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.     

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.     

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.     

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     

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.     

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     

Percolation and Julia sets

The types of Julia sets for the renormalization group (RG) transformation for percolation on the hierarchial model are derived. The RG transformation for the concentration for two-dimensional triangular lattice site percolation problems and two-dimensional square lattice bond percolation problems are generalized to the complex plane. Julia sets for both cases are found.     

Strange objects in the complex plane

Julia sets are examined as examples of strange objects which arise in the study of long time properties of simple dynamical systems. Technically they are the closure of the set of unstable cycles of analytic maps. Physically, they are sets of points which lead to chaotic behavior. The mapf(z)=z²+p is analyzed for smallp where the Julia set is a closed curve, and for largep where the Julia set is completely disconnected. In both cases the Hausdorff dimension is calculated in perturbation theory and numerically. An expression for the rate at which points escape from the neighborhood of the Julia set is derived and tested in a numerical simulation of the escape.     

Multi-goal Control of Chaotic Connected Complex Networks

Beam transport network （BTN） with small world （SW） （so-called BTN-SW） and Lorenz chaotic connected network with scale-free （SF） are taken as two typical examples, we proposed a global linear coupling and combined with local error feedback methods in sub-networks to realize multi-goal control method of halo and chaos in two networks above. The simulation results show that the methods above is effective for any chaotic connected networks and has a potential of applications in based-halo-chaos secure communication.     

On Computational Complexity of Siegel Julia Sets

It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high. The first and third authors are partially supported by NSERC Discovery grants. The second author is partially supported by NSERC Postgraduate Scholarship     

Modified Projective Synchronization of a New Hyperchaotic System via Nonlinear Control

In this paper, a nonlinear control scheme of two identical hyperchaotic Chert systems is developed to realize their modified projective synchronization. We achieve modified projective synchronization between the two identical hyperchaotic systems by directing the scaling factor onto the desired value. With symbolic computation system Maple and Lyapunov stability theory, numerical simulations are given to perform the process of the synchronization.     

On Connectivity of Julia Sets of Yang--Lee Zeros

We prove that the Julia sets of the Yang–Lee zeros of the Potts model on the diamond hierarchical lattice are connected sets on the complex plane. Received: 23 May 2000 / Accepted: 9 May 2001     

Reproducing Kernels and Coherent States on Julia Sets

We construct classes of coherent states on domains arising from dynamical systems. An orthonormal family of vectors associated to the generating transformation of a Julia set is found as a family of square integrable vectors, and, thereby, reproducing kernels and reproducing kernel Hilbert spaces are associated to Julia sets. We also present analogous results on domains arising from iterated function systems. The research of the first two authors was supported by Natural Sciences and Engineering Research Council of Canada.     

Constructing Locally Connected Non-Computable Julia Sets

A locally connected quadratic Siegel Julia set has a simple explicit topological model. Such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution. We constructively produce parameter values for Siegel quadratics for which the Julia sets are non-computable, yet locally connected. This research was partially conducted during the period the first author was employed by the Clay Mathematics Institute as a Liftoff Fellow. The second author’s research is supported by NSERC operating grant.     

Feedback control and synchronization of Mandelbrot sets

The movement of a particle could be depicted by the Mandelbrot set from the fractal viewpoint. According to the requirement, the movement of the particle needs to show different behaviors. In this paper, the feedback control method is taken on the classical Mandelbrot set. By amending the feedback item in the controller, the control method is applied to the generalized Mandelbrot set and by taking the reference item to be the trajectory of another system, the synchronization of Mandelbrot sets is achieved.     

Unraveling the primary mechanisms leading to synchronization response in dissimilar oscillators

We study how the phenomenon of response to synchronization arises in sets of pulse-coupled dissimilar oscillators. One of the sets is constituted by oscillators that can easily synchronize. Conversely, the oscillators of the other set do not synchronize. When the elements of the first set are not synchronized, they induce oscillation death in the constituents of the second set. By contrast, when synchronization is achieved in oscillators of the first set, those of the second set recover their oscillatory behavior and thus, responding to synchronization. Additionally, we found another interesting phenomenon in this type of systems, namely, a new control of simultaneous firings in a population of similar oscillators attained by means of the action of a dissimilar oscillator.     

Timing system of HIRFL-CSR

The national science project HIRFL-CSR has recently been officially accepted. As a cyclotron and synchrotron complex, it puts some particularly high demands on the control system. There are hundreds of pieces of equipment that need to be synchronized. An integrated timing control system is built to meet these demands. The output rate and the accuracy of the controller are 16 bit/μs. The accuracy of the time delay reaches 40 ns. The timing control system is based on a typical event distribution system, which adopts the new event generation and the distribution scheme. The scheme of the timing control system with innovation points, the architecture and the implemented method are presented in the paper.     

Generalized dimensions,entropies, and Liapunov exponents from the pressure function for strange sets

For conformal mixing repellers such as Julia sets and nonlinear one-dimensional Cantor sets, we connect the pressure of a smooth transformation on the repeller with its generalized dimensions, entropies, and Liapunov exponents computed with respect to a set of equilibrium Gibbs measures. This allows us to compute the pressure by means of simple numerical algorithms. Our results are then extended to axiom-A attractors and to a nonhyperbolic invariant set of the line. In this last case, we show that a first-order phase transition appears in the pressure.     

On the iteration of a rational function: Computer experiments with Newton's method

Using Newton's method to look for roots of a polynomial in the complex plane amounts to iterating a certain rational function. This article describes the behavior of Newton iteration for cubic polynomials. After a change of variables, these polynomials can be parametrized by a single complex parameter, and the Newton transformation has a single critical point other than its fixed points at the roots of the polynomial. We describe the behavior of the orbit of the free critical point as the parameter is varied. The Julia set, points where Newton's method fail to converge, is also pictured. These sets exhibit an unexpected stability of their gross structure while the changes in small scale structure are intricate and subtle.     

Experimental results on quadratic maps using generalized complex numbers

Summary Some results obtained introducing two-unit generalized complex numbers to determine Mandelbrot and Julia sets for quadratic maps are presented. We also highlight some analogies and differences between classic complex numbers and one of the proposed extensions. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Capanello, Italy, July 4–8, 1994.