The J Sets Of Newton's Method Of Complex Exponential Function And Synchronization Of Chaotic System

Non-linear theory contains three important parts: Fractal, Chaos and Soliton theory. They compose the theoretical foundation of non-linear theory which describes the complex systematic structure shape that has a random structure. Based on non-linear theoretical analysis, the research has studied into some chaotic and fractal problems, such as modeling and showing J Sets of Newton's method of a family of exponential functions, presenting a systematic design procedure to anti-synchronize chaotic system, and researching into the adaptive synchronization problem of two Rikitake systems in the presence of unknown system parameters.On Fractal: Newton's method is an important technique which is used to find the solutions of nonlinear equation or equations. It transforms the solution of the equation f(x) = 0 into a dynamical process, and the solution process is related to the initial value. The paper develops Figen Cilinger's studies, make use of the relaxed Newton's method to polynomials and simple exponential functions F(z) = ze^zw (w∈C), construct J Sets of relaxed Newton's method, research into the attractive basin of two fixed points 0 and ∞ according to different w values, analyze characters of the structures of the J Sets, and find out that they're symmetrical, periodic and conform to fission evolvement laws. The results are: ① If w = 2n(n = 0, ±2, ±4, …), the attractive basins of 0 and ∞ is symmetrical about axis x and axis y.Choose the range of main angle as [-π,π), for a random w = α(α∈R), theattractive basins of 0 and ∞ is symmetrical about axis x. ②The attractive basin of two fixed points 0 and ∞ is circumgyrated symmetrical with a η angle when the parameter w is an integer. ③Set the parameters some particular values such as w = -4.7,h = 0.8, the figuresare quite similar with figures of different zoom scales. ④If w is a complex, the choosed main angle θ_z is discontinuousness on the minus axis x, so the rupture of attractive basin of two fixed points 0 and ∞ only occure on the minus axis x.On Chaos: A systematic procedure to anti-synchronize chaotic coupled dynamo systems is designed, the method is simpler and faster to achieve anti-synchronization in contrast with the conventional synchronization approaches, and numerical simulations show the the proposed scheme is effective. The adaptive synchronization problem of two Rikitake systems is also been researched in the presence of unknown system parameters. The parameters update...