Fractal Control of Planar Complex Dynamical Systems

Abstract

Until now, there are lots of outcomes about fractal basic theory, especially for the studies of Julia sets in fractal. However, the focus of their discussions are on the drawing of graphics and qualitative characters of Julia sets for various types of functions. In fact, the performances of Julia sets need to be changed continually involved extensive applications of Julia sets in engineering and technology, which is the control problem of Julia sets, that is fractal control. This chapter firstly introduces the control ideas and methods into fractal controls of Julia sets, and achieves expected targets of various performance by using various methods [324–375]. Thus, it helps us to comprehend applications of Julia sets in various fields and explain variety of related phenomenons, such as the basic comprehension for the famous issue of small denominators which makes the Hamiltonian mechanics distressed, as well as the basic comprehension for scaling behaviors of the circumferential mappings, moreover, the modeling about beginnings of chaos performed smoothly in many practical physical scenarios. With the continuous deepening of fractal theory in engineering and technology as well as the actual requirements, systems need to represent different behaviors or properties according to different requirements, nonlinear behaviors which have such requirements, that is the control of Julia sets.