复系统Julia集的同步

非线性系统中的分形集——Julia集, 在工程技术中有着十分重要的应用,定义了不同系统间的Julia集同步的概念, 并引入一种非线性耦合的方法, 对同一系统不同参数的Julia集进行了有效的同步.并以多项式形式和三角函数形式的Julia集同步为例验证了该方法的有效性. 关键词： Julia集 同步 分形     

广义Julia 集的参数辨识

本文研究了一类相当广泛的复系统Julia 集的参数辨识问题.基于非线性反馈控制器和差分方程稳定性理论, 设计了普遍适用的自适应同步控制器和参数自适应律的解析表达式.理论证明设计的控制器可使得此类广义复系统Julia 集达到同步,并且可以辨识广义Julia 集的未知参数.通过仿真实例验证了该方法的有效性.另外, 本文特别地讨论了最基本的Julia 集的参数辨识问题. 关键词： Julia集 参数辨识 同步     

空间交替Julia集的反馈控制与线性广义同步

讨论了由空间交替系统迭代产生的空间交替Julia集.空间交替Julia集是由交替的复动力系统zm+1,n+azm,n+1=(1+a)2z2mn+ci,i=1,2迭代产生的,且通过辅助反馈控制方法实现了对空间交替Julia集的控制.同时利用线性反馈的方法,实现了不同空间交替Julia集之间的线性广义同步.仿真结果验证了控制和同步方法的有效性.     

基于Julia分形的多涡卷忆阻混沌系统

忆阻器作为一种非线性电子元件,能用作混沌系统中的非线性项,从而提高系统的复杂度.分形与混沌是密切相连的,分别对两者的研究都已成熟,却鲜有将分形过程应用到混沌系统中,以产生丰富的混沌吸引子.为了探索将分形与混沌系统相结合的可能性,本文首先提出了一个新的忆阻混沌系统,并从对称性、耗散性、平衡点稳定性、功率谱、Lyapunov指数和分数维等方面探讨了系统的动力学特性;紧接着,把经典的Julia分形过程应用到该忆阻混沌系统中,产生了新的混沌吸引子,并将几种由Julia分形衍生的变形Julia分形过程应用于文中提出的忆阻混沌系统,获得了丰富的混沌吸引子;最后,讨论了分形过程中的复常数对系统的影响.从仿真结果可以看出,分形过程与混沌系统的结合能产生丰富的多涡卷混沌吸引子.这不仅为产生多涡卷混沌吸引子提供了一种新方法,还弥补了使用功能函数方法造成混沌系统不光滑的不足.     

势垒隧穿含时演化的Julia数值模拟

势垒隧穿是初等量子力学中的一个重要模型,但由于求解其波函数涉及超越方程,因此在许多初等量子力学教材中往往着重对透射系数的讲解,很少提及其波函数演化,在部分教材中虽有提及,但往往采用图解法,不利于初学者对该过程的理解。本文针对这一问题,提出了一种便于初学者理解的数值计算方法。该方法根据矩阵力学的向量化思想,将薛定谔方程中的波函数与算符分别以向量和矩阵的形式进行离散化,并利用Julia编程对几种势垒情况下的波函数隧穿的含时演化进行数值模拟。     

噪声诱导的二维复时空系统的同步研究

研究了一类噪声诱导的二维复时空系统的同步问题.首先讨论了二维复Ginzburg-Laudau(CGL) 方程随时间和空间变化的时空混沌特性;其次,研究了时空噪声驱动下CGL系统的同步问题.理论上利用线性稳定性分析,得到了常数激励下CGL系统达到稳定态的临界强度;结合噪声的随机性和非零均值特性, 揭示了噪声诱导同步的机理;并从理论上和数值上分别给出了达到同步所需要的控制参数和噪声强度满足的条件,实现了两个非耦合CGL系统的完全同步.结果表明,数值模拟和理论分析有很好的一致性.     

定义在分形集上函数的H-导数

 分形的生成含有降维与升维两种形式,采用适当的方式总可以对分形集加以定向化,分形的内禀测度是Hausdorff测度。因此,定义在分形集上的函数可以看作是在Hausdorff测度下定义在区间[0, h]。在函数图像也是分形的前提下,依函数的Hausdorff测度的定义,给出了定义在分形集上的函数f(p)关于Hausdorff测度p的导数以及一些性质。     

A new identification control for generalized Julia sets

In this paper, we propose a new method to realize drive-response system synchronization control and parameter identification for a class of generalized Julia sets. By means of this method, the zero asymptotic sliding variables are applied to control the fractal identification. Furthermore, the problems of synchronization control are solved in the case of a drive system with unknown parameters, and the unknown parameters of the drive system can be identified in the asymptotic synchronization process. The results of simulation examples demonstrate the effectiveness of this new method. Particularly, the basic Julia set is also discussed.     

