具有非线性控制的Chua电路的混沌同步

Chua电路是一个非光滑系统.本文通过广义哈密顿系统和观测器方法,将具有非线性控制的Chua电路的混沌同步问题转化成研究具有非线性控制的光滑系统的零解稳定性;进而利用滑模控制对该光滑系统的零解稳定性进行了研究,从而使得Chua电路达到了混沌同步.最后,将上述方法应用到具体系统,数值结果也表明其正确性.     

Stationary Classical Chaos of Trapped Two-Component Bose-Einstein Condensates in 1-D Optical Lattice Potentials

We study the nonlinear dynamics of two-component Bose-Einstein condensates in one-dimensional periodic optical lattice potentials. The stationary state perturbation solutions of the coupled two-component nonlinear Schrödinger/Gross-Pitaevskii equations are constructed by using the direct perturbation method. Theoretical analysis revels that the perturbation solution is the chaotic one, which indicates the existence of chaos and chaotic region in parameter space. The corresponding numerical calculation results agree well with the analytical results. By applying the chaotic perturbation solution, we demonstrate the atomic spatial population and the energy distribution of the system are chaotic generally.     

模糊熵算法在混沌序列复杂度分析中的应用

为了准确分析混沌序列的复杂性, 采用模糊熵算法(FuzzyEn) 对典型离散混沌系统和连续混沌系统的复杂度进行分析. 与近似熵(ApEn)、 样本熵(SampEn) 和强度统计复杂度算法相比, FuzzyEn算法是一种更有效的混沌复杂度测度算法, 且对相空间维数(m)、 相似容限度(r) 和序列长度(N) 的敏感性、 依赖性更低, 鲁棒性和测度值的连续性更好. 对混沌系统的复杂性分析表明, 连续混沌系统的复杂度远小于离散混沌系统, 但是如果利用高复杂度的离散混沌伪随机序列或经典 m序列对连续混沌系统产生的伪随机序列进行扰动, 则能大大提高混沌序列的复杂性. 为混沌序列在密码学和混沌保密通信中的应用提供了理论依据.     

基于强度统计算法的混沌序列复杂度分析

为了分析混沌序列的复杂度,文中采用强度统计复杂度算法分别对离散混沌系统(TD-ERCS)和连续混沌系统(简化Lorenz系统)进行复杂度分析,计算了混沌序列随参数变化的复杂度,分析了连续混沌系统产生的伪随机序列分别进行m序列和混沌伪随机序列扰动后的复杂度.研究表明,强度统计复杂度算法是一种有效的复杂度分析方法,离散混沌序列复杂度大于连续混沌序列复杂度,但对连续混沌系统的伪随机序列进行m序列和混沌伪随机序列扰动后可大大增加复杂度,为混沌序列在信息加密中的应用提供了理论依据. 关键词： 强度统计复杂度算法 TD-ERCS系统 简化Lorenz系统 序列扰动     

Stationary Classical Chaos of Trapped Two-Component Bose-Einstein Condensates in 1-D Optical Lattice Potentials

We study the nonlinear dynamics of two-component Bose-Einstein condensates in one-dimensional periodic optical lattice potentials. The stationary state perturbation solutions of the coupled two-component nonlinear Schr(o)dinger/Gross-Pitaevskii equations are constructed by using the direct perturbation method. Theoretical analysis revels that the perturbation solution is the chaotic one, which indicates the existence of chaos and chaotic region in parameter space. The corresponding numerical calculation results agree well with the analytical results. By applying the chaotic perturbation solution, we demonstrate the atomic spatial population and the energy distribution of the system are chaotic generally.     

