Two Kinds Of Generalized M-J Sets And Synchronization Of Uncertain Different-Structural Chaotic Systems

Non-linear theory contains three important parts: Fractal, Chaos and Soliton theory. They are theoretical foundation of non-linear theory. Based on non-linear theoretical analysis, the research has studied some chaotic and fractal problems as follows:(1) We used the doubling and truncation techniques to generate a hypercomplex number system of any dimension. We discussed the precondition of that addition and multiplication are closed in hypercomplex number system, and listed out the definition and constructing arithmetic of the hyperdimensional generalized Mandelbrot-Julia sets (in abbreviated form generalized M-J sets) in hypercomplex number system. Using the 2-D and 3-D cross sections of the hyperdimensional generalized M-J sets, the author studied the fractal feature of 2-D and 3-D cross sections, and used theories to prove the symmetry of 2-D and 3-D cross sections.(2) The theory of M-J sets of high degree complex polynomials is introduced, and the method to calculate the positions, the sizes, and the orientations of these M-like sets of high degree complex polynomials in parameter space is put forward. The fractal feature of M sets and generalized M sets corresponding to different critical points is studied. The study results are stated as follows: ① The symmetry of M sets and generalized M sets corresponding to different critical points has been proved; ② M-like sets have different type, and they are the small copies of M sets generated from f:z←z~n+ c(n = 2,3,4,...); ③ The type of M-likesets depends on the multiplicity of critical points; ④ M-like sets appear only in the M sets of critical points; ⑤ With the increase of period, the M-like sets will become more and more, which implies the distribution of M-like sets is fractal.(3) Synchronization and parameters identification of uncertain different-structural chaotic system via active control is researched. Based on the Lyapunov stability theory, an active controller and the parameters update rule are designed. It is proved that the controller can make the states of different-structural drive and response systems, such as Rossler system and uncertain Genesio system, Chen-Lee system and uncertain Genesio system, asymptotically synchronized, and identify the parameters of response system. Numerical simulations have shown the effectiveness of the active controller.