Renormalization Group for Strongly Coupled Maps

Systems of strongly coupled chaotic maps generically exhibit collective behavior emerging out of extensive chaos. We show how the well-known renormalization group (RG) of unimodal maps can be extended to the coupled systems, and in particular to coupled map lattices (CMLs) with local diffusive coupling. The RG relation derived for CMLs is nonperturbative, i.e., not restricted to a particular class of configurations nor to some vanishingly small region of parameter space. After defining the strong-coupling limit in which the RG applies to almost all asymptotic solutions, we first present the simple case of coupled tent maps. We then turn to the general case of unimodal maps coupled by diffusive coupling operators satisfying basic properties, extending the formal approach developed by Collet and Eckmann for single maps. We finally discuss and illustrate the general consequences of the RG: CMLs are shown to share universal properties in the space-continuous limit which emerges naturally as the group is iterated. We prove that the scaling properly ties of the local map carry to the coupled systems, with an additional scaling factor of length scales implied by the synchronous updating of these dynamical systems. This explains various scaling laws and self-similar features previously observed numerically.     

Periodic orbits in a two-variable coupled map

Periodic orbits are calculated for a linear transformation composed of two coupled tent maps using a symbolic dynamics defined as the direct product of the single-map symbols {0,1,2}. As the coupling strength is increased orbits are pruned and a crossover to one-dimensional behavior is observed. The disallowed binary orbits containing only symbols {0,1} form a connected region in a binary symbol plane. Stable orbits may appear for strong couplings.     

Periodic attractors of random truncator maps

This paper introduces the truncator map as a dynamical system on the space of configurations of an interacting particle system. We represent the symbolic dynamics generated by this system as a non-commutative algebra and classify its periodic orbits using properties of endomorphisms of the resulting algebraic structure. A stochastic model is constructed on these endomorphisms, which leads to the classification of the distribution of periodic orbits for random truncator maps. This framework is applied to investigate the periodic transitions of Bornholdt's spin market model.     

Statistical properties of actions of periodic orbits

We investigate statistical properties of unstable periodic orbits, especially actions for two simple linear maps (p-adic Baker map and sawtooth map). The action of periodic orbits for both maps is written in terms of symbolic dynamics. As a result, the expression of action for both maps becomes a Hamiltonian of one-dimensional spin systems with the exponential-type pair interaction. Numerical work is done for enumerating periodic orbits. It is shown that after symmetry reduction, the dyadic Baker map is close to generic systems, and the p-adic Baker map and sawtooth map with noninteger K are also close to generic systems. For the dyadic Baker map, the trace of the quantum time-evolution operator is semiclassically evaluated by employing the method of Phys. Rev. E 49, R963 (1994). Finally, using the result of this and with a mathematical tool, it is shown that, indeed, the actions of the periodic orbits for the dyadic Baker map with symmetry reduction obey the uniform distribution modulo 1 asymptotically as the period goes to infinity. (c) 2000 American Institute of Physics.     

A method of estimating initial conditions of coupled maplattices based on time-varying symbolic dynamics

A novel computationally efficient algorithm in terms of the time-varying symbolic dynamic method is proposed to estimate the unknown initial conditions of coupled map lattices (CMLs). The presented method combines symbolic dynamics with time-varying control parameters to develop a time-varying scheme for estimating the initial condition of multi-dimensional spatiotemporal chaotic signals. The performances of the presented time-varying estimator in both noiseless and noisy environments are analysed and compared with the common time-invariant estimator. Simulations are carried out and the obtained results show that the proposed method provides an efficient estimation of the initial condition of each lattice in the coupled system. The algorithm cannot yield an asymptotically unbiased estimation due to the effect of the coupling term, but the estimation with the time-varying algorithm is closer to the Cramer--Rao lower bound (CRLB) than that with the time-invariant estimation method, especially at high signal-to-noise ratios (SNRs).     

A linear code for the sawtooth and cat maps

The sawtooth maps are a one-parameter set of piecewise linear area preserving maps on the torus. For positive integer values of the parameter K they are automorphisms of the torus, known as the cat maps. We present a symbolic dynamics for these maps in which the symbols are integers. This code is related to a practical problem of the stabilisation of a system which is perturbed by impulses. The code is linear in the sense that an orbit and its code are linearly related, so it is not difficult to obtain a good approximation to one from the other in practice. A stationary stochastic process for generating the code is given explicitly. The theory uses Green function methods, which are also used to study ordered periodic orbits and cantori. The problems of using a similar code for arbitrary area preserving twist maps on the torus are briefly discussed.     

