Adopting the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, in this paper on the structure characteristics and the discontinuity evolution law of the additive noise perturbed generalized Mandelbrot sets (M-sets) was studied. On the influence of stochastic perturbed parameters of the structure of generalized M-sets was analyzed. The physical meaning of the additive noise perturbed generalized M-sets was expounded. IntroductionIn recent 20 years, people have lucubrated on the generalized Mandelbrot-sets (M-sets in short) constructed from theand found there existed orderly structure in it [1,2]. For example, Gujar et al. proposed several assumptions based on the visually structural characteristics of generalized M-sets [3]; Glynn found the symmetrical evolution of generalized M-sets when the phase angle θ ∈ [−π , π) [4]; the authors put forward the embedded topological distribution theorem of the generalized M-sets and discussed the different selections of the principal value range of the phase angle θ and the fission-evolution rule of the generalized M-sets for decimal index number [5]; Sasmor analyzed the fission of generalized M-sets for rational index number when the phase angle θ ∈ [−π , π) [6]; Romera and Pastor et al. researched on the embedded-layer relationship of bud of generalized M-sets in the Misiurewicz points [7,8]; Geum and the authors both studied the structure and distribution of the periodic bud of generalized M-sets and the topological rule of their periodic trajectories [9,10]; Beck and the authors both discussed the physical meaning of generalized M-sets [11,12]. Furthermore, Argyris, Lasota, Kapitaniak, et al. discussed the classification and affection of noise in complex dynamical system [13-15]; Argyris et al. studied the structural characteristic of M-sets containing noise after importing additive noise and multiplicative noise into the complex map z n+1 = z 2 n + c [16-18]. Based on above researches, this paper studied the structural characteristic and fission-evolution law of stochastic perturbed generalized M-sets, analyzed the influence of stochastic perturbed parameters of the structure of generalized M-sets and expatiated on the physical meaning of a type of generalized M-sets.

In this paper a fast fractal coding method based on fractal dimension is proposed. Image texture is an important content in image analysis and processing which can be used to describe the extent of irregular surface. The fractal dimension in fractal theory can be used to describe the image texture, and it is the same with the human visual system. The higher the fractal dimension, the rougher the surface of the corresponding graph, and vice versa. Therefore in this paper a fast fractal encoding method based on fractal dimension is proposed. During the encoding process, using the fractal dimension of the image, all blocks of the given image first are defined into three classes. Then each range block searches the best match in the corresponding class. The method is based on differential box counting which is chosen specifically for texture analysis. Since the searching space is reduced and the classification operation is simple and computationally efficient, the encoding speed is improved and the quality of the decoded image is preserved. Experiments show that compared with the full search method, the proposed method greatly reduced the encoding time, obtained a rather good retrieved image, and achieved the stable speedup ratio.

In this paper a chaotic system is proposed via modifying hyperchaotic Chen system. Some basic dynamical properties, such as Lyapunov exponents, fractal dimension, chaotic behaviors of this system are studied. The conventional feedback, linear function feedback, nonlinear hyperbolic function feedback control methods are applied to control chaos to unstable equilibrium point. The conditions of stability to control the system is derived according to the Routh–Hurwitz criteria. Numerical results have shown the validity of the proposed schemes.