On Melnikov's method in the study of chaotic motions of gyrostat

The homoclinic solutions of the attitude motion of a gyrostat with wheels along its principal axes are formulated using the procedure developed by Wittenburg. The generic Melnikov function for the attitude motion of a gyrostat subject to small non-linear damping torques and small periodic torques is derived. The derivation is based on the chaotic theory for a 'one and half' degrees of freedom system established by Wiggins and Shaw. The conditions for the physical parameters at the onset of the homoclinic solutions are given in detail. The chaotic criteria are investigated using Melnikov's function. In particular, the conditions of the physical parameters (moments of inertia, damping coefficients, amplitudes and frequencies of external excitation, moments of momentum of wheels) that ensure the convergence of Melnikov's integral are also worked out. In order to acquire some knowledge of long-term behaviours of the chaotic attitude motion, the fourth-order Runge-Kutta numerical algorithms are utilized to simulate its dynamics. The numerical experiment shows that the chaotic motions of the gyrostat are bounded, non-periodic and sensitive to initial conditions. One of the practical mechanical models of the gyrostat is the artificial satellite. The attitude motion of the three-axis stabilized gyrostat satellite, whose torque-free motion is the homoclinic orbits, will oscillate chaotically when it is subjected to the appropriate external disturbances.     

离散混沌网络系统中共同噪声诱导同步的条件

噪声普遍存在于各种真实系统和人造系统中,它对系统的同步动力学具有多重影响.本文探索外部噪声对高维离散复杂网络系统同步行为的积极作用.首先构建外部共同噪声驱动下两个参数相同、未耦合的离散混沌网络模型,然后利用Birkhoff遍历定理与矩阵论等相关理论,严格证明了噪声诱导两个离散混沌网络系统取得同步的充分条件,进一步借助于具体的混沌网络模型,利用数值仿真验证了理论分析的有效性.数值模拟的结果表明当网络模型参数满足理论分析的充分条件时,共同噪声可以诱导两个参数相同、未耦合的离散混沌网络在随机意义下取得同步,而且同步效果不依赖于复杂网络拓扑结构的选取.     

天气系统的混沌特性

运用非线性动力学工具分析天气系统,着重从数据角度描述天气系统的特性,包括重建系统相空间、计算系统吸引子的维数以及Lyapunov指数,从而论证系统的混沌特性．目的在于以混沌动力学为工具。将天气系统作为非线性动力学系统。从实际的天气数据分析天气系统的特性,从数据的角度论证天气系统的混沌特性,并计算系统的混沌参数．     

液位控制系统的混沌故障建模分析

正常的控制系统可能会由于某些非线性故障使系统转变为一个混沌动力学系统,针对实际的液位控制系统的一般数学模型进行仿真分析,仿真过程中通过改变系统的比例以及微分环节的反馈参数以及在此回路中加入一些非线性环节,使液位控制系统的运动转变为混沌运动,同时计算系统的最大Lyapunov指数,通过判断李雅普诺夫指数的正负,从而说明了液位控制系统在满足某些条件下,会转变为混沌运动.研究结果说明,研究控制系统中混沌运动的必要性.     

基于混沌映射的HMAC算法设计与分析

分析了HMAC（Hash-based Message Authentication Code）算法存在的固有缺陷,给出了针对HMAC参数的伪造攻击实例。在此基础上,提出了一种采用混沌映射的构造HMAC的算法,该算法通过混沌迭代生成HMAC参数值,混沌系统的初值敏感和不可预测性确保了参数值的动态性,从而有效抵抗伪造攻击,提高HMAC算法的安全性能。算法仿真与分析表明构造HMAC算法需要满足的安全性要求及嵌入的hash函数需满足的安全性条件。     

Complex dynamics and synchronization of a duopoly game with bounded rationality

A dynamic Cournot game characterized by players with bounded rationality is modeled by two non-linear difference equations. The stability of the equilibria of the discrete dynamical system is analyzed. As some parameters of the model are varied, the stability of Nash equilibrium is lost and the complex chaotic behavior occurs. Synchronization of two dynamic Cournot duopoly games are considered. In the case of identical players, such dynamical system becomes symmetric, and this implies that synchronized dynamics can be obtained by a simpler one-dimensional model whose dynamics summarizes the common behavior of the two identical players.     

Complex dynamics and the structure of small neural networks

The discrete-time dynamics of small neural networks is studied empirically, with emphasis laid on non-trivial bifurcation scenarios. For particular two- and three-neuron networks interesting dynamical properties like periodic, quasi-periodic and chaotic attractors are observed, many of them co-existing for one and the same set of parameters. An appropriate equivalence class of networks is defined, describing them as parametrized dynamical systems with identical dynamical capacities. Combined symmetries in phase space and parameter space are shown to generate different representatives of such a class. Moreover, conditions on the connectivity structure are suggested, which guarantee the existence of complex dynamics for a considered equivalence class of network configurations.     