Generalized analytical solutions for certain coupled simple chaotic systems

We present a generalized analytical solution to the normalized state equations of a class of coupled simple secondorder non-autonomous circuit systems. The analytical solutions thus obtained are used to study the synchronization dynamics of two different types of circuit systems, differing only by their constituting nonlinear element. The synchronization dynamics of the coupled systems is studied through two-parameter bifurcation diagrams, phase portraits, and time-series plots obtained from the explicit analytical solutions. Experimental figures are presented to substantiate the analytical results. The generalization of the analytical solution for other types of coupled simple chaotic systems is discussed. The synchronization dynamics of the coupled chaotic systems studied through two-parameter bifurcation diagrams obtained from the explicit analytical solutions is reported for the first time.     

Chaotic solitons in the quadratic-cubic nonlinear Schrödinger equation under nonlinearity management

We analyze the response of rational and regular (hyperbolic-secant) soliton solutions of an extended nonlinear Schro?dinger equation (NLSE) which includes an additional self-defocusing quadratic term, to periodic modulations of the coefficient in front of this term. Using the variational approximation (VA) with rational and hyperbolic trial functions, we transform this NLSE into Hamiltonian dynamical systems which give rise to chaotic solutions. The presence of chaos in the variational solutions is corroborated by calculating their power spectra and the correlation dimension of the Poincare? maps. This chaotic behavior (predicted by the VA) is not observed in the direct numerical solutions of the NLSE when rational initial conditions are used. The solitary-wave solutions generated by these initial conditions gradually decay under the action of the nonlinearity management. On the contrary, the solutions of the NLSE with exponentially localized initial conditions are robust solitary-waves with oscillations consistent with a chaotic or a complex quasiperiodic behavior.     

基于高斯伪谱方法的混沌系统最优控制

针对混沌系统最优控制问题,提出一种基于高斯伪谱方法的数值求解新算法. 首先在勒让德-高斯点上构造Lagrange插值多项式并近似表示混沌系统最优控制中的状态变量和控制变量;接着将连续空间的最优控制问题转化为非线性规划问题;然后通过序列二次规划（SQP）算法获得最优解;最后对三个典型混沌系统的仿真实验结果表明,新方法能有效和快速地实现混沌系统的最优控制. 关键词： 混沌系统 最优控制 高斯伪谱法 非线性规划     

一个具有完全四翼形式的四维混沌

本文通过引进一个非线性状态反馈控制器, 提出了一个新的四维混沌系统, 该混沌吸引子能在任何方向上都表现出四翼形式. 由于存在一个大的正李雅普诺夫指数, 混沌系统具有一些非常有趣和复杂的动力学行为. 对系统的一些基本动力学特性进行了数值模拟和理论分析, 如平衡点、耗散性、Poincaré映射、频谱、时域谱和混沌行为等. 通过对Lyapunov指数谱和分岔图的分析, 进一步研究了混沌行为的系统参数敏感性. 最后, 设计了一个实现四翼混沌系统的振荡电路, EWB观察结果与数值模拟结果具有良好的一致性.     

延迟-非线性反馈控制混沌

提出了基于稳定性准则的延迟非线性反馈控制混沌的方法，即SC延迟非线性反馈控制法. 通过对混沌系统的适当分离，得到一个特殊的非线性函数，并利用混沌输出信号与其延迟信号的非线性函数的差，构造了连续反馈输入干扰，以控制混沌轨到某一期望的不稳周期轨上. 该方法继承了延迟反馈控制方法的优点，实现了自-控制过程. 另外由于该方法基于线性系统的稳定性准则，保证了控制的有效性. 控制过程可随时开始，具有简便、灵活性. 给出耦合Duffing振子的例子，数值模拟结果显示了SC延迟反馈方法控制的有效性. 关键词： 稳定性准则 混沌控制 延迟反馈 干扰     

