From the Perron-Frobenius equation to the Fokker-Planck equation

We show that for certain classes of deterministic dynamical systems the Perron-Frobenius equation reduces to the Fokker-Planck equation in an appropriate scaling limit. By perturbative expansion in a small time scale parameter, we also derive the equations that are obeyed by the first- and second-order correction terms to the Fokker-Planck limit case. In general, these equations describe non-Gaussian corrections to a Langevin dynamics due to an underlying deterministic chaotic dynamics. For double-symmetric maps, the first-order correction term turns out to satisfy a kind of inhomogeneous Fokker-Planck equation with a source term. For a special example, we are able solve the first- and second-order equations explicitly.     

一阶时滞混沌的参数辨识

估计混沌系统的未知参数是混沌控制与同步中必须解决的关键问题.针对两种不同类型的时滞混沌系统中的未知参数的辨识问题，将系统的未知参数看作系统的未知状态，利用非线性状态观测器理论进行参数估计，通过选取观测器中非线性增益函数，使得闭环误差系统指数或渐进稳定，从而给出参数估计器存在的充分条件.以典型的时滞Logistic系统为例进行了数值模拟，数值仿真结果表明，使用该方法可以对时滞混沌系统的未知参数进行有效地估计.     

A new deterministic model for chaotic reversals

We present a new chaotic system of three coupled ordinary differential equations, limited to quadratic nonlinear terms. A wide variety of dynamical regimes are reported. For some parameters, chaotic reversals of the amplitudes are produced by crisis-induced intermittency, following a mechanism different from what is generally observed in similar deterministic models. Despite its simplicity, this system therefore generates a rich dynamics, able to model more complex physical systems. In particular, a comparison with reversals of the magnetic field of the Earth shows a surprisingly good agreement, and highlights the relevance of deterministic chaos to describe geomagnetic field dynamics.     

混沌信号自适应协同滤波去噪

混沌信号协同滤波去噪算法充分利用了混沌信号的自相似结构特征,具有良好的信噪比提升性能.针对该算法的滤波参数优化问题,考虑到最优滤波参数的选取受到信号特征、采样频率和噪声水平的影响,为提高该算法的自适应性使其更符合实际应用需求,基于排列熵提出一种滤波参数自动优化准则.依据不同噪声水平的混沌信号排列熵的不同,首先选取不同滤波参数对含噪混沌信号进行去噪,然后计算各滤波参数对应重构信号的排列熵,最后通过比较各重构信号的排列熵,选取排列熵最小的重构信号对应的滤波参数为最优滤波参数,实现滤波参数的优化.分析了不同信号特征、采样频率和噪声水平情况下滤波参数的选取规律.仿真结果表明,该参数优化准则能在不同条件下对滤波参数进行有效的自动最优化,提高了混沌信号协同滤波去噪算法的自适应性.     

Symbolic dynamics of one-dimensional maps: Entropies,finite precision,and noise

In the study of nonlinear physical systems, one encounters apparently random or chaotic behavior, although the systems may be completely deterministic. Applying techniques from symbolic dynamics to maps of the interval, we compute two measures of chaotic behavior commonly employed in dynamical systems theory: the topological and metric entropies. For the quadratic logistic equation, we find that the metric entropy converges very slowly in comparison to maps which are strictly hyperbolic. The effects of finite precision arithmetric and external noise on chaotic behavior are characterized with the symbolic dynamics entropies. Finally, we discuss the relationship of these measures of chaos to algorithmic complexity, and use algorithmic information theory as a framework to discuss the construction of models for chaotic dynamics.     

Chaos and chaotic control in a relative rotation nonlinear dynamical system under parametric excitation

This paper studies the chaotic behaviours of a relative rotation nonlinear dynamical system under parametric excitation and its control. The dynamical equation of relative rotation nonlinear dynamical system under parametric excitation is deduced by using the dissipation Lagrange equation. The criterion of existence of chaos under parametric excitation is given by using the Melnikov theory. The chaotic behaviours are detected by numerical simulations including bifurcation diagrams, Poincar map and maximal Lyapunov exponent. Furthermore, it implements chaotic control using non-feedback method. It obtains the parameter condition of chaotic control by the Melnikov theory. Numerical simulation results show the consistence with the theoretical analysis. The chaotic motions can be controlled to period-motions by adding an excitation term.     