A note on chaotic unimodal maps and applications

Based on the word-lift technique of symbolic dynamics of one-dimensional unimodal maps, we investigate the relation between chaotic kneading sequences and linear maximum-length shift-register sequences. Theoretical and numerical evidence that the set of the maximum-length shift-register sequences is a subset of the set of the universal sequence of one-dimensional chaotic unimodal maps is given. By stabilizing unstable periodic orbits on superstable periodic orbits, we also develop techniques to control the generation of long binary sequences.     

Globally enumerating unstable periodic orbits for observed data using symbolic dynamics

The unstable periodic orbits of a chaotic system provide an important skeleton of the dynamics in a chaotic system, but they can be difficult to find from an observed time series. We present a global method for finding periodic orbits based on their symbolic dynamics, which is made possible by several recent methods to find good partitions for symbolic dynamics from observed time series. The symbolic dynamics are approximated by a Markov chain estimated from the sequence using information-theoretical concepts. The chain has a probabilistic graph representation, and the cycles of the graph may be exhaustively enumerated with a classical deterministic algorithm, providing a global, comprehensive list of symbolic names for its periodic orbits. Once the symbolic codes of the periodic orbits are found, the partition is used to localize the orbits back in the original state space. Using the periodic orbits found, we can estimate several quantities of the attractor such as the Lyapunov exponent and topological entropy.     

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     

Targeting of Kolmogorov-Arnold-Moser Orbits by the Bailout Embedding Method in Two Coupled Standard Maps

A bailout embedding method for controlling chaos can make the chaotic orbits targeting into Kolmogorov- Arnold-Moser orbits. We apply this method to a high-dimensional system with two coupled standard maps. The numerical simulation shows that this method could obtain target islands in order and hence could be used to control chaos. Moreover, it is robust in the presence of weak external noise.     

A method of recovering the initial vectors of globally coupled map lattices based on symbolic dynamics

Based on symbolic dynamics, a novel computationally efficient algorithm is proposed to estimate the unknown initial vectors of globally coupled map lattices (CMLs). It is proved that not all inverse chaotic mapping functions are satisfied for contraction mapping. It is found that the values in phase space do not always converge on their initial values with respect to sufficient backward iteration of the symbolic vectors in terms of global convergence or divergence (CD). Both CD property and the coupling strength are directly related to the mapping function of the existing CML. Furthermore, the CD properties of Logistic, Bernoulli, and Tent chaotic mapping functions are investigated and compared. Various simulation results and the performances of the initial vector estimation with different signal-to-noise ratios (SNRs) are also provided to confirm the proposed algorithm. Finally, based on the spatiotemporal chaotic characteristics of the CML, the conditions of estimating the initial vectors using symbolic dynamics are discussed. The presented method provides both theoretical and experimental results for better understanding and characterizing the behaviours of spatiotemporal chaotic systems.     

Unstable periodic orbits and templates of the Rossler system: Toward a systematic topological characterization

The Rossler system has been exhaustively studied for parameter values (a in [0.33,0.557],b=2,c=4). Periodic orbits have been systematically extracted from Poincare maps and the following problems have been addressed: (i) all low order periodic orbits are extracted, (ii) encoding of periodic orbits by symbolic dynamics (from 2 letters up to 11 letters) is achieved, (iii) some rules of growth and of pruning of the periodic orbits population are obtained, and (iv) the templates of the attractors are elaborated to characterize the attractors topology. (c) 1995 American Institute of Physics.     