输入受限的混沌系统同步控制

在混沌系统的同步控制中, 由于混沌系统对初始状态的敏感性, 一旦两个混沌系统的状态初值偏差大, 其状态同步往往需要高幅值的控制律来达到, 这给同步控制实现带来了困难, 并且在同步控制中, 两个混沌系统的初始值通常是未知的. 本文考虑控制输入受限情况下的混沌同步控制问题, 基于符号函数的近似表示式, 将受限的控制输入建模为连续可微的光滑函数, 在每一个采样点将同步控制误差系统近似为局部最优线性模型并设计连续型线性二次型调节器(LQR)最优控制律. 为降低混沌同步控制律的幅值和维持同步系统采样时刻之间的动态, 设计了等价的离散最优控制律, 并通过调整LQR性能加权矩阵值, 确保同步控制信号不会超出其受限的上界. 最后对统一混沌模型下的三种不同混沌系统同步控制进行了仿真研究. 仿真结果验证了方法的有效性. 关键词： 统一混沌模型 符号函数 输入受限 同步控制     

Nonlinear coupled-mode equations

Nonlinear coupled-mode equations governing the modal coupling of a two-mode coupled system (such as twin core couplers) are integrable; power swapping in such a system follows a periodical manner and can be expressed analytically. When three or more modes (for systems such as multiple-core couplers) are involved, the nonlinear coupled-mode equations are no longer integrable and chaotic power swapping is expected. A numerical approach is required, in general, to solve such nonlinear coupled systems involving the coupling of three or more modes. We find, however, that for certain structural configurations, such as triple-core couplers with the cores arranged in the shape of an isosceles triangle, the nonlinear coupled-mode equations for multiple-core couplers can be solved analytically under a resonant condition. The analytical solution indicates that power swapping among, for example, the three cores placed in the shape of an isosceles triangle can be aperiodic at high power, although power may flow from core to core periodically at low power.     

Generalized analytical solutions for secure transmission of signals using a simple communication scheme with numerical and experimental confirmation

A novel explicit analytical solution is reported for the transmission and recovery of information signals using a simple communication scheme. Analytical solutions are obtained for the normalized state equations of coupled second-order chaotic transmitter and receiver systems embedding the information signal. The analytical solution of the difference system obtained from the state equations of the transmitter and receiver systems has been identified as a measure of the recovered information signal which is transmitted securely by chaotic masking. The analytical solutions are used to reveal the nature of synchronization and the enhancement of the amplitude of recovered information signal. The difference signal of the coupled state variables indicating the recovered information signal obtained through numerical simulations is presented to validate the analytical results. The electronic circuit experimental results are presented to confirm the analytical and numerical results of the communication scheme discussed.     

A linear stability analysis for nonlinear,grey, thermal radiative transfer problems

We present a new linear stability analysis of three time discretizations and Monte Carlo interpretations of the nonlinear, grey thermal radiative transfer (TRT) equations: the widely used “Implicit Monte Carlo” (IMC) equations, the Carter Forest (CF) equations, and the Ahrens–Larsen or “Semi-Analog Monte Carlo” (SMC) equations. Using a spatial Fourier analysis of the 1-D Implicit Monte Carlo (IMC) equations that are linearized about an equilibrium solution, we show that the IMC equations are unconditionally stable (undamped perturbations do not exist) if α, the IMC time-discretization parameter, satisfies 0.5 < α ? 1. This is consistent with conventional wisdom. However, we also show that for sufficiently large time steps, unphysical damped oscillations can exist that correspond to the lowest-frequency Fourier modes. After numerically confirming this result, we develop a method to assess the stability of any time discretization of the 0-D, nonlinear, grey, thermal radiative transfer problem. Subsequent analyses of the CF and SMC methods then demonstrate that the CF method is unconditionally stable and monotonic, but the SMC method is conditionally stable and permits unphysical oscillatory solutions that can prevent it from reaching equilibrium. This stability theory provides new conditions on the time step to guarantee monotonicity of the IMC solution, although they are likely too conservative to be used in practice. Theoretical predictions are tested and confirmed with numerical experiments.     

Chaos and unpredictability in evolutionary dynamics in discrete time

A discrete-time version of the replicator equation for two-strategy games is studied. The stationary properties differ from those of continuous time for sufficiently large values of the parameters, where periodic and chaotic behavior replace the usual fixed-point population solutions. We observe the familiar period-doubling and chaotic-band-splitting attractor cascades of unimodal maps but in some cases more elaborate variations appear due to bimodality. Also unphysical stationary solutions can have unusual physical implications, such as the uncertainty of the final population caused by sensitivity to initial conditions and fractality of attractor preimage manifolds.     

