保守系统的混沌控制

保守系统的混沌控制是一个重要而富有挑战性的研究课题。由于Liouville定理的限制和初始条件的特殊作用，使得适用于耗散系统的混沌控制方法不能直接用于保守系统。本文通过对耗散系统和保守系统混沌运动的特征进行分析和比较，阐述了保守系统混沌运动的规律，总结了近期研究过程中一些典型的基本理论和方法，综述了近年来保守系统混沌控制的相关进展和我们在保守系统的混沌控制方面所做的工作，并对保守系统混沌控制的应用和发展方向进行了展望。     

Chaotic memristor

We suggest and experimentally demonstrate a chaotic memory resistor (memristor). The core of our approach is to use a resistive system whose equations of motion for its internal state variables are similar to those describing a particle in a multi-well potential. Using a memristor emulator, the chaotic memristor is realized and its chaotic properties are measured. A Poincaré plot showing chaos is presented for a simple nonautonomous circuit involving only a voltage source directly connected in series to a memristor and a standard resistor. We also explore theoretically some details of this system, plotting the attractor and calculating Lyapunov exponents. The multi-well potential used resembles that of many nanoscale memristive devices, suggesting the possibility of chaotic dynamics in other existing memristive systems.     

一种基于保守混沌的密钥分发协议及图像加密算法

耗散混沌系统可以通过时滞嵌入法重构混沌吸引子,因而耗散混沌在基于混沌的信息加密技术中存在一定隐患.针对这一问题提出一种基于保守混沌的密钥分发协议及图像加密算法,该算法将图像数据通过Hash算法转换为保守混沌系统的初始值,形成一次一密的加密结构.然后利用保守混沌信号结合密钥分发协议生成二进制密钥流,该过程由发送方和接受方...     

Nonlinear dynamics analysis of cluster-shaped conservative flows generated from a generalized thermostatted system

The thermostatted system is a conservative system different from Hamiltonian systems,and has attracted much attention because of its rich and different nonlinear dynamics.We report and analyze the multiple equilibria and curve axes of the cluster-shaped conservative flows generated from a generalized thermostatted system.It is found that the cluster-shaped structure is reflected in the geometry of the Hamiltonian,such as isosurfaces and local centers,and the shapes of cluster-shaped chaotic flows and invariant tori rely on the isosurfaces determined by initial conditions,while the numbers of clusters are subject to the local centers solved by the Hessian matrix of the Hamiltonian.Moreover,the study shows that the cluster-shaped chaotic flows and invariant tori are chained together by curve axes,which are the segments of equilibrium curves of the generalized thermostatted system.Furthermore,the interesting results are vividly demonstrated by the numerical simulations.     

基于忆阻器的多涡卷混沌系统及其脉冲同步控制

忆阻器是一种具有记忆功能和纳米级尺寸的非线性元件,作为混沌系统的非线性部分,能够提高混沌系统的信号随机性和复杂度.本文基于增广Lü系统设计了一个三维忆阻混沌系统.仅仅通过改变系统的一个参数,该系统能产生单涡巻、双涡卷和四涡巻的混沌吸引子,说明该系统具有丰富的混沌特性.首先对该忆阻混沌系统的基本动力学行为进行了理论分析和数值仿真,如平衡点稳定性、对称性,Lyapunov指数和维数,分岔图和Poincare截面等.同时,建立了模拟该忆阻混沌系统的SPICE(simulation program with integrated circuit emphasis)电路,给出了不同参数下的电路实验相图,其仿真结果与数值分析相符,从而验证了该忆阻混沌系统的混沌产生能力.由于脉冲同步只在离散时刻传递信息,能量消耗小,同步速度快,易于实现单信道传输,因而在混沌保密通信中更具有实用性.因此,本文从最大Lyapunov指数的角度实现了该忆阻混沌系统的脉冲混沌同步,数值仿真证实了忆阻混沌系统的存在性以及脉冲同步控制的可行性,为进一步研究该忆阻混沌系统在语音保密通信和信息处理中的应用提供了实验基础.     

