A Kind Of Rational Fractal In Control

Fractal is an active branch of the modern mathematics and nonlinear science and is also geometrical description of chaotic motion. The nature of the world is nonlinear, and chaotic phenomena exist everywhere, so the application of fractal geometry is very extensive and is a hot topic of current research.Julia set theory is an important aspect of the fractal theory and is an important study for the non-linear theory. The study of plane Julia set is relatively mature, and that one of spatial Julia set is relatively little. Especially, the study about spatial Julia sets of rational function is much less. Theoretical analysis of the Rational Fractal, controls and spatial simulations are our research objects. We will apply three control methods to achieve controls of Rational Fractal. Mainly involves the following aspects:One The definition and simulation about Julia sets from Rational FractalFirstly, we give the definition of Rational Fractal. Also the Rational Fractal has two variables, so it means there are four variable. In order to portray the Julia set via its simulations, we fix one of the variables. So we can observe the Julia set from one side and it is simulated in the three-dimensional space.Two Control theory of Julia sets from Rational FractalFrom the definition of Rational Fractal, we know the bounded orbits of the system play a very important role in the construction process of the Julia set. We can achieve controls of Julia set via the boundedness of iterative rails. So we can consider the stability of the fixed points. It means that we can make the unstable fixed points stable. Hence, the control purpose is realizied.Three Control methods of Julia sets from Rational FractalBased on the control purpose of Julia sets from Rational Fractal, we adopted three control methods to achieve the control of the Julia sets, which also are used to achieving chaos control, such as auxiliary reference feedback control, gradient control and optimal function control. According to the characteristics of Rational Fractal, we discuss the cases of b=0 and b≠0. According to the theoretical analysis, we verify their feasibility and effectiveness via the simulation from the three methods.Finally, we point out the insufficiency about the thesis, propose some problems that need to be solved further and prospect the development of Julia sets from Rational Fractal.