Stability analysis of coupled map lattices at locally unstable fixed points

Numerical simulations of coupled map lattices (CMLs) and other complex model systems show an enormous phenomenological variety that is difficult to classify and understand. It is therefore desirable to establish analytical tools for exploring fundamental features of CMLs, such as their stability properties. Since CMLs can be considered as graphs, we apply methods of spectral graph theory to analyze their stability at locally unstable fixed points for different updating rules, different coupling scenarios, and different types of neighborhoods. Numerical studies are found to be in excellent agreement with our theoretical results.     

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.     

Lattice limit cycle dynamics: Influence of long-distance reactive and diffusive mixing

The properties of global oscillations produced by coupled reactive stochastic discrete systems on a 2D lattice support are studied, taking into account the competitive influence of local and global mixing processes. Two types of global mixing are considered: reactive and diffusive. It is shown that in the case of diffusive mixing the increase in the diffusive coupling leads to a corresponding increase in the amplitude of the global oscillations. In the case of reactive mixing the competition of local-to-global effects leads to unexpected complex phenomena. Kinetic Monte Carlo simulations demonstrate that the amplitude of oscillations as a function of the mixing-reactive coupling presents an optimal value, which is attributed to the competitive effects between the local and global processes.     

Spectral Properties of the Renormalization Group at Infinite Temperature

The renormalization group (RG) approach is largely responsible for the considerable success that has been achieved in developing a quantitative theory of phase transitions. Physical properties emerge from spectral properties of the linearization of the RG map at a fixed point. This article considers RG for classical Ising-type lattice systems. The linearization acts on an infinite-dimensional Banach space of interactions. At a trivial fixed point (zero interaction), the spectral properties of the RG linearization can be worked out explicitly, without any approximation. The results are for the RG maps corresponding to decimation and majority rule. They indicate spectrum of an unusual kind: dense point spectrum for which the adjoint operators have no point spectrum at all, only residual spectrum. This may serve as a lesson in what one might expect in more general situations.     

Dynamics of unidirectionally coupled bistable Hénon maps

We study dynamics of two bistable Hénon maps coupled in a master-slave configuration. In the case of coexistence of two periodic orbits, the slave map evolves into the master map state after transients, which duration determines synchronization time and obeys a −1/2 power law with respect to the coupling strength. This scaling law is almost independent of the map parameter. In the case of coexistence of chaotic and periodic attractors, very complex dynamics is observed, including the emergence of new attractors as the coupling strength is increased. The attractor of the master map always exists in the slave map independently of the coupling strength. For a high coupling strength, complete synchronization can be achieved only for the attractor similar to that of the master map.     

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.     

Coupled Iterated Map Models of Action Potential Dynamics in a One-dimensional Cable of Cardiac Cells

Low-dimensional iterated map models have been widely used to study action potential dynamics in isolated cardiac cells. Coupled iterated map models have also been widely used to investigate action potential propagation dynamics in one-dimensional (1D) coupled cardiac cells, however, these models are usually empirical and not carefully validated. In this study, we first developed two coupled iterated map models which are the standard forms of diffusively coupled maps and overcome the limitations of the previous models. We then determined the coupling strength and space constant by quantitatively comparing the 1D action potential duration profile from the coupled cardiac cell model described by differential equations with that of the coupled iterated map models. To further validate the coupled iterated map models, we compared the stability conditions of the spatially uniform state of the coupled iterated maps and those of the 1D ionic model and showed that the coupled iterated map model could well recapitulate the stability conditions, i.e., the spatially uniform state is stable unless the state is chaotic. Finally, we combined conduction into the developed coupled iterated map model to study the effects of coupling strength on wave stabilities and showed that the diffusive coupling between cardiac cells tends to suppress instabilities during reentry in a 1D ring and the onset of discordant alternans in a periodically paced 1D cable.     

Periodic Windows in Weakly Coupled Map Lattices

The periodic windows in weakly coupled map lattices with both diffusive and gradient couplings are studied. By using the mode analysis method, which reduces the behavior of the coupled systems to a few numbers of independent modes, we theoretically analyze the detailed structures of the periodic windows. We find that the gradient coupling greatly enlarges the width of the periodic windows, compared with the diffusive coupling.     

Uniqueness of the SRB Measure for Piecewise Expanding Weakly Coupled Map Lattices in Any Dimension

We prove the existence of a unique SRB measure for a wide range of multidimensional weakly coupled map lattices. These include piecewise expanding maps with diffusive coupling. The essential part of this research was done during an ESF explorative workshop at the Max-Planck-Institute for Mathematics, Bonn. We thank both institutions for their support.     

Electronically-implemented coupled logistic maps

The logistic map is a paradigmatic dynamical system originally conceived to model thediscrete-time demographic growth of a population, which shockingly, shows that discretechaos can emerge from trivial low-dimensional non-linear dynamics. In this work, we designand characterize a simple, low-cost, easy-to-handle, electronic implementation of thelogistic map. In particular, our implementation allows for straightforwardcircuit-modifications to behave as different one-dimensional discrete-time systems. Also,we design a coupling block in order to address the behavior of two coupled maps, although,our design is unrestricted to the discrete-time system implementation and it can begeneralized to handle coupling between many dynamical systems, as in a complex system. Ourfindings show that the isolated and coupled maps’ behavior has a remarkable agreementbetween the experiments and the simulations, even when fine-tuning the parameters with aresolution of ~10^-3. We support these conclusions by comparing the Lyapunovexponents, periodicity of the orbits, and phase portraits of the numerical andexperimental data for a wide range of coupling strengths and map’s parameters.     