耦合不连续系统同步转换过程中的多吸引子共存

本文研究了耦合不连续系统的同步转换过程中的动力学行为, 发现由混沌非同步到混沌同步的转换过程中特殊的多吸引子共存现象. 通过计算耦合不连续系统的同步序参量和最大李雅普诺夫指数随耦合强度的变化, 发现了较复杂的同步转换过程: 临界耦合强度之后出现周期非同步态(周期性窗口); 分析了系统周期态的迭代轨道,发现其具有两类不同的迭代轨道: 对称周期轨道和非对称周期轨道, 这两类周期吸引子和同步吸引子同时存在, 系统表现出对初值敏感的多吸引子共存现象. 分析表明, 耦合不连续系统中的周期轨道是由于局部动力学的不连续特性和耦合动力学相互作用的结果. 最后, 对耦合不连续系统的同步转换过程进行了详细的分析, 结果表明其同步呈现出较复杂的转换过程.     

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.     

Periodic orbits in magnetic billiards

We propose a simple method to calculate periodic orbits in two-dimensional systems with no symbolic dynamics. The method is based on a line by line scan of the Poincaré surface of section and is particularly useful for billiards. We have applied it to the Square and Sinai's billiards subjected to a uniform orthogonal magnetic field and we obtained about 2000 orbits for both systems using absolutely no information about their symbolic dynamics. Received 21 September 1999 and Received in final form 13 April 2000     

Normally attracting manifolds and periodic behavior in one-dimensional and two-dimensional coupled map lattices

We consider diffusively coupled logistic maps in one- and two-dimensional lattices. We investigate periodic behaviors as the coupling parameter varies, i.e., existence and bifurcations of some periodic orbits with the largest domain of attraction. Similarity and differences between the two lattices are shown. For small coupling the periodic behavior appears to be characterized by a number of periodic orbits structured in such a way to give rise to distinct, reverse period-doubling sequences. For intermediate values of the coupling a prominent role in the dynamics is played by the presence of normally attracting manifolds that contain periodic orbits. The dynamics on these manifolds is very weakly hyperbolic, which implies long transients. A detailed investigation allows the understanding of the mechanism of their formation. A complex bifurcation is found which causes an attracting manifold to become unstable. (c) 1994 American Institute of Physics.     

Coupled map lattices as computational systems

The coupled map lattice (CML) as a mathematical model for a computer is considered. Using the theory of synchronous concurrent algorithms, it is shown that the CML is a valid new model for a parallel deterministic analog machine, but that, in principle, such a CML computer does not generate computations that cannot be reproduced by the standard mathematical models for computing on real numbers. The analysis is based on new general mathematical definitions of CMLs, and an axiomatic approach to determining which models of computation can be used to simulate CMLs.     

Periodic orbits in coupled Henon maps: Lyapunov and multifractal analysis

A powerful algorithm is implemented in a 1-d lattice of Henon maps to extract orbits which are periodic both in space and time. The method automatically yields a suitable symbolic encoding of the dynamics. The arrangement of periodic orbits allows us to elucidate the spatially chaotic structure of the invariant measure. A new family of specific Lyapunov exponents is defined, which estimate the growth rate of spatially inhomogeneous perturbations. The specific exponents are shown to be related to the comoving Lyapunov exponents. Finally, the zeta-function formalism is implemented to analyze the scaling structure of the invariant measure both in space and time.     

Strange Attractors with One Direction of Instability

We give simple conditions that guarantee, for strongly dissipative maps, the existence of strange attractors with a single direction of instability and certain controlled behaviors. Only the d= 2 case is treated in this paper, although our approach is by no means limited to two phase-dimensions. We develop a dynamical picture for the attractors in this class, proving they have many of the statistical properties associated with chaos: positive Lyapunov exponents, existence of SRB measures, and exponential decay of correlations. Other results include the geometry of fractal critical sets, nonuniform hyperbolic behavior, symbolic coding of orbits, and formulas for topological entropy. Received: 25 April 2000 / Accepted: 17 October 2000     

Symbolic dynamics and synchronization of coupled map networks with multiple delays

We use symbolic dynamics to study discrete-time dynamical systems with multiple time delays. We exploit the concept of avoiding sets, which arise from specific non-generating partitions of the phase space and restrict the occurrence of certain symbol sequences related to the characteristics of the dynamics. In particular, we show that the resulting forbidden sequences are closely related to the time delays in the system. We present two applications to coupled map lattices, namely (1) detecting synchronization and (2) determining unknown values of the transmission delays in networks with possibly directed and weighted connections and measurement noise. The method is applicable to multi-dimensional as well as set-valued maps, and to networks with time-varying delays and connection structure.