Predictability and unpredictability in Kalman filtering

The authors study the dynamical behavior of the Kalman filter when the given parameters are allowed to vary in a way which does not necessarily correspond to an underlying stochastic system. This may correspond to situations in which the basic parameters are chosen incorrectly through estimates. The authors show that, as has been suggested by Kalman, the filter equations converge to a limit (corresponding to a steady-state filter) for a subset of the parameter space which is much larger than that corresponding to bona fide stochastic systems. More surprisingly, in the complement of this subset, the filtering equations behave in both a regular and an unpredictable manner, representative of some of the basic aspects of chaotic dynamics. This interesting dynamical behavior occurs already for one-dimensional filters, and a complete phase portrait in this case is given     

Recovery of hidden information through synaptic dynamics

The role of synaptic dynamics in processing neural information is investigated in a neural information channel with realistic model neurons having chaotic intrinsic dynamics. Our neuron models are realized in simple analogue circuits, and our synaptic connections are realized both in analogue circuits and through a dynamic clamp program. The information which is input to the first chaotic neuron in the channel emerges partially absent and partially 'hidden'. Part is absent because of the dynamical effects of the chaotic oscillation that effectively acts as a noisy channel. The 'hidden' part is recoverable. We show that synaptic parameters, most significantly receptor binding time constants, can be tuned to enhance the information transmission by the combination of a neuron plus a synapse. We discuss how the dynamics of the synapse can be used to recover 'hidden' information using average mutual information as a measure of the quality of information transport.     

Structure and stability analysis of a Takagi–Sugeno fuzzy PI controller with application to tissue hyperthermia therapy

 In this paper, we first reveal the analytical structure of a simple Takagi–Sugeno (TS) fuzzy PI controller relative to the linear PI controller. The fuzzy controller consists of two linear input fuzzy sets, four TS fuzzy rules with linear consequent, Zadeh fuzzy logic AND and the centroid defuzzifier. We prove that the fuzzy controller is actually a nonlinear PI controller with the gains changing with process output. Utilizing the well-known small Gain Theorem in control theory, we then derive sufficient conditions for global stability of the fuzzy control systems involving the TS fuzzy PI controller. Finally, as an application demonstration, we apply the fuzzy PI controller to control issue temperature, in computer simulation, during hyperthermia therapy. The relationship between heat energy and tissue temperature is represented by a linear time-varying model with a time delay. The sufficient conditions for global stability are used to design a stable fuzzy control system. Our simulation results show that the fuzzy PI control system achieves satisfactory temperature control performance. The control system is robust and stable even when the model parameters are changed suddenly and significantly.     

蔡氏混沌电路频谱分布与元件参数的选择

以蔡氏电路为研究对象,依据电路方程从理论上给出了在保持混沌特性不变时的三种改变电路元件参数的途径,并进行了Multisim仿真实验.比较不同情况下混沌系统各电路变量的频谱分布曲线,发现在一定参数空间内仿真实验的结果和理论分析一致.这将为设计不同频率需求的混沌电路提供很好的设计思路,也可以在一定范围内选择不同电路元件参数,来改善所需混沌系统的频率特性,设计出更符合需求的混沌电路.     

Development of PI training algorithms for neuro-wavelet control on the synchronization of uncertain chaotic systems

This paper investigates a neuro-wavelet control (NWC) system to address the problem of synchronization control of uncertain chaotic systems. In this NWC system, a wavelet neural network (WNN) controller is the principal tracking controller designed to mimic the perfect control law and an auxiliary compensation controller is used to recover the residual approximation error so that the favorable synchronization can be achieved. Moreover, the proportional-integral (PI) training algorithms of the control system are derived from the Lyapunov stability theorem, which are utilized to update the adjustable parameters of WNN controller on-line for further assuring system stability and obtaining a fast convergence. In addition, to relax the requirement of unknown uncertainty bound, a bound estimation law is derived to estimate the uncertainty bound. Finally, some numerical simulations are presented to illustrate the effectiveness of the proposed control strategy. The simulation results demonstrate that the proposed NWC with PI training algorithms can synchronize the chaotic systems more accurately than the other control strategies.     

State estimation of dynamical systems with nonlinearities by using proportional-integral observer

In this paper the application of a PI observer technique to dynamical systems with nonlinearities is proposed. The PI observer has two feedback loops, a proportional loop and an integral loop of the estimation error. In this way the PI observer combines the structures of the practical orientated nonlinearity observer developed by the third author and the classical Luenberger observer. The structure and the estimation performance of the PI observer are discussed and analysed. The results show that the PI observer can estimate the states not only of linear systems, but also, more significantly, of systems with any arbitrary external input which appear as unknown input, nonlinearity or unmodelled dynamics. It is shown that the PI observer works with weak assumptions, which can be fulfilled by many classes of systems to be observed. Owing to the weak assumptions it can improve many observer-based technical solutions as diagnosis or control based on observers. In the paper the conditions are given and proved. The design method is declared and carried out with illustrative examples of a linear system and of a nonlinear system of a link manipulator with flexible joints. The results are good and they show the efficiency of the PI observer. In the case of nonlinear systems the advantages of ‘robustness’ and the model independency of the proposed observer scheme can be shown clearly.     

