''sign-persistence,'' defined as the probability that the local map keeps the sign of the local variable in one iteration, plays a fundamental role in the presence of continuous phase transitions in coupled map lattices, and is a basic ingredient for models that belong to this weak Ising universality. However, the fact that the second stochastic map, which has an extremely simple local dynamics, seems to fall in the 2D Ising universality class, suggests that some minimal local complexity is also needed to generate the specific correlations that end up giving non-Ising critical behavior.

Synchronization and suppression of chaos in non-locally coupled map lattices

Pramana, 2009

We considered coupled map lattices with long-range interactions to study the spatiotemporal behaviour of spatially extended dynamical systems. Coupled map lattices have been intensively investigated as models to understand many spatiotemporal phenomena observed in extended system, and consequently spatiotemporal chaos. We used the complex order parameter to quantify chaos synchronization for a one-dimensional chain of coupled logistic maps with a coupling strength which varies with the lattice in a powerlaw fashion. Depending on the range of the interactions, complete chaos synchronization and chaos suppression may be attained. Furthermore, we also calculated the Lyapunov dimension and the transversal distance to the synchronization manifold.

Spatial correlations and synchronization in coupled map lattices with long-range interactions

Physica A: Statistical Mechanics and its Applications, 2004

We used numerical diagnostics to quantify spatial disorder, and its relation with temporal chaos, for a one-dimensional chain of coupled logistic maps with a coupling strength which varies with the lattice distance in a power-law fashion. The main tool is spatial return plots, whose properties are used to obtain information about the chaotic synchronized states of the system. A spatial correlation integral is introduced to characterize the clustering of points in the spatial return plots. r

Spatiotemporal structure of Lyapunov vectors in chaotic coupled-map lattices

Physical Review E, 2007

The spatiotemporal dynamics of Lyapunov vectors (LVs) in spatially extended chaotic systems is studied by means of coupled-map lattices. We determine intrinsic length scales and spatiotemporal correlations of LVs corresponding to the leading unstable directions by translating the problem to the language of scale-invariant growing surfaces. We find that the so-called characteristic LVs exhibit spatial localization, strong clustering around given spatiotemporal loci, and remarkable dynamic scaling properties of the corresponding surfaces. In contrast, the commonly used backward LVs (obtained through Gram-Schmidt orthogonalization) spread all over the system and do not exhibit dynamic scaling due to artifacts in the dynamical correlations by construction.

Synchronous chaos in coupled map lattices with small-world interactions

Physical Review E, 2000

In certain physical situations, extensive interactions arise naturally in systems. We consider one such situation, namely, small-world couplings. We show that, for a fixed fraction of nonlocal couplings, synchronous chaos is always a stable attractor in the thermodynamic limit. We point out that randomness helps synchronization. We also show that there is a size dependent bifurcation in the collective behavior in such systems.

Extinction and Chaotic Patterns in Map Lattices Under Hostile Conditions

Bulletin of Mathematical Biology, 2009

Population dynamics in spatially extended systems can be modeled by Coupled Map Lattices (CML). We employ such equations to study the behavior of populations confined to a finite patch surrounded by a completely hostile environment. By means of the Galerkin projection and the normal solution ansatz, we are able to find analytical expressions for the critical patch size and show the existence of chaotic patterns. The analytical solutions provided are shown to fit, under the appropriate approximations, the dynamics of a logistic map. This interesting result, together with our discussion, suggests the existence of a universal class of spatially extended systems directly linked to the wellknown characteristics of the logistic map.

Lyapunov spectrum of a lattice of chaotic systems with local and non-local couplings

Chaos Solitons & Fractals, 2007

We consider a one-dimensional chaotic piecewise linear map lattice with periodic boundary conditions and two types of interactions: (i) local couplings between nearest and next-to-the-nearest neighbors; and (ii) non-local couplings randomly chosen along the lattice according to a specified probability. The chaoticity of the lattice is described by means of its Lyapunov spectrum, which furnishes also information about the system global attractor in a high-dimensional phase space. We study in particular the dependence of this spectrum with the coupling parameters, as well as make comparisons with limiting cases, for which the Lyapunov spectrum is known.

Influence of solitons on the transition to spatiotemporal chaos in coupled map lattices

Physical review. E, Statistical, nonlinear, and soft matter physics, 2003

We study the transition from laminar to chaotic behavior in deterministic chaotic coupled map lattices and in an extension of the stochastic Domany-Kinzel cellular automaton [E. Domany and W. Kinzel, Phys. Rev. Lett. 53, 311 (1984)]. For the deterministic coupled map lattices, we find evidence that "solitons" can change the nature of the transition: for short soliton lifetimes it is of second order, while for longer but finite lifetimes, it is more reminiscent of a first-order transition. In the second-order regime, the deterministic model behaves like directed percolation with infinitely many absorbing states; we present evidence obtained from the study of bulk properties and the spreading of chaotic seeds in a laminar background. To study the influence of the solitons more specifically, we introduce a soliton including variant of the stochastic Domany-Kinzel cellular automaton. Similar to the deterministic model, we find a transition from second- to first-order behavior ...