Research On Complex Dynamical System Of Time-delay Model And McMullen Family Maps

Since an American mathematician Mandelbrot proposed "fractal" in 1975, the fractal has already become a hot subject. Many scientists have solved some problems and phenomenona which can not be done by traditional subjects in the nature. According to the computer technology, specially for the computer graphic, we can make complex and beautiful fractal graphs on the computer. Now, it is a primary way to study the fractal theory using complex dynamical system theory and computer graphic. We study the nature of complex dynamical system of time-delay model and McMullen family maps as below.Firstly, we research complex time-delay dynamical systems of quadratic polynomials mapping. We study the time-delay mapping, and make short-lived time-delay and sustained time-delay both happen on ordinate axis and abscissa axis. The time when the time-delay happens is initial status, unstable status and stable status. We make the Julia set using the escape time algorithm. After numerical experiments and analysis, stable conditions of complex dynamic systems when time-delay happens on ordinate axis and abscissa axis ordinate axis and abscissa axis are given. At the same time, we get some properties of no time-delay Julia sets according the study of time-delay Julia sets.Secondly, period region distributions of McMullen family M sets are discussed. The methods of the calculation of the numbers of period-n steady regions are presented and the center points and boundaries of period-1 are studied by complex dynamical system theory. In addition, the problem about the free critical points is discussed. The experimental results show that the free critical points do not influence the distributions of period-n steady regions when m=d, which is proved theoretically. Then, analyses of the influence of period-1 steady regions by the free critical points are given.The last, McMullen family Julia sets are analysed. We study connected McMullen family Julia sets importantly, and colour different colours for different regions of Julia sets which can describe the inside of Julia sets more carefully. Then we compute the geometry central points of the largest steady regions of Julia set which is homeomorphism to. At last, the property which the center points of steady regions own to m and d..