Controlling Spatiotemporal Chaos in Coupled Map Lattices to Periodic Orbits

Two methods are presented for controlling spatiotemporal chaotic motion in coupled map lattices to a kind of periodic orbit where the dynamicM variables of all lattice sites are equM and act periodically as time evolves. Stability analysis of the periodic orbits is presented. We prove that especially the second controlling method can stabilize all the periodic orbits we concern. Basin of attraction and noise problem are discussed.     

A Cryptographic Scheme Based on Spatiotemporal Chaos of Coupled Map Lattices

We propose a cryptographic scheme based on spatiotemporal chaos of coupled map lattices (CML) ,which is based on one-time pad. The structure of the cryptosystem determines that the progress in decryption implies the progress in exploring the dynamical behavior of spatiotemporal chaos in CML. A part of the initial condition of CML is used as a secret key, and the recovery of the secret key by exhaustive search is impossible due to the sensitivity to the initial condition in spatiotemporal chaos system. Specially the software implementation of the scheme is easy.     

耦合双稳映象格子模型的时空混沌控制

变量反馈技术实现了耦合双稳映象格子模型的时空混沌控制.数值实验结果表明,利用不同的反馈技术和不同的反馈强度,可以将双稳映象系统的混沌及耦合双稳映象格子模型的时空混沌控制到不动点或周期轨道.变量反馈控制法除了局域双稳映象系统的定态点外,不需要先获取耦合双稳映象格子时空系统的动力学信息,它对抑制耦合双稳映象系统中的湍流具有一定的指导作用.     

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.     

Synchronization of Spatiotemporal Chaos in Coupled Complex Ginzburg-Landau Oscillators

Synchronization of spatiotemporal distributed system is investigated by considering the model of 1D dif-fusively coupled complex Ginzburg-Landau oscillators. An itinerant approach is suggested to randomly move turbulentsignal injections in the space of spatiotemporal chaos. Our numerical simulations show that perfect turbulence synchro-nization can be achieved with properly selected itinerant time and coupling intensity.     

时空混沌系统的主动-间隙耦合同步

提出了离散系统中的主动-间隙耦合同步方法。该方法由同步相和自治相组成,在同步相,同步方案使得混沌系统趋于同步,而在自治相,两系统间的误差将迅速放大,导致同步失去。但只要同步相足够大,最终可实现系统的完全同步。从理论上讨论了同步条件,并在数值实验上讨论了同步相与耦合强度的关系。     

A Self-synchronizing Stream Encryption Scheme Based on One-Dimensional Coupled Map Lattices

We present a self-synchronizing stream encryption scheme based on one-dimensional coupled map lattices which is introduced as a model with the essential features of spatiotemporal chaos, and of great complexity and diffusion capability of the little disturbance in the initial condition. To evaluate the scheme, a series of statistical tests are employed, and the results show good random-look nature of the ciphertext. Furthermore, we apply our algorithm to encrypt a grey-scale image to show the key sensitivity.     

Controlling Chaos with Rectificative Feedback Injections in 2D Coupled Complex Ginzburg-Landau Oscillators

A model of two-dimensional coupled complex Ginzburg-Landau oscillators driven by a rectificative feedbackcontroller is used to study controlling spatiotemporal chaos without gradient force items. By properly selecting the signalinjecting position with considering the maximum gap between signals and targets, and adjusting the control time interval,we have finally obtained the efficient chaos control via numerical simulations.     

Controlling Chaos with Rectificative Feedback Injections in 2D Coupled Complex Ginzburg-Landau Oscillators

A model of two-dimensional coupled complex Ginzburg-Landau oscillators driven by a rectificative feedback controller is used to study controlling spatiotemporal chaos without gradient force items. By properly selecting the signal injecting position with considering the maximum gap between signals and targets, and adjusting the control time interval,we have finally obtained the efficient chaos control via numerical simulations.     

Ising-Type and Other Transitions in One-Dimensional Coupled Map Lattices with Sign Symmetry

We consider a one-dimensional lattice of expanding antisymmetric maps [–1, 1][–1, 1] with nearest neighbor diffusive coupling. For such systems it is known that if the coupling parameter is small there is unique stationary (in time) state, which is chaotic in space-time. A disputed question is whether such systems can exhibit Ising-type phase transitions as grows beyond some critical value _c. We present results from computer experiments which give definite indication that such a transition takes place: the mean square magnetization appears to diverge as approaches some critical value, with a critical exponent around 0.9. We also study other properties of the coupled map system.     