一种最简的并行忆阻器混沌系统

在提出的一种压控忆阻器的基础上, 构造了最简的并联忆阻器混沌系统, 分析其动力学特性, 得到了该系统的Lyapunov指数和Lyapunov维数, 给出了时域波形、相图、Lyapunov指数谱、分岔图、Poincaré映射等. 利用EWB软件设计了该新混沌系统的振荡电路并进行了仿真实验. 研究结果表明, 忆阻器的i-v特性在参数的变化时, 并不保持斜“8”字形, 会变为带尾巴的扇形. 该混沌系统与磁控忆阻器混沌系统不同, 系统只有一个平衡点, 初始条件在系统能振荡的情况下不影响系统状态. 电路实验仿真结果和数值仿真具有很好的一致性, 证实了该系统的存在性和物理上可实现性. 关键词： 忆阻器 混沌电路 并联 动力学行为     

Design and multistability analysis of five-value memristor-based chaotic system with hidden attractors

A five-value memristor model is proposed, it is proved that the model has a typical hysteresis loop by analyzing the relationship between voltage and current. Then, based on the classical Liu–Chen system, a new memristor-based four-dimensional(4D) chaotic system is designed by using the five-value memristor. The trajectory phase diagram, Poincare mapping, bifurcation diagram, and Lyapunov exponent spectrum are drawn by numerical simulation. It is found that, in addition to the general chaos characteristics, the system has some special phenomena, such as hidden homogenous multistabilities, hidden heterogeneous multistabilities, and hidden super-multistabilities. Finally, according to the dimensionless equation of the system, the circuit model of the system is built and simulated. The results are consistent with the numerical simulation results, which proves the physical realizability of the five-value memristor-based chaotic system proposed in this paper.     

基于Chua电路的四维超混沌忆阻电路

近年来,忆阻混沌电路受到国内外学者的广泛关注,然而目前四维忆阻系统往往只存在普通混沌(仅有一个正Lyapunov指数),本文通过用忆阻替换Chua电路中电阻的新途径,得出一个简单的四维忆阻电路,与仅含有限个孤立不稳定平衡点的大多已知系统不同,本系统存在无穷多个稳定和不稳定平衡点,研究发现该系统存在着极限环、混沌、超混沌等丰富的复杂行为,通过进一步数值分析和电路仿真实验,证实了超混沌吸引子的存在,从而解决了关于四维忆阻电路是否存在超混沌的疑问。     

Integrable discrete static Hamiltonian with a parametric double-well potential

A one-dimensional discrete conservative Hamiltonian with a generalized form of the Schmidt potential, is constructed with the help of a non-integrable discrete Hamiltonian whose parametrized double-well potential can be reduced to the ?⁴ potential. The new conservative Hamiltonian is completely integrable in the discrete static regime, and the associate exact nonlinear solution is shown to coincide with the continuum nonlinear periodic solution of the non-integrable Hamiltonian. Numerical simulations and nonlinear stability analysis suggest that the discrete mapping derived from the completely integrable Hamiltonian undergoes a bifurcation which does not leads to the chaotic phase with randomly pinned states, but instead to a phase where real solutions become rare forming a cluster of periodic points around an elliptic fixed point.     

Stationary solutions of Liouville equations for non-Hamiltonian systems

We consider the class of non-Hamiltonian and dissipative statistical systems with distributions that are determined by the Hamiltonian. The distributions are derived analytically as stationary solutions of the Liouville equation for non-Hamiltonian systems. The class of non-Hamiltonian systems can be described by a non-holonomic (non-integrable) constraint: the velocity of the elementary phase volume change is directly proportional to the power of non-potential forces. The coefficient of this proportionality is determined by Hamiltonian. The constant temperature systems, canonical-dissipative systems, and Fermi-Bose classical systems are the special cases of this class of non-Hamiltonian systems.     

