耦合映像格子中时空混沌的状态反馈控制

利用状态反馈的方法，实现了耦合映像格子中时空混沌的稳定控制.反馈方式可以是没有延迟的，也可以是有延迟的；控制方式可以是连续的，也可以是脉冲的.数值模拟结果证明了状态反馈方法的有效性. 关键词： 耦合映像格子 时空混沌 状态反馈控制     

Extremal coupled map lattices

Multifractal critical phenomena with infinite-temperature critical point and with complex coexistence of the infinite and finite temperature critical points are considered and it is shown that strange attractors generated by cascades of period-doubling bifurcations (Feigenbaum scenario) as well as fields of velocity differences in fluid turbulence belong to the former subclass of the multifractal critical phenomena, while the real traffic processes and real currency exchange processes belong to the last (complex) subclass of the multifractal critical phenomena. Data obtained by different authors are used for this purpose. Received 5 February 1999     

Overview of coupled map lattices

Studies in coupled map lattices are briefly surveyed in connection with the papers in the present focus issue.     

Entropy of coupled map lattices

The entropy of coupled map lattices with respect to the group of space-time translations is considered. We use the notion of generalized Lyapunov spectra to prove the analogue of the Ruelle inequality and the Pesin formula.     

Bifurcations in globally coupled map lattices

The dynamics of globally coupled map lattices can be described in terms of a nonlinear Frobenius-Perron equation in the limit of large system size. This approach allows for an analytical computation of stationary states and their stability. The bifurcation behavior of coupled tent maps near the chaotic band merging point is presented. Furthermore, the time-independent states of coupled logistic equations are analyzed. The bifurcation diagram of the uncoupled map carries over to the map lattice. The analytical results are supplemented with numerical simulations     

Self-organized memories in coupled map lattices

Spatially extended dynamical systems subjected to a periodic external input are able to store short-term memories as a result of self-organization. This phenomenon has been used to interpret sliding charge density wave experiments, and has been found in lattices of coupled maps and oscillators. In this paper, we aim to describe the influence of a stochastic perturbation in the self-organized memories in a lattice of coupled linear and weakly nonlinear maps. We consider the case in which the coupling between maps depends on the lattice distance as a power law.     

Ising-type transitions in coupled map lattices

We study, by means of computer simulations, some models of coupled map lattices (CML) with symmetry, subject to diffusive nearest neighbor coupling, with the purpose of providing, a better understanding of the occurrence of Isingtype transitions of the type found by Miller and Huse. We argue, on the basis of numerical evidence, that such transitions are connected to the appearance of a minimum in the Lyapunov dimension of the system as a function of the coupling parameter. Two-dimensional CMLs similar to the one in Miller and Huse, but with no minimum in the Lyapunov dimension plot, have no Ising transition. The condition seems to be necessary, though by no means sufficient. We also argue, relying on the analysis of Bunimovich and Sinai, that coupled map lattices should behave differently, with respect to dimension, than Ising models.Dedicated to Yakov Grigorievich Sinai on his 60th birthday.     

Spatio-temporal intermittency in coupled map lattices

The transition to turbulence in a one-dimensional array of maps coupled by diffusion is shown to display critical properties resembling those of directed percolation. The analogy is supported by the reconstruction of a probabilistic cellular automation with closely similar statistical properties. Numerical results suggest however that spatio-temporal intermittency does not belong to the same universality class as directed percolation.     

Experimental system of coupled map lattices

We design an optical feedback loop system consisting of a liquid-crystal spatial light modulator (SLM), a lens, polarizers, a CCD camera, and a computer. The system images every SLM pixel onto one camera pixel. The light intensity on the camera pixel shows a nonlinear relationship with the phase shift applied by the SLM. Every pixel behaves as a nonlinear map, and we can control the interaction of pixels. Therefore, this feedback loop system can be regarded as a spatially extended system. This experimental coupled map has variable dimensions, which can be up to 512 by 512. The system can be used to study high-dimensional problems that computer simulations cannot handle.     

Kink dynamics in one-dimensional coupled map lattices

We examine the problem of the dynamics of interfaces in a one-dimensional space-time discrete dynamical system. Two different regimes are studied: the non-propagating and the propagating one. In the first case, after proving the existence of such solutions, we show how they can be described using Taylor expansions. The second situation deals with the assumption of a travelling wave to follow the kink propagation. Then a comparison with the corresponding continuous model is proposed. We find that these methods are useful in simple dynamical situations but their application to complex dynamical behaviour is not yet understood. (c) 1995 American Institute of Physics.     

Chimera states in Gaussian coupled map lattices

We study chimera states in one-dimensional and two-dimensional Gaussian coupled map lattices through simulations and experiments. Similar to the case of global coupling oscillators, individual lattices can be regarded as being controlled by a common mean field. A space-dependent order parameter is derived from a self-consistency condition in order to represent the collective state.     

耦合映象格子系统的相互同步化

研究了两个耦合格子动力系统的非线性相互作用。当两个耦合格子系统一样时,则导致完全同步化。而当两个耦合格子系统的参数不一样或者这两个系统不相同时,则导致广义同步化。计算了Lyapunov指数谱。     

Error growth patterns in systems with spatial chaos: from coupled map lattices to global weather models

Error growth in spatiotemporal chaotic systems is investigated by analyzing the interplay between temporal and spatial dynamics. The spatial correlation and localization of relative fluctuations grow and decay indicating two different regimes, before and after saturation by nonlinear effects. This general behavior is shown to hold both in simple coupled map lattices and in global weather models. This explains the increasing or decreasing trends previously observed in the exponential growth rate of these spatiotemporal systems.     

Metastability and functional integration in anisotropically coupled map lattices

Metastability is a property of systems composed of many interacting parts wherein the parts exhibit simultaneously a tendency to function autonomously (local segregation) and a tendency to cooperate (global integration). We study anisotropically coupled map lattices and discover that for specific values of the coupling control parameters the entire system transits to a metastable regime. We show that this regime manifests a quasi-stable state in which the system can flexibly switch to another such state. We briefly discuss the relevance of our findings for information processing, functional integration, metastability in the brain, and phase transitions in complex systems.     

Noise induced resonance phenomena in coupled map lattices

We analyse different types of resonance phenomena that can occur in a coupled map lattice in the presence of noise with a subthreshold signal. The onsite dynamics considered here is different from previous such studies, namely, a bimodal cubic map capable of bistability in its dynamics. In addition to the resonance observed in the temporal iterates (the conventional stochastic resonance), we establish the possibility of resonance patterns in spatial sequences along the lattice, which we refer to as “Lattice Stochastic Resonance”. The characterising features of both are investigated in detail, under different types of signals with nearest neighbour coupling between lattice points. Possible practical applications are in signal detection, image processing and in communication networks.     

The emergence of coherent structures in coupled map lattices

We show how increasing spatial interaction leads to the merging of coherent structures from chaos in some systems of coupled map lattices. This phenomenon reflects the arising of new ground states in the corresponding model of statistical mechanics. If we further increase the coupling then, new ground states appear showing the coexistence of a large-scale coherent structure with a small-scale chaotic motion. This allows us to propose a generalization of the notion of spatial intermittency.