Dynamics of inhomogeneous one--dimensional coupled map lattices (original) (raw)
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Spatiotemporal intermittency and scaling laws in inhomogeneous coupled map lattices
Physical Review E, 2002
We discuss the spatiotemporal intermittency (STI) seen in coupled map lattices (CML-s). We identify the types of intermittency seen in such systems in the context of several specific CML-s. The Chaté-Manneville CML is introduced and the ongoing debate on the connection of the spatiotemporal intermittency seen in this model with the problem of directed percolation is summarised. We also discuss the STI seen in the sine circle map model and its connection with the directed percolation problem, as well as the inhomogenous logistic map lattice which shows the novel phenomenon of spatial intermittency and other types of behaviour not seen in the other models. The connection of the bifurcation behaviour in this model with STI is touched upon. We conclude with a discussion of open problems.
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.
Periodic lattices of chaotic defects
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A novel type of self-organized lattice in which chaotic defects are arranged periodically is reported for a coupled map model of open ow. We nd that temporally chaotic defects are followed by spatial relaxation to an almost periodic state when suddenly a next defect appears. The distance between successive defects is found to be generally predetermined and diverging logarithmically when approaching a certain critical point. The phenomena are analyzed and shown to be explicable as the results of a boundary crisis for the spatially extended system.
Model of a spatially inhomogeneous one-dimensional active medium
Theoretical and Mathematical Physics, 2000
We investigate the dynamics of one-dimensional discrete models of a one-component active medium analyt-icalh~. The models represent spatially inhomogeneous diffusively concatenated systems of one-dimensional piecewise-continuous maps. The discontinuities (the defects) are interpreted as the differences in the parameters of the maps constituting the model. Two classes of defects are considered: spatially periodic defects and localized defects. The area of regular dynamics in the space of the parameters is estimated analytically. For the model with a periodic inhomogeneity, an exact analytic partition into domains with regular and with chaotic types of behavior is found. Numerical results are obtained for the model with a single defect. The possibility of the occunence of each behavior type for the svstem as a whole is investigated.
Dynamical behavior of the multiplicative diffusion coupled map lattices
Chaos: An Interdisciplinary Journal of Nonlinear Science, 1996
We report a dynamical study of multiplicative diffusion coupled map lattices with the coupling between the elements only through the bifurcation parameter of the mapping function. We discuss the diffusive process of the lattice from an initially random distribution state to a homogeneous one as well as the stable range of the diffusive homogeneous attractor. For various coupling strengths we find that there are several types of spatiotemporal structures. In addition, the evolution of the lattice into chaos is studied. A largest Lyapunov exponent and a spatial correlation function have been used to characterize the dynamical behavior.
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).
Phase transitions in 2D linearly stable coupled map lattices
Physica D: Nonlinear Phenomena, 1997
Interlace dynamics separating homogeneous phases is shown to be the main mechanism underlying irregular evolution in 2D linearly stable, coupled map lattices. In a fully deterministic model belonging to this class, we find evidence of at least two different regimes that we call weak and strong turbulence. The transition between the two regimes is carefully investigated revealing a direct connection with the destabilization of the interfaces separating homogeneous phases. The critical behaviour i~ analysed and compared with that of stochastic models like directed percolation.
Physica A: Statistical Mechanics and its Applications, 2006
Spatially extended dynamical systems may exhibit intermittent behavior in both spatial and temporal scales, characterized by repeated conversions from spatially localized transient chaos into global laminar patterns. A simple model, yet retaining some features of more complex systems, consists of a lattice of a class of tent maps with an escaping region. The coupling prescription we adopt in this work considers the interaction of a site with all its neighbors, the corresponding strength decaying with the lattice distance as a power-law. This makes possible to pass continuously from a local (nearest-neighbor) to a global kind of coupling. We investigate statistical properties of both the chaotic transient bursts and the periodic laminar states, with respect to the coupling parameters.
Dynamics of chaos-order interface in coupled map lattices
Physica D: Nonlinear Phenomena, 1997
We study a coupled map lattice model with two states: a simple fixed point and spatio-temporal chaos. Preparing properly initial conditions, we investigate the dynamics of the interface between order and chaos. In the one-dimensional lattice regimes of irregular and regular front propagation behavior are observed and analyzed by introducing a local front map and a front Lyapunov exponent. Corresponding to these different regimes of front propagation we can characterize different types of transitions from laminar state to chaos using comoving Lyapunov exponents. In the two-dimensional lattice these types of front motion are related to regimes of roughening and flattening of the interface.
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.