改进的保群算法及其在混沌系统中的应用

混沌系统的跟踪控制是近年来非线性控制领域研究的热点之一. 本文提出了一种基于快速下降控制方法的保群算法, 此方法使受控混沌系统能够快速稳定到相空间的一个不动点; 另外提出一种基于滑模控制方法的保群算法, 此方法使受控混沌系统能够快速跟踪一个确定的运动轨迹. 应用这两种新方法分别对两个经典的混沌系统(Lorenz系统和Duffing系统)进行了相应的数值实验, 实验结果表明这两种方法都具用较高的精度和稳定性.     

Tracking controlled chaos: Theoretical foundations and applications

Tracking controlled states over a large range of accessible parameters is a process which allows for the experimental continuation of unstable states in both chaotic and non-chaotic parameter regions of interest. In algorithmic form, tracking allows experimentalists to examine many of the unstable states responsible for much of the observed nonlinear dynamic phenomena. Here we present a theoretical foundation for tracking controlled states from both dynamical systems as well as control theoretic viewpoints. The theory is constructive and shows explicitly how to track a curve of unstable states as a parameter is changed. Applications of the theory to various forms of control currently used in dynamical system experiments are discussed. Examples from both numerical and physical experiments are given to illustrate the wide range of tracking applications. (c) 1997 American Institute of Physics.     

双向耦合混沌系统的反同步性研究

针对两个双线性耦合的统一混沌系统,通过运用自适应方法实现它们的完全反同步.自适应控制器和耦合系数均由Lyapunov稳定性理论和最小值定理估计出来,这种方法并不需具体求出耦合系数范围,也无需事先知道耦合混沌系统的最大值,就能实现反同步.将这种方法应用到Chen-Chen混沌系统的双向耦合情况,数值仿真表明该方法的有效性和实用性. 关键词： 统一混沌系统 双线性耦合 自适应反同步     

Approximation-error-ADP-based optimal tracking control for chaotic systems with convergence proof

In this paper, an optimal tracking control scheme is proposed for a class of discrete-time chaotic systems using the approximation-error-based adaptive dynamic programming （ADP） algorithm. Via the system transformation, the optimal tracking problem is transformed into an optimal regulation problem, and then the novel optimal tracking control method is proposed. It is shown that for the iterative ADP algorithm with finite approximation error, the iterative performance index functions can converge to a finite neighborhood of the greatest lower bound of all performance index functions under some convergence conditions. Two examples are given to demonstrate the validity of the proposed optimal tracking control scheme for chaotic systems.     

Analysis of some recently proposed chaos-based encryption algorithms

We analyze the security and performance of two recently proposed encryption algorithms designed using deterministic chaotic dynamical systems. The analysis shows not only that the encryption rates of these algorithms are not competitive to the encryption rates of the standard algorithms, but that the algorithms can be easily broken using known-plaintext attack.     

Fuzzy Modeling and Impulsive Control of a Memristor-Based Chaotic System

We mainly investigate the issues of fuzzy modeling and impulsive control of a memristor-based chaotic system and present a memristor-based chaotic system as the Takagi-Sugeno model-based fuzzy system. Then, based on the impulsive control theory of dynamical systems, a criterion ensuring impulsive stabilization of the memristorbased chaotic system is derived for the first time. An illustrative example is given to verify the effectiveness of the control scheme.     

Hyperbolification of dynamical systems: The case of continuous-time systems

We present a new method to generate chaotic hyperbolic systems. The method is based on the knowledge of a chaotic hyperbolic system and the use of a synchronization technique. This procedure is called hyperbolification of dynamical systems. The aim of this process is to create or enhance the hyperbolicity of a dynamical system. In other words, hyperbolification of dynamical systems produces chaotic hyperbolic (structurally stable) behaviors in a system that would not otherwise be hyperbolic. The method of hyperbolification can be outlined as follows. We consider a known n-dimensional hyperbolic chaotic system as a drive system and another n-dimensional system as the response system plus a feedback control function to be determined in accordance with a specific synchronization criterion. We then consider the error system and apply a synchronization method, and find sufficient conditions for the errors to converge to zero and hence the synchronization between the two systems to be established. This means that we construct a 2n-dimensional continuous-time system that displays a robust hyperbolic chaotic attractor. An illustrative example is given to show the effectiveness of the proposed hyperbolification method.     

Synchronization of spatiotemporal chaos in a class of complex dynamical networks

This paper studies the synchronization of complex dynamical networks constructed by spatiotemporal chaotic systems with unknown parameters. The state variables in the systems with uncertain parameters are used to construct the parameter recognizers, and the unknown parameters are identified. Uncertain spatiotemporal chaotic systems are taken as the nodes of complex dynamical networks, connection among the nodes of all the spatiotemporal chaotic systems is of nonlinear coupling. The structure of the coupling functions between the connected nodes and the control gain are obtained based on Lyapunov stability theory. It is seen that stable chaos synchronization exists in the whole network when the control gain is in a certain range. The Gray--Scott models which have spatiotemporal chaotic behaviour are taken as examples for simulation and the results show that the method is very effective.     