离子迁移忆阻混沌电路及其在语音保密通信中的应用

忆阻器是一种具有记忆功能和纳米级尺寸的非线性元件, 作为混沌系统的非线性部分, 能够使系统的物理尺寸大大减小, 同时可以得到各种丰富的非线性曲线, 提高混沌系统的复杂度和信号的随机性. 因此, 本文采用离子迁移忆阻器的磁控模型设计了一个新的混沌系统. 通过理论推导、数值仿真、Lyapunov指数谱、分岔图和Poincaré截面图研究了系统的基本动力学特性, 并分析了改变不同参数时系统动力学行为的变化. 同时, 建立了模拟该系统的SPICE电路, SPICE仿真结果与数值分析相符, 从而验证该混沌系统的混沌产生能力. 最后, 利用线性反馈同步控制方法实现了新构造的离子迁移忆阻混沌系统的同步, 并且采用该同步方法有效实现了语音信号的保密通信. 数值仿真证实了新混沌系统的存在性以及同步控制应用的可行性.     

An h-adaptive mesh method for Boltzmann-BGK/hydrodynamics coupling

We introduce a coupled method for hydrodynamic and kinetic equations on 2-dimensional h-adaptive meshes. We adopt the Euler equations with a fast kinetic solver in the region near thermodynamical equilibrium, while use the Boltzmann-BGK equation in kinetic regions where fluids are far from equilibrium. A buffer zone is created around the kinetic regions, on which a gradually varying numerical flux is adopted. Based on the property of a continuously discretized cut-off function which describes how the flux varies, the coupling will be conservative. In order for the conservative 2-dimensional specularly reflective boundary condition to be implemented conveniently, the discrete Maxwellian is approximated by a high order continuous formula with improved accuracy on a disc instead of on a square domain. The h-adaptive method can work smoothly with a time-split numerical scheme. Through h-adaptation, the cell number is greatly reduced. This method is particularly suitable for problems with hydrodynamics breakdown on only a small part of the whole domain, so that the total efficiency of the algorithm can be greatly improved. Three numerical examples are presented to validate the proposed method and demonstrate its efficiency.     

一个一维周期驱动哈密顿系统的实例及混沌控制

提出一种新的周期驱动非线性不可积哈密顿系统模型,并对其特性进行了讨论.通过简单的非反馈控制装置对这一系统进行混沌控制,将其混沌轨道分别控制在周期,准周期及指定混沌轨道上.与以往的控制方法不同的是,控制项仅是一结构简单、可调节的限位装置.为保守系统混沌控制的实际应用提供可供选择的途径     

Stabilization of coherent breathers in perturbed Hamiltonian coupled oscillators

We propose and study a type of non-Hamiltonian, power-preserving perturbation of Hamiltonian systems of nonlinear coupled oscillators with a global phase invariance symmetry. Our results highlight the role of non-conservative perturbations in selecting a preferred dynamically attracting mode which can be chosen to be highly coherent. The proof of principle described above should be of interest for optical systems such as fiber laser arrays whose objective is to produce high power coherent light.     

旋转致密双星后牛顿轨道的引力波研究

本文利用辛算法和功率谱研究旋转致密双星保守的后牛顿哈密顿系统的引力辐射, 讨论了系统的动力学参量、旋转-轨道耦合、旋转-旋转耦合效 应及轨道类型对后牛顿近似引力波形的影响. 数值结果表明有序轨道的引力波随时间呈周期性地变化, 而混沌轨道引力波的变化具有混沌性, 并且轨道的混沌特性可提高引力波的辐射能量, 尤其指出的是旋转参量大小对引力波形的变化发挥至关重要的作用.     