用直线稳定方法控制耦合哈密顿系统的混沌运动

讨论了不稳定不动点邻域的不稳定轨道的稳定问题．通过对系统施加外部的控制信号,将直线稳定方法推广到控制高维保守系统一耦合标准映象的混沌运动．通过对外加控制信号的调整,使系统不稳定不动点邻域的不稳定轨道沿着连接任意时刻轨道点和该不动点的直线趋向不动点,从而使难于控制的高维保守系统的不稳定轨道趋于稳定．这种方法不需要事先掌握系统动力行为,而且具有较强的抗干扰能力。     

Non-Equilibrium Behaviour in Unidirectionally Coupled Map Lattices

A spatially one dimensional coupled map lattice with a local and unidirectional coupling is introduced. This model is studied analytically by a perturbation theory that is valid for small coupling strength. In parameter space three phases with different ergodic behaviour are observed. Via coarse graining the deterministic model is mapped to a stochastic spin model that can be described by a master equation. Because of the anisotropic coupling non-equilibrium behaviour is found on the coarse grained level. However, the stationary statistical properties of the spin dynamics can still be described with a nearest neighbour Ising model whereby the ordering is predominantly antiferromagnetic.     

Equilibrium Phase Transitions in Coupled Map Lattices: A Pedestrian Approach

A class of piecewise linear coupled map lattices with simple symbolic dynamics is constructed. It can be solved analytically in terms of the statistical mechanics of spin lattices. The corresponding Hamiltonian is written down explicitly in terms of the parameters of the map. The approach follows the line of recent mathematical investigations. But the presentation is kept elementary so that phase transitions in the dynamical model can be studied in detail. Although the method works only for map lattices with repelling invariant sets some of the conclusions, i.e., the role of local curvature of the single site map and properties of the nearest neighbour coupling might play an important role for phase transitions in general dynamical 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     

Controlling Strong Chaos by an Aperiodic Perturbation in Area Preserving Maps

We demonstrate a method for controlling strong chaos by an aperiodic perturbation in two-dimensional Hamiltonian systems.The method has the advantages that the controlled system remains conservative property and the selection of the perturbation has a considerable diversity.We illustrate this method with two area preserving maps:the non-monotonic twist map which is a mixed system and the perturbed cat map which exhibits hard chaos.Numerical results show that the strong chaos can be effectively controlled into regular motions,and the final states are always quasiperiodic ones.The method is robust against the presence of weak external noise.     

Spatiotemporal Chaos Control with Pinning Signals in Distributed Systems

We suggest a local pinning feedback control for stabilizing periodic pattern in spatially extended systems. Analytical and numerical investigations of this method for a system described by the one-dimensional complex Ginzburg-Landau equation are carried out. We found that it is possible to suppress spatiotemporal chaos by using a few pinning signals in the presence of a large gradient force. Our analytical predictions well coincide with numerical observations.     

Evaluating the dynamical coupling between spatiotemporally chaotic signals via an information theory approach

An information-theoretic measure is introduced for evaluating the dynamical coupling of spatiotemporally chaotic signals produced by extended systems. The measure of the one-way coupled map lattices and the one-dimensional, homogeneous, diffusively coupled map lattices is computed with the symbolic analysis method. The numerical results show that the information measure is applicable to determining the dynamical coupling between two directly coupled or indirectly coupled chaotic signals.     

用常数周期脉冲方法控制耦合标准映象的混沌

将用常数周期脉冲方法控制保守系统的混沌运动的方法，用于控制高维系统耦合标准映象的混沌运动，计算了系统参数k=0.8和耦合常数β=0.04的有限时间李雅普诺夫指数，考察了不同系统参数和耦合常数下的有限时间李雅普诺夫指数随迭代次数的演化规律，从各次迭代的有限时间收敛区选择满足期望动力学特征的轨道片段，施加适当的脉冲强度找出其对应的周期不动点，从而得到期望的周期转道。     

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.