Showing papers on "Coupled map lattice published in 1991"

Spatiotemporal analysis of complex signals: Theory and applications

Abstract: We present a space-time description of regular and complex phenomena which consists of a decomposition of a spatiotemporal signal into orthogonal temporal modes that we call chronos and orthogonal spatial modes that we call topos. This permits the introduction of several characteristics of the signal, three characteristic energies and entropies (one temporal, one spatial, and one global), and a characteristic dimension. Although the technique is general, we concentrate on its applications to hydrodynamic problems, specifically the transition to turbulence. We consider two cases of application: a coupled map lattice as a dynamical system model for spatiotemporal complexity and the open flow instability on a rotating disk. In the latter, we show a direct relation between the global entropy and the different instabilities that the flow undergoes as Reynolds number increases.

Weak chaos and quasi-regular patterns

Abstract: Part I. General Concepts: Hamiltonian dynamics Stability and chaos Part II. Dymanic Order and Choas: The stochastic layer Stochastic layer - stochastic sea transition The stochastic web Uniform web Part III. Spatial Patterns: Two-dimensional patterns with quasi-symmetry Two-dimensional hydrodynamic patterns with symmetry and quasi-symmetry Chaos and streamlines Part IV. Miscellanea: Patterns in art and nature References Index.

Coupled Map Lattice

Abstract: In the first two lectures of the summer school, I have reported the studies of spatiotemporal chaos with the use of coupled map lattices (CML). On the first lecture, I have shown some qualitative results of CMLs. Since some of the inovating features in CML modelling are not recognized, some questions and answers are listed up here, starting from the most trivial one.

Associative dynamics in chaotic neural networks

Abstract: The authors apply a model of a chaotic neural network composed of neuron models with chaotic dynamics to associative memory, the stored patterns of which are mutually orthogonal, and analyze its dynamical behavior quantitatively. In order to clarify the chaotic dynamics, they calculate the Lyapunov spectrum, the temporal changes of the distance between the output pattern of the chaotic neural network and the stored patterns, and the local divergence rate. It is demonstrated that dynamical associative memory can be realized with the chaotic neural network. >

Pattern dynamics of chaotic neural networks with nearest-neighbor couplings

Abstract: The authors introduce chaos into simple mathematical neuron models which are deterministic rather than probabilistic. The authors apply chaotic dynamics to artificial neural networks, using a chaotic neuron model based on electrophysiological experiments with squid giant axons and on numerical experiments with the Hodgkin-Huxley equations. First, the authors explain the chaotic neuron model and its dynamics. The authors also demonstrate spatio-temporal pattern dynamics of chaotic neural networks with nearest-neighbor couplings. It is shown that the chaotic neural networks with nearest-neighbor couplings have abundant spatio-temporal dynamics with a possible applicability to dynamical spatio-temporal memory. >

Symbol dynamic maps of spatial‐temporal chaotic vibrations in a string of impact oscillators

Abstract: Spatially complex, temporally chaotic dynamics of N‐coupled impact oscillators connected by a string are studied experimentally using a discrete measure of the motion for each of the masses. For N=8, a binary assignment of symbols, corresponding to whether or not the masses impact an amplitude constraint, is used to code the spatial pattern as a binary number and to store its change in time in a computer. A spatial pattern return map is then used to observe the change in spatial patterns with time. Bifurcations in spatial impact patterns are observed in this experiment. An entropy measure is also used to characterize the dynamics. Numerical simulation shows behavior similar to the experimental system.

Chemical turbulence and phase resetting dynamics

Abstract: The onset of chemical turbulence in the 2‐D complex Ginzburg–Landau equation is studied for a class of finite‐amplitude, inhomogeneous, random initial conditions. Numerical simulations on a coupled map lattice corresponding to this reaction–diffusion equation are carried out in order to characterize the nature of the turbulent state. The phase diagram giving the zone of chemical turbulence in the parameter space that specifies the initial condition is determined numerically and compared with an analytical estimate.

Excitable Media as Computational Systems

Abstract: A number of mathematical approaches may be used to model a given excitable system. For an excitable system that is not spatially extensive a map, or a system of nonlinear ordinary differential equations may be appropriate. For a spatially extensive excitable medium a system of partial differential equations, or a coupled map lattice, or a cellular automaton, might be an appropriate model. These different types of model are all nonlinear, and are often intractable, and so their behaviour is usually investigated by numerical methods on a digital computer, using some appropriate algorithms.

Coupled Map Lattices: Abstract Dynamics and Models for Physical Systems

Abstract: Complicated spatio-temporal structures can arise when large numbers of simple dynamical elements are coupled. There are many physically interesting systems that fall into this category. Numerous examples can be found in biology where self-organization occurs at the cellular level, the brain where interactions among neurons are responsible for its activity, the heart where patterned excitation leads to normal rhythms and the converse to fibrillation. One can include the equations of continuum hydrodynamics and reaction-diffusion equations in this category since they can be considered to arise from a coupling among local fluid elements. A range of descriptions has been used to study the dynamical behavior of such systems.

Ordering Process in the Diffusively Coupled Logistic Lattice

Abstract: Coupled map lattices have been studied in the last years as an attempt to bridge the gap between low-dimensional chaotic systems and spatially extended systems showing turbulent behaviour [1, 2, 3, 4]. For recent reviews see [5, 6]. In a chaotic coupled map lattice a large number of maps, each of which can generate chaotic behaviour, are coupled together as a coarse model of turbulence. From this approach we can hope to learn something about the characteristic length and time scales in ”spatio-temporally chaotic” systems and about the emergence of ordered states in strongly fluctuating systems.