Control And Application Of Fractal Growth

Fractal growth theory is one of important parts of the fractal theory. There are many real growth phenomenon which can be explained by fractal theory, for example, aggregation of soot, formation of lighting and diffusion and aggregation of cancer cells. Therefore, the theory of fractal growth and its application have theoretical significance and useful value.However, many complex environmental disturbance, such as variation of magnetic field, temperature or distribution of latent heat, etc., will have complex effect on the aggregation of aggregation growth in reality so that the complex variation of growth mor-phology and quick growth velocity of growth particles bring deleterious effect for nature and human society. Hence, the prediction and control of the morphology of aggregation growth and growth velocity under complex disturbance become important in reality. Un-til now, most works focus on the methods that adjust some parameters of simulations or experiments to control the morphology or growth velocity of aggregation growth. Fewr study the prediction and control of the morphology of aggregation growth and growth velocity from the point of mathematical model analytically.In this paper, we achieve the following details:1 Control and Application of Fractal Growth with Environmental Distur-banceBased on the two famous models, Diffusion Limited Aggregation (DLA) model and Dielectirc Breakdown (DB) model, the control system of fractal growth with environmen-tal disturbance is given and then the quantitative relationship between the aggregation probability (or aggregation concentration) and environmental disturbance is obtained, which predicts that the morphology of fractal growth will be controlled to aggregate in the disturbance regions. The prediction is verified by simulation choosing the cycle con-cussion function as the nonlinear term and the constant or random number as the source term. Furthermore, the variation of scaling dimension of growth morphology illustrates that the environmental disturbance has effect on the complexity of real growth. In addi-tion, we use the quantitative relationship to control the fractal growth of thermal diffusion of thin plate. The conclusion provides proper theoretical foundation for understanding deeply the physical mechanism of non-equilibrium growth in reality and application in many fields, such as physics, biology, medicine and materials science, etc.2 Directed Control and Application of Fractal GrowthFor the fractal growth in real environment, we obtain the directed control model of fractal growth. Using the norm theory, a quantitative relationship for generalized function as the environmental disturbance is acquired. which predicts that under some certain conditions, the source term can suppress the nonlinear term, meaning that the source term can control the growth particles to aggregate in different direction or region named directed control. For to simulation, we choose cycle concussion function or quadratic polynomial function as the nonlinear term and constant as the source term. In addition, the fractal dimension of round region of fractal growth is larger than that of piecewise region of fractal growth, meaning that the complexity of latter is smaller than that of former. Furthermore, the directed control relationship is used to control the fractal growth of thermal diffusion of thin plate. Simulations indicate that the method is effective. We believe that the conclusion will contribute to further understand and apply the fractal growth theory in many related fields.3 Control of Velocity of Fractal GrowthWe introduce a reaction-diffusion system to analyze the effect of different noise on growth velocity in reality. Using the related theory of functional partial differential, we ob-tain the quantitative conclusions that different noise can make the growth velocity tend to zero,to constant, or tend to infinity at different velocity which predict that the noise has direct influence on velocity of fractal growth. Simulations taking noise as constant func-tion,inverse function,cycle concussion function exponential function or power function, respectively, validate the feasibility of these conclusions. Also conclusions supply proper theory foundation for controlling the real related growth velocity and understanding the erupting growth phenomenon.4 Fractal Growth and Spatial Chaos and Bifurcation Behavior of Inter-laminar Displacement in Composite MaterialsIn addition, the one dimension Logistic system which is nonlinear and discrete, will be chaos when the parameters of this system in some field change. And the bifurcation behavior is one characteristic of the chaos. The self similar is shown in the bifurcation behavior of the chaos, and then the bifurcation is part of the fractal growth. So that we obtain the elastic system of interlaminar displacement of composite materials and study its spatial chaos and bifurcation behavior using the quantitative relations between displacemn,tstrain and stress in theory of elastic mechanics. We hope that the nonlin-ear behavior analysis will be useful for understanding the destabilization of composite material.