基于LaSalle不变集定理自适应控制永磁同步电动机的混沌运动

基于LaSalle不变集定理,设计自适应混沌控制器对永磁同步电动机中的混沌进行控制.由于该控制器能自动跟踪平衡点,无需预先知道系统平衡点位置,不需要改变受控系统原有的结构, 因此设计直接简便,控制代价小,易于工程实现.仿真结果表明了该方法的有效性. 关键词： LaSalle不变集定理 自适应律 混沌控制 永磁同步电动机     

不同结构混沌系统的自适应同步和反同步

针对不同结构混沌系统的同步与反同步问题进行了研究.在系统参数已知时,采用主动控制法实现混沌系统的同步与反同步,并将主动控制器的设计方法进行了推广.在参数未知时,基于Lyapunov稳定性理论和自适应控制方法,给出了自适应控制器和参数自适应律,实现了参数均未知且结构不同的驱动系统和响应系统的同步与反同步.在控制器的设计过程中,将驱动系统和响应系统进行互换,讨论了互换前后的控制器和自适应律之间的关系.数值仿真结果说明了所提出设计方法的有效性. 关键词： 混沌同步 反同步 主动控制法 自适应控制法     

Gradient control and synchronization of Julia sets

This paper firstly introduces the control methods to fractals and give the definition of synchronization of Julia sets between two different systems. Especially, the gradient control method is taken on the classic Julia sets of complex quadratic polynomial Zn＋1 = zn^2＋ c, which realizes its Julia sets control and synchronization. The simulations illustrate the effectiveness of the method.     

Control and synchronization of second Julia sets

A visualization of Julia sets of the complex Henon map system with two complex variables is introduced in this paper.With this method,the optimal control function method is introduced to this system and the control and synchronization of its Julia sets are achieved.Control and synchronization of generalized Julia sets are also achieved with this optimal control method.The simulations illustrate the efficacy of this method.     

Geometry and combinatorics of Julia sets of real quadratic maps

For real a correspondence is made between the Julia setB forz(z–)², in the hyperbolic case, and the set of-chains±(±(±..., with the aid of Cremer's theorem. It is shown how a number of features ofB can be understood in terms of-chains. The structure ofB is determined by certain equivalence classes of-chains, fixed by orders of visitation of certain real cycles; and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of-chains. The functional equations obeyed by attractive cycles are investigated, and their relation to-chains is given. The first cascade of period-doubling bifurcations is described from the point of view of the associated Julia sets and-chains. Certain Julia sets associated with the Feigenbaum function and some theorems of Lanford are discussed.Supported by NSF grant No. MCS-8104862.Supported by NSF grant No. MCS-8203325.     

Meblin transforms associated with Julia sets and physical applications

We introduce the Mellin transform of the balanced invariant measure associated to the Julia set generated by a rational transformation. We show that its analytic continuation is a meromorphic function, the poles of which are on a semi-infinite periodic lattice. This allows one to have an understanding of the behavior of the measure near a repulsive fixed point. Trace identities corresponding to the fact that the analytically continued Mellin transform vanishes at negative integers are derived for the polynomial case. The quadratic map is first analyzed in detail, and the analytic properties of the inverse of the Green's function are exhibited. Of interest is the appearance of a dense set of spikes at dyadic points when the Julia set is disconnected. These results are used to study the residues of the Mellin transform. A certain number of physically interesting consequences are derived for the spectral dimensionality of quantum mechanical systems, the excitation spectrum of which displays unusual oscillations. The appearance of complex critical indices for thermodynamical systems is also discussed in the conclusion.Supported in part by a N.A.T.O. Postdoctoral fellowship.     

Iterated networks and the spectra of renormalizable electromechanical systems

Beginning with anLRC network with impedance functionZ(), a sequence of iterated networksN _k with impedance functionsZ ^k(), k= 1, 2, 3,..., is introduced. The asymptotic comportment ofZ ^k() and the spectra ofN _k are analyzed in terms of the Julia set ofZ. An example is given of an iterated network associated with a cascade of period-doubling bifurcations.     

Synchronization of oscillators in complex networks

Theory of identical or complete synchronization of identical oscillators in arbitrary networks is introduced. In addition, several graph theory concepts and results that augment the synchronization theory and a tie in closely to random, semirandom, and regular networks are introduced. Combined theories are used to explore and compare three types of semirandom networks for their efficacy in synchronizing oscillators. It is shown that the simplest k-cycle augmented by a few random edges or links are the most efficient network that will guarantee good synchronization.