Research On Fractal Structure Of Noise Perturbations Of Generalized M-J Sets

Noneliner science is a new crossed discipline, which concerns the common properties of nonlinear phenomena. It nearly involves to each domain of natural sciences and social sciences. Chaos theory, Fractal theory and Soliton theory make up of theoretical basis of non-linear science together. Fractal is of great challenge and bright application future in nonlinear science. Then research on fractal has both important theoretical significance and extensive applied value.As one brand-new world view and methodology, fractal theory deeply opens out unification of regularity-irregularity and determinism-randomicity. Speaking of one monomer fractal set, it is one entia of regularity-irregularity, determinism-randomicity and chaotic-fractal. As for determinate fractal sets (e.g. generalized M-J sets) and stochastic fractal sets (e.g. brownian motion), whether between of them have some kind of inner link each other?Is precisely ponders based on above, based on the analyses and studies of the generalized M-J sets fractal theory and noise theory, the stochastic noise was introduced to research of the generalized M-J sets. The method constructing stochastic perturbed M-J set from the quadriccomplex mapping z ←z~2 +c developed by Argyris, Karakasidis and Andreadis was expanded. According to the complex mapping z ←z~a + c(a∈R) expanded by the author, a series of the stochastic perturbed generalized M-J sets for real index number were constructed. Definitions of the stochastic perturbed generalized M-J sets are theoretically produced separately. Using the experimental mathematics method combing the theory of analytic function of one complex variable with computer aided drawing, the fractal features and evolutions of the stochastic perturbed generalized M-J sets are studied. The following conclusions are shown:1. On the structure characteristics and the discontinuity evolution law of the additive and multiplicative noise perturbed generalized 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.2. On the structure characteristics and the discontinuity evolution law of the additive and multiplicative noise perturbed generalized J sets was studied. On the influence of stochastic perturbed parameters of the structure of generalized J sets was analyzed. The physical meaning of the additive noise perturbed generalized J sets was expounded. The followingconclusions are shown: (1) Characteristics of anti-noise of the generalized J sets correlates to the index of a value. When the noise strength is invariable, a value is smaller, anti-noise characteristics of the generalized J sets are weaker; Otherwise, anti-noise characteristics of the generalized J sets are stronger. (2) Under the low strength noise perturbation, fractal growth of the generalized J sets is stable, along with the noise strength increases, fractal growth of the generalized J sets from the stable state transits to the chaotic state. This indicates the Brownian motion may be chaotic; In the chaotic area that periodic windows exist indicates again the Brownian motion has a characteristic of organizational structure and highly regularity.Above studies related to the papers have contributed to Applied Mathematics A Journal of Chinese Universities Series A and Numerical Mathematics A Journal of Chinese University, and so on the publications.