Probabilistic signatures of spatiotemporal intermittency in the coupled sine circle map lattice

The phase diagram of the coupled sine circle map system exhibits a variety of interesting phenomena including spreading regions with spatiotemporal intermittency, non-spreading regions with spatial intermittency, and coherent structures termed solitons. A spreading to non-spreading transition is seen in the system. A cellular automaton version of the coupled system maps the spreading to non-spreading transition to a transition from a probabilistic to a deterministic cellular automaton. The solitonic sector of the system shows spatiotemporal intermittency with soliton creation, propagation and absorption. A probabilistic cellular automaton mapping is set up for this sector which can identify each one of these phenomena.      

一个全局耦合不连续映像格子中的奇异态

文章研究了一类由既不可逆又不连续映像构成的全局耦合映像格子系统中的奇异态行为,计算了系统的同步序参量和空间振幅变化图.结果表明,在某些特定的参数区间内,耦合映像格子系统会出现奇异态或团簇态,并且敏感地依赖于耦合强度的选择.上述丰富的动力学现象是由于单映像中不连续、不可逆性以及空间耦合相互作用的结果.通过数值模拟找到了奇异态或团簇态出现的特定参数区域.     

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     

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.     

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.     

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.     

Analytical solutions of weakly coupled map lattices using recurrence relations

By using asymptotic methods recurrence relations are found that rule weakly CML evolution, with both global and diffusive coupling. The solutions obtained from these relations are very general because they do not hold restrictions about boundary conditions, initial conditions and number of oscilators in the CML. Furthermore, oscillators are ruled by an arbitraty C² function.     

Coupled map lattices as cellular automata

We approach the problem of the complex dynamics of coupled map lattices (CML) by proposing a reduction to deterministic cellular automata (CA) with more than two states per site. The reduction scheme replaces the local map by an approximation in terms of a step function based on a straightforward analysis of the local dynamics. The variation of the spatial coupling in the CML then translates itself as a path in the spaces of rules for the equivalent deterministic CA. The transition to turbulence via spatiotemporal intermittency in the CML is then interpreted as a transition in the space of rules. The observed nonuniversality of this transition can be traced back to the nature of the rules involved on both sides of the transition region and to the character of the escape process from the turbulent state, either strongly deterministic or quasiprobabilistic. The relation between CML, deterministic, and probabilistic CA and the possibility of a mean-field treatment of the dynamics of CML are discussed at a more formal level.     

Local dimension and finite time prediction in coupled map lattices

Forecasting, for obvious reasons, often become the most important goal to be achieved. For spatially extended systems (e.g. atmospheric system) where the local nonlinearities lead to the most unpredictable chaotic evolution, it is highly desirable to have a simple diagnostic tool to identify regions of predictable behaviour. In this paper, we discuss the use of the bred vector (BV) dimension, a recently introduced statistics, to identify the regimes where a finite time forecast is feasible. Using the tools from dynamical systems theory and Bayesian modelling, we show the finite time predictability in two-dimensional coupled map lattices in the regions of low BV dimension.     

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.     

一个全局耦合不连续映像格子中的冻结化随机图案模式

本文研究了一类既不连续又不可逆分段线性映像构成的全局耦合映像格子系统中的一类典型集体动力学行为, 即冻结化随机图案模式. 计算了平均同步序参量和最大李雅普诺夫指数随耦合强度的变化. 结果显示, 当耦合强度超过某个阈值后, 在给定动力学变量的初始下, 系统几乎都能达到完全或部分同步状态, 出现冻结化随机图案. 这些现象表明, 耦合映像格子系统中存在着多个共存的吸引子. 因此, 其冻结化图案的结构和分布敏感地依赖于格点动力学变量初始值的选取. 感兴趣地是, 即使当单映像处于混沌状态时, 格点间的耦合仍能将系统调制到规则的运动状态, 这种特征对于混沌控制具有重要的利用价值. 上述丰富动力学行为的出现是由于单映像中不连续性和不可逆性相互作用的结果.     

