Xingyuan Wang

Module-phase synchronization in complex dynamic system

The concept of module-phase synchronization was proposed. The chaos synchronization between drive system and response system was achieved in module space and phase space respectively (module-phase synchronization). Different from the evolutions in real space, there is no pseudorandom behavior in phase space when module-phase synchronization achieve. All the phases of complex state variables switched between two fixed values which are determined by initial values of drive system. And the modules varied within a bounded field. The theoretical analysis and the simulations were also given.

Additive perturbed generalized Mandelbrot–Julia sets

Abstract Adopting the experimental mathematics method combining complex variable function theory with computer aided drawing, this paper researches on the structural characteristic and the fission-evolution law of additive perturbed generalized Mandelbrot–Julia sets (generalized M–J sets in short). The corresponding relationship between point coordinates in generalized M set and the general structure of generalized J sets has been founded qualitatively and the physical meaning of the generalized M–J sets has been expounded. The following conclusions are deduced: (1) Chaotic patterns of fractal structure of generalized J sets may emerge out of double-periodic bifurcation, which shows that Brownian movement can be chaotic. (2) Experimental evidence of Li–Yorke theorem is given out. (3) The additive perturbed generalized M set contains abundant information on the construction of generalized J sets. (4) Resemble logistic map, in the process of a series of double-periodic bifurcation coming into chaos, generalized J sets also present self-similarity in parameter space.

Research on fractal structure of generalized M–J sets utilized Lyapunov exponents and periodic scanning techniques

Abstract In order to show details of fractal structure of Mandelbrot set precisely, Lyapunov exponents and periodic scanning techniques have been brought forward by Shirriff and Welstead. This paper generalizes these two techniques and puts forward periodicity orbit search and comparison technique which can be used to discuss the relationship of the generalized Mandelbrot–Julia sets (the generalized M–J sets). Adopting the techniques mentioned above and the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, this paper researches on the structure topological inflexibility and the discontinuity evolution law of the generalized M–J sets generated from the complex mapping z xa0→xa0 z α xa0+xa0 c ( α xa0∈xa0 R ), and explores structure and distributing of periodicity “petal” and topological law of periodicity orbits of the generalized M sets, and finds that the generalized M set contains abundant information of structure of the generalized J sets by founding the whole portray of the generalized J sets based on the generalized M set qualitatively. Furthermore, the physical meaning of the generalized M–J sets have been expounded.

Topology Identification and Module–Phase Synchronization of Neural Network With Time Delay

This paper presents the module–phase synchronization for a class of neural networks with weights identification and time delays. In module–phase synchronization, complex-valued node states are taken into consideration. The topology weights considered here are uncertain and the time delays are bounded. By constructing a Lyapunov–Krasovskii functional and employing adaptive feedback control, sufficient conditions for module–phase synchronization are derived. After the general synchronization theory, the synchronization with topology identification is discussed. Finally, pertinent examples are given to demonstrate the effectiveness of the obtained results.

Module-phase synchronization in hyperchaotic complex Lorenz system after modified complex projection

Abstract This paper studies the modified complex projective synchronization which has plural projective factors and is meaningful to complex systems, and applies this kind of synchronization in a hyperchaotic complex Lorenz system. Based on the proposed synchronization, we also study the hybrid synchronization which contains modified complex projection and module-phase synchronization because hybrid synchronization not only deal with the synchronization real part and imaginary part, respectively, but also take module and phase of complex system into consideration. Owing to it is difficult to design the controller directly, our work involve an intermediary system. The asymptotic convergence of the errors between the states is proven and the computer simulation results present the effectiveness of our method.

Julia sets for the standard Newton’s method, Halley’s method, and Schröder’s method

Abstract We analyze the Julia sets theory for the standard Newton’s method, Halley’s method, and Schroder’s method, study structural characteristics of Julia sets, show the properties and conditions of fixed points for the standard Newton’s method, Halley’s method, and Schroder’s method. We observe the relation between roots of polynomial and Julia sets structure, namely if keeping the relative position of the roots of a polynomial invariable, then the topological structure is also invariable. If there is an extraneous fixed point, then the extraneous fixed point is also invariable. Otherwise, the structures of Julia sets and fixed points would be changed.

Stability and synchronization for discrete-time complex-valued neural networks with time-varying delays.

In this paper, the synchronization problem for a class of discrete-time complex-valued neural networks with time-varying delays is investigated. Compared with the previous work, the time delay and parameters are assumed to be time-varying. By separating the real part and imaginary part, the discrete-time model of complex-valued neural networks is derived. Moreover, by using the complex-valued Lyapunov-Krasovskii functional method and linear matrix inequality as tools, sufficient conditions of the synchronization stability are obtained. In numerical simulation, examples are presented to show the effectiveness of our method.

A method of quickly calculating the number of pinning nodes on pinning synchronization in complex networks

In this paper, we propose one adaptive pinning synchronization scheme in complex networks, whose adaptive law is simple and flexible. Most of all, we find the decreasing law of maximum eigenvalues of the principal submatrices for coupling matrix, and give a method of quickly calculating pinning nodes in complex networks with binary search algorithm. Furthermore, we extend the method to the pinning synchronization scheme with linear feedback control. Numerical simulations give effectiveness of the adaptive pinning synchronization in a scale-free network, and the relations between maximum eigenvalue sequence of the principal submatrices and the number of pinning nodes under the conditions of three pinning strategies in a scale-free network and a small-world network, respectively.

Dynamics of the generalized M set on escape-line diagram

The basic contour diagrams of the M set are briefly reviewed. With the method combining escape-line diagram of the generalized M sets and bifurcation diagram of a family of one-dimensional maps, the dynamics of a family of one-dimensional maps are studied and the graphic method to determine the period of midgets corresponding to the map is given.

Generalized Julia sets from a non-analytic complex mapping

Abstract The method constructing the Julia sets from a simple non-analytic complex mapping developed by Michelitsch and Rossler was expanded. According to the complex mapping expanded by the author, a series of the generalized Julia sets for real index number were constructed. Using the experimental mathematics method combining the theory of analytic function of one complex variable with computer aided drawing, the fractal features and evolutions of the generalized Julia sets are studied. The results show: (1) the geometry structure of the generalized Julia sets depends on the parameters α , R and c ; (2) the generalized Julia sets have symmetry and fractal feature; (3) the generalized Julia sets for decimal index number have discontinuity and collapse, and their evolutions depend on the choice of the principal range of the phase angle.