The M-J Sets Of Compound Complex Map And Projective Synchronization Of A Class Of Chaos System

The nonlinear theory is a new developing frontier science which describes the complex systematic structure shape which has a random structure. The nonlinear theory contains three important concepts: Fractal, Chaos and Soliton. Here we lay a particular emphasis on the studying of the theory and methods of the M-J sets (abbreviated to M-J sets