一种新型混沌系统及其DSP实现

混沌系统被广泛应用于军事保密通信和信息安全领域,该文在Liu混沌系统基础上设计了一个新型混沌系统,并重点对新型混沌系统的基本动力学特性进行了分析,包括系统的分岔图、相图、Lyapunov指数图分析。在保留了Liu混沌系统的优点基础上,增加了参数个数,扩大了参数范围,最后利用DSP实现了该新型混沌系统,并对Liu混沌系统生成的数字序列和改进后的序列进行NIST测试,对比测试结果表明改进后的数字序列性能更优。     

Periodic and chaotic dynamics of a switched-server system undercorridor policies

We consider a fluid model of a manufacturing system serving three buffers (part-types). The system operates under a feedback control policy called a corridor policy. This policy determines the times to switch between buffers and to which buffers to switch. The machine is assumed to be perfectly flexible, i.e., the switching times between buffers are zero. Open and closed versions of the system are considered. We give two sets of conditions on the parameters of the corridor: one guarantees periodic dynamics and the other chaotic behavior     

Neural network-based synchronization of uncertain chaotic systems with unknown states

In this paper, synchronization of chaotic systems with unknown parameters and unmeasured states is investigated. Two nonidentical chaotic systems in the framework of a master and a slave are considered for synchronization. It is assumed that both systems have uncertain dynamics, and states of the slave system are not measured. To tackle this challenging synchronization problem, a novel neural network-based adaptive observer and an adaptive controller have been designed. Moreover, a neural network is utilized to approximate the unknown dynamics of the slave system. The proposed method imposes neither restrictive assumption nor constraint on the dynamics of the systems. Furthermore, the stability of the entire closed-loop system in the presence of the observer dynamics has been established. Finally, effectiveness of the proposed scheme is demonstrated via computer simulation.     

混沌系统不稳定平衡点的镇定及其在蔡氏电路中的应用

基于混沌系统的遍历性和状态PI调节器理论, 提出一类混沌系统不稳定平衡点的镇定和设定点跟踪新方法, 给出用于控制器参数设计的Lyapunov矩阵不等式. 对于分段线性混沌系统, 如蔡氏电路, 可通过控制理论中的极点配置技术来设计控制器参数. 该方法对系统参数变化具有很强的鲁棒性, 能够消除外部定值扰动. 将该方法用于蔡氏混沌电路不稳定平衡点的镇定, 取得了满意的结果.     

Compressive sensing based image compression-encryption using Novel 1D-Chaotic map

Compressive sensing based encryption achieves simultaneous compression-encryption by utilizing a low complex sampling process, which is computationally secure. In this paper, a new novel 1D–chaotic map is proposed that is used to construct an incoherence rotated chaotic measurement matrix. The chaotic property of the proposed map is experimentally analysed. The linear measurements obtained are confused and diffused using the chaotic sequence generated using the proposed map. The chaos based measurement matrix construction results in reduced data storage and bandwidth requirements. As it needs to store only the parameters required to generate the chaotic sequence. Also, the sensitivity of the chaos to the parameters makes the data transmission secure. The secret key used in the encryption process is dependent on both the input data and the parameter used to generate the chaotic map. Hence the proposed scheme can resist chosen plaintext attack. The key space of the proposed scheme is large enough to thwart statistical attacks. Experimental results and the security analysis verifies the security and effectiveness of the proposed compression-encryption scheme.     

Real-time computation at the edge of chaos in recurrent neural networks

Depending on the connectivity, recurrent networks of simple computational units can show very different types of dynamics, ranging from totally ordered to chaotic. We analyze how the type of dynamics (ordered or chaotic) exhibited by randomly connected networks of threshold gates driven by a time-varying input signal depends on the parameters describing the distribution of the connectivity matrix. In particular, we calculate the critical boundary in parameter space where the transition from ordered to chaotic dynamics takes place. Employing a recently developed framework for analyzing real-time computations, we show that only near the critical boundary can such networks perform complex computations on time series. Hence, this result strongly supports conjectures that dynamical systems that are capable of doing complex computational tasks should operate near the edge of chaos, that is, the transition from ordered to chaotic dynamics.     

A tuning strategy for multivariable PI and PID controllers using differential evolution combined with chaotic Zaslavskii map

A technique for tuning of decoupled proportional-integral (PI) and proportional-integral-derivative (PID) multivariable controllers based on a chaotic differential evolution (DE) approach is presented in this paper. Due to the simple concept, easy implementation and quick convergence, nowadays DE has gained much attention and wide application in solving continuous non-linear optimization problems. However, the performance of DE greatly depends on its control parameters and it often suffers from being trapped in local optimum. The application of chaotic sequences based on chaotic Zaslavskii map instead of random sequences in DE is a powerful strategy to diversify the population and improve the DE’s performance in preventing premature convergence to local optima. The optimized PD and PID controllers shows good closed-loop responses in control of the binary Wood–Berry distillation column, a multivariable process with strong interactions between input and output pairs. Some comparison results of PD and PID tuning using chaotic DE, classical DE and genetic algorithm are presented and discussed.