Quasilinear Approximation and WKB

Quasilinear solutions of the radial Schrödinger equation for different potentials are compared with corresponding WKB solutions. For this study, the Schrödinger equation is first cast into a nonlinear Riccati form. While the WKB method generates an expansion in powers of , the quasi-linearization method (QLM) approaches the solution of the Riccati equation by approximating its nonlinear terms by a sequence of linear iterates. Although iterative, the QLM is not perturbative and does not rely on the existence of any kind of smallness parameters. If the initial QLM guess is properly chosen, the usual QLM solution, unlike the WKB, displays no unphysical turning-point singularities. The first QLM iteration is given by an analytic expression. This allows one to estimate analytically the role of different parameters, and the influence of their variation on the boundedness or unboundedness of a critically stable quantum system, with much more precision than provided by the WKB approximation, which often fails miserably for systems on the border of stability. It is therefore demonstrated that the QLM method is preferable over the usual WKB method.     

A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions

The subject of this paper is the development of a general solution procedure for the vibrations (primary resonance and nonlinear natural frequency) of systems with cubic nonlinearities, subjected to nonlinear and time-dependent internal boundary conditions—this is a commonly occurring situation in the vibration analysis of continuous systems with intermediate elements. The equations of motion form a set of nonlinear partial differential equations with nonlinear, time-dependent, and coupled internal boundary conditions. The method of multiple timescales, an approximate analytical method, is applied directly to each partial differential equation of motion as well as coupled boundary conditions (i.e. on each sub-domain and the corresponding internal boundary conditions for a continuous system with intermediate elements) which ultimately leads to approximate analytical expressions for the frequency-response relation and nonlinear natural frequencies of the system. These closed-form solutions provide direct insight into the relationship between the system parameters and vibration characteristics of the system. Moreover, the suggested solution procedure is applied to a sample problem which is discussed in detail.     

研究强迫非线性振子中倍周期分岔和“混乱”现象的分频采样方法

采用分频采样方法对于在周期外力作用下的非线性振子进行数值研究,达到可与离散映象相比拟的高分辨力。首次为常微分方程组描述的系统确定了高达8192分频的倍周期分岔序列和相应的“混乱”带的序列,并证实存在着嵌在混乱带中的二阶和三阶分岔序列。讨论了分频采样方法的优点和局限性,以及使用这一方法时应注意的问题。 关键词：     

分片光滑物态方程的能量方程非线性迭代解法

实际应用中的物态方程由分片光滑曲面拼接而成,拼接处存在间断.隐式求解相应的能量方程时,经常出现迭代收敛慢的情况和非物理解.本文通过构造对应的新的非线性问题,提出一种非线性迭代算法.该算法适用于求解有间断的分片光滑物态方程的非线性能量方程,其中引入一个度量能量变化的参数用于自动判断跳段是否发生,在求解时无需事先知道物态方程间断的位置,且能精确计算物态方程间断带来的能量盈亏,用于评估物态方程间断对能量的影响.典型算例验证了新算法具有稳定的收敛性,并给出符合物理规律的解.     

非线性动力系统分岔图中的暗线

基于非线性动力系统混沌运动的回归特性，构造了一种对分岔图中穿过混沌区的暗线进行研究的数值回归算法。运用该算法求得抛物线映射的暗线，并与通过暗线方程精确求得的暗线进行比较，验证了算法的有效性。对Brussel振子系统和分段线性单级齿轮动力系统的暗线进行了研究。通过对非线性动力系统分岔图中暗线的研究，由其切点可以得到嵌在混沌区中的周期窗口，由其交点可以得到混沌吸引子的激变点。研究结果表明该算法有助于分析系统的动力学行为和控制混沌运动。