Coherence resonance in coupled chaotic oscillators

Existing works on coherence resonance, i.e., the phenomenon of noise-enhanced temporal regularity, focus on excitable dynamical systems such as those described by the FitzHugh-Nagumo equations. We extend the scope of coherence resonance to an important class of dynamical systems: coupled chaotic oscillators. In particular, we show that, when a system of coupled chaotic oscillators is under the influence of noise, the degree of temporal regularity of dynamical variables characterizing the difference among the oscillators can increase and reach a maximum value at some optimal noise level. We present numerical results illustrating the phenomenon and give a physical theory to explain it.     

A new approach of optimal control for a class of continuous-time chaotic systems by an online ADP algorithm

We develop an online adaptive dynamic programming （ADP） based optimal control scheme for continuous-time chaotic systems. The idea is to use the ADP algorithm to obtain the optimal control input that makes the performance index function reach an optimum. The expression of the performance index function for the chaotic system is first presented. The online ADP algorithm is presented to achieve optimal control. In the ADP structure, neural networks are used to construct a critic network and an action network, which can obtain an approximate performance index function and the control input, respectively. It is proven that the critic parameter error dynamics and the closed-loop chaotic systems are uniformly ultimately bounded exponentially. Our simulation results illustrate the performance of the established optimal control method.     

Nonlinear observer for synchronization of chaotic systems with application to secure data transmission

The main issue of this work is related with the design of a class of nonlinear observer in order to synchronize chaotic dynamical systems in a master-slave scheme, considering different initial conditions. The oscillator of Chen is proposed as a benchmark model and a bounded-type observer is proposed to reach synchronicity between both two chaotic systems. The proposed observer contains a proportional and sigmoid form of a bounded function of the synchronization error in order to provide asymptotic synchronization with a satisfactory performance. Some numerical simulations were carrying out in order to show the operation of the proposed methodology, with possible applications to secure data communications issues.     

Hybrid Control of Digital Baker Map with Application to Pseudo-Random Number Generator

Dynamical degradation occurs when chaotic systems are implemented on digital devices, which seriously threatens the security of chaos-based cryptosystems. The existing solutions mainly focus on the compensation of dynamical properties rather than on the elimination of the inherent biases of chaotic systems. In this paper, a unidirectional hybrid control method is proposed to improve the dynamical properties and to eliminate the biases of digital chaotic maps. A continuous chaotic system is introduced to provide external feedback control of the given digital chaotic map. Three different control modes are investigated, and the influence of control parameter on the properties of the controlled system is discussed. The experimental results show that the proposed method can not only improve the dynamical degradation of the digital chaotic map but also make the controlled digital system produce outputs with desirable performances. Finally, a pseudorandom number generator (PRNG) is proposed. Statistical analysis shows that the PRNG has good randomness and almost ideal entropy values.     

Policy iteration optimal tracking control for chaotic systems by using an adaptive dynamic programming approach

A policy iteration algorithm of adaptive dynamic programming(ADP) is developed to solve the optimal tracking control for a class of discrete-time chaotic systems. By system transformations, the optimal tracking problem is transformed into an optimal regulation one. The policy iteration algorithm for discrete-time chaotic systems is first described. Then,the convergence and admissibility properties of the developed policy iteration algorithm are presented, which show that the transformed chaotic system can be stabilized under an arbitrary iterative control law and the iterative performance index function simultaneously converges to the optimum. By implementing the policy iteration algorithm via neural networks,the developed optimal tracking control scheme for chaotic systems is verified by a simulation.     

On the Validity of Linear Response Theory in High-Dimensional Deterministic Dynamical Systems

This theoretical work considers the following conundrum: linear response theory is successfully used by scientists in numerous fields, but mathematicians have shown that typical low-dimensional dynamical systems violate the theory’s assumptions. Here we provide a proof of concept for the validity of linear response theory in high-dimensional deterministic systems for large-scale observables. We introduce an exemplary model in which observables of resolved degrees of freedom are weakly coupled to a large, inhomogeneous collection of unresolved chaotic degrees of freedom. By employing statistical limit laws we give conditions under which such systems obey linear response theory even if all the degrees of freedom individually violate linear response. We corroborate our result with numerical simulations.