从耦合映象格子中粗略恢复初值的统计特性

耦合映象格子用于信号处理研究时，从中恢复出初始条件是一个非常重要的问题.提出一种粗略恢复格点初值的方法，数值实验表明，动力学函数使用Logistic映射时，在映象过程不存在噪声的情况下，恢复的整个格子初始信号平均值等于给定信号分布的真实平均值，而恢复信号的方差小于给定信号的真实方差.将耦合看作是对独立映射的一种变换，对此作了初步解释，同时发现Logistic映射不同参数下的符号序列排序存在一些有趣的规律.对耦合格子映射研究、非线性耦合则量等是非常有启发意义的. 关键词： 耦合映射格子 信号恢复的统计特性     

Clustering Behavior in Globally Coupled Rotator Systems

Spatial clustering behaviors in globally coupled rotator systems are explored, statistical properties of cluster and disordered states are analyzed, scaling relations between the averaged lifetime of the clustering states and the coupling parameter are obtained. It is pointed out that clustering behavior also exists in the weakly antiferromagnetic case. Clustering is also investigated in terms of averaged localization in tangent space, it is shown that localization in tangent space corresponds to broken ergodicity. Influence of local coupling is studied, it is found that there exists a best local coupling which possesses the longest lifetime.     

Renormalization group operators for maps and universal scaling of universal scaling exponents

Renormalization group (RG) methods provide a unifying framework for understanding critical behaviour, such as transition to chaos in area-preserving maps and other dynamical systems, which have associated with them universal scaling exponents. Recently, de la Llave et al. (2007) [10] have formulated the Principle of Approximate Combination of Scaling Exponents (PACSE for short), which relates exponents for different criticalities via their combinatorial properties. The main objective of this paper is to show that certain integrable fixed points of RG operators for area-preserving maps do indeed follow the PACSE.     

Synchronization of spatiotemporal chaotic systems and application to secure communication of digital image

Coupled map lattices (CMLs) are taken as examples to study the synchronization of spatiotemporal chaotic systems.In this paper,we use the nonlinear coupled method to implement the synchronization of two coupled map lattices.Through the appropriate separation of the linear term from the nonlinear term of the spatiotemporal chaotic system,we set the nonlinear term as the coupling function and then we can achieve the synchronization of two coupled map lattices.After that,we implement the secure communication of digital image using this synchronization method.Then,the discrete characteristics of the nonlinear coupling spatiotemporal chaos are applied to the discrete pixel of the digital image.After the synchronization of both the communication parties,the receiver can decrypt the original image.Numerical simulations show the effectiveness and the feasibility of the proposed program.     

A coupled map lattice model for rheological chaos in sheared nematic liquid crystals

A variety of complex fluids under shear exhibit complex spatiotemporal behavior, including what is now termed rheological chaos, at moderate values of the shear rate. Such chaos associated with rheological response occurs in regimes where the Reynolds number is very small. It must thus arise as a consequence of the coupling of the flow to internal structural variables describing the local state of the fluid. We propose a coupled map lattice model for such complex spatiotemporal behavior in a passively sheared nematic liquid crystal using local maps constructed so as to accurately describe the spatially homogeneous case. Such local maps are coupled diffusively to nearest and next-nearest neighbors to mimic the effects of spatial gradients in the underlying equations of motion. We investigate the dynamical steady states obtained as parameters in the map and the strength of the spatial coupling are varied, studying local temporal properties at a single site as well as spatiotemporal features of the extended system. Our methods reproduce the full range of spatiotemporal behavior seen in earlier one-dimensional studies based on partial differential equations. We report results for both the one- and two-dimensional cases, showing that spatial coupling favors uniform or periodically time-varying states, as intuitively expected. We demonstrate and characterize regimes of spatiotemporal intermittency out of which chaos develops. Our work indicates that similar simplified lattice models of the dynamics of complex fluids under shear should provide useful ways to access and quantify spatiotemporal complexity in such problems, in addition to representing a fast and numerically tractable alternative to continuum representations.     

Crossover scale of the fixed point with replica symmetry breaking in the random Potts model

We study the scaling properties of the renormalization group (RG) flows in the two-dimensional random Potts model, assuming a general type of replica symmetry breaking (RSB) in the renormalized coupling matrix. It is shown that in the asymptotic regime the RG flows approach the non-trivial RSB fixed point algebraically slowly, which reflects the fact that this type of the fixed point is marginally stable. As a consequence, the crossover spatial scale corresponding to the critical regime described by this fixed point turns out to be exponentially large. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 11, 718–723 (10 December 1997) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Emergence and evolution of multiple clusters of attracting agents

We investigate a multi-agent system with a behavior akin to the cluster formation in systems of coupled oscillators. The saturating attractive interactions between an infinite number of non-identical agents, characterized by a multimodal distribution of their natural velocities, lead to the emergence of clusters. We derive expressions that characterize the clusters, and calculate the asymptotic velocities of the agents and the critical value for the coupling strength under which no clustering can occur. The results are supported by mathematical analysis.For the particular case of a symmetric and unimodal distribution of the natural velocities, the relationship with the Kuramoto model of coupled oscillators is highlighted. While in the generic case the emergence of a cluster corresponds to a second-order phase transition, for a specific choice of the natural velocity distribution a first-order phase transition may occur, a phenomenon recently observed in the Kuramoto model. We also present an example for which the clustering behavior is quantitatively described in terms of the coupling strength.As an illustration of the potential of the model, we discuss how it applies to the dynamic process of opinion formation.     

Roughening of spatial profiles in the presence of parametric noise

We have studied the influence of parametric noise on the spatially homogeneous phase of a generalized coupled map lattice with varying ranges of interaction. We show that synchronicity is completely stable under perturbations of the coupling strength, while variations in the local nonlinearity parameter lead to a coarsening of the spatial profile, well characterised by a host of scaling laws relating spatial roughness to range of disorder and strength of coupling.