Existence and stability of steady fronts in bistable coupled map lattices

We prove the existence and we study the stability of the kinklike fixed points in a simple coupled map lattice (CML) for which the local dynamics has two stable fixed points. The condition for the existence allows us to define a critical value of the coupling parameter where a (multi) generalized saddle-node bifurcation occurs and destroys these solutions. An extension of the results to other CMLs in the same class is also displayed. Finally, we emphasize the property of spatial chaos for small coupling.     

Collective behavior in coupled chaotic map lattices with random perturbations

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).     

Tolerance of edge cascades with coupled map lattices methods

This paper studies the cascading failure on random networks and scale-free networks by introducing the tolerance parameter of edge based on the coupled map lattices methods. The whole work focuses on investigating some indices including the number of failed edges, dynamic edge tolerance capacity and the perturbation of edge. In general, it assumes that the perturbation is attributed to the normal distribution in adopted simulations. By investigating the effectiveness of edge tolerance in scale-free and random networks, it finds that the larger tolerance parameter λ can more efficiently delay the cascading failure process for scale-free networks than random networks. These results indicate that the cascading failure process can be effectively controlled by increasing the tolerance parameter λ. Moreover, the simulations also show that, larger variance of perturbation can easily trigger the cascading failures than the smaller one. This study may be useful for evaluating efficiency of whole traffic systems, and for alleviating cascading failure in such 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.     

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     

基于符号向量动力学的耦合映像格子参数估计

本文,将符号动力学推广到耦合映像格子中,以Logistic映射下耦合映像格子为研究对象,研究控制参数对符号向量序列动力学特性的影响.通过研究耦合映像格子逆函数,给出耦合映像格子的遍历条件.进一步,将给出系统初始向量,禁止字以及控制参数的符号向量序列描述方法,并最终给出基于符号向量动力学的耦合映像格子控制参数估计方法.实验结果表明,根据本文算法可以有效建立符号序列和耦合映像格子控制参数之间的对应关系,能够更好地刻画了实际模型的物理过程. 关键词： 符号向量动力学 耦合映像格子 参数估计 遍历性     

Directed percolation phase transition at the onset of STI in an inhomogeneous coupled map lattice

A model for inhomogeneously coupled logistic maps is considered to find some critical exponents in the transition from inhomogeneous steady state to spatiotemporal chaos through spatiotemporal intermittency. The laminar state in the model is described by inhomogeneous steady state with spatial period two. We obtain a complete set of static exponents which match with the corresponding directed percolation (DP) values in (1+1) dimension. We also find four nonuniversal spreading exponents in which three exponents are in agreement with DP values. The model in which absorbing state is inhomogeneous steady state, contributes a new example in evidence of Pomeau's [18] conjecture that the onset of STI in a deterministic system belongs to DP universality class.     

Application of coupled map lattice with parameter q in image encryption

In this paper, a novel coupled map lattice (CML) with parameter q is applied to image encryption to get higher security. The CML with parameter q is provided with Euler method and Adams–Bashforth–Moulton predictor–corrector method. In the new CML, dynamical properties are improved because the coupled strength can decrease the periodic dynamical behaviors which are caused by finite-precision. What's more, the CML changes system parameters from one-dimensional to two-dimensional. Two-dimensional parameters and coupling strengths provide researchers a possibility to improve the performance in image encryption. Finally, from numerical simulation results, it can be found that the CML improves the effectiveness and security.     

Differentiation in a globally coupled circle map with growth and death

A key characteristic of biological systems is the continuous life cycle where cells are born, grow and die. From a dynamical point of view the events of cell division and cell death are of paramount importance and constitute a radical departure from systems with a fixed size. In this paper, a globally coupled circle map where elements can dynamically be added and removed is investigated for the conditions under which differentiation of roles can occur. In the presence of an external source, it is found that populations of very long-living cells are sustained by short-living cells. In the case without an external source, it is found that at higher nonlinearities of the local map, large populations cannot be sustained with a previously employed division strategy but that a different and conceptually equally natural division strategy allows for differentiation of roles.