非保守力和非完整约束对Hamilton系统Lie对称性的影响

研究非保守力和非完整约束对Hamilton系统的Lie对称性和守恒量的影响.分别研究了Hamilt on系统受到非保守力和非完整约束作用时，系统的Lie对称性保持不变的条件，同时给出了 系统的结构方程和守恒量保持不变的条件.以著名的Emden方程和Appell-Hamel模型为例进行 了分析讨论. 关键词： 分析力学 Hamilton系统 非保守力 非完整约束 对称性 守恒量     

Quantum signatures of chaos or quantum chaos?

A critical analysis of the present-day concept of chaos in quantum systems as nothing but a “quantum signature” of chaos in classical mechanics is given. In contrast to the existing semi-intuitive guesses, a definition of classical and quantum chaos is proposed on the basis of the Liouville–Arnold theorem: a quantum chaotic system featuring N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) specified by the symmetry of the Hamiltonian of the system. Quantitative measures of quantum chaos that, in the classical limit, go over to the Lyapunov exponent and the classical stability parameter are proposed. The proposed criteria of quantum chaos are applied to solving standard problems of modern dynamical chaos theory.     

Three types of chaos in the forced nonlinear Schrödinger equation

Three different types of chaotic behavior and instabilities (homoclinic chaos, hyperbolic resonance, and parabolic resonance) in Hamiltonian perturbations of the nonlinear Schr?dinger (NLS) equation are described. The analysis is performed on a truncated model using a novel framework in which a hierarchy of bifurcations is constructed. It is demonstrated numerically that the forced NLS equation exhibits analogous types of chaotic phenomena. Thus, by adjusting the forcing frequency, the behavior near the plane wave solution may be set to any one of the three different types of chaos for any periodic box length.     

Non-linear diffusion in Hamiltonian systems exhibiting chaotic motion

The exact evolution equation for the angle averaged phase space density in action-angle space is derived from the Liouville equation using projection operator techniques. This equation involves a correlation function of the initial value of the phase space density with the angle dependent part of the Hamiltonian and a correlation function of the angle dependent part of the Hamiltonian and a correlation function of the angle dependent part of the Hamiltonian with itself. Each of these correlation functions develops in time with angle projected dynamics. We show their relation to the correlation functions which develop in time with usual Hamiltonian dynamics. These correlation functions are then studied in the standard model of Chirikov, and we conclude that they behave as $\text{e}{}^{\mathrm{-\sigma t}}\text{cos}\text{(Ωt + φ)}$ in regions of irregular motion. We conjecture that angle averaged correlation functions behave this way in general, and we give an argument based on the mixing property of the Hamiltonian system. Our argument goes beyond the usual mixing, so we regard it as a quasi-mixing hypothesis. Under this hypothesis the equation for the angle averaged phase space density becomes a diffusion equation which incorporates much of the non-linear dynamics of Hamiltonian systems exhibiting chaotic motion.     

基于哈密顿函数的永磁同步电机混沌系统鲁棒控制

哈密顿系统理论是研究非线性系统的一种重要工具, 近年来在电机调速、控制等方面得到广泛应用. 本文针对永磁同步电机运行中存在的混沌现象, 提出一种基于哈密顿函数的永磁同步电机混沌系统鲁棒控制器设计方法. 将永磁同步电机动态模型变换为类Lorenz混沌方程, 在特定参数下, 通过Lyapunov指数和Lyapunov维数的计算可知系统是混沌的. 令电机转速跟踪给定值得误差方程. 由于误差方程并不具有标准哈密顿函数形式, 将其转化为具有扰动不确定项的哈密顿系统, 并与负载扰动一起作为系统的总扰动量, 设计了一种鲁棒控制器. 控制器由两部分组成, 一部分基于互联与阻尼配置法, 实现任意转速的有效跟踪, 另一部分实现扰动补偿. 仿真表明, 控制器使电机迅速脱离混沌状态, 并能实现转速趋近跟踪, 验证了控制器的可行性与有效性. 该方法扩展了哈密顿函数的适用范围, 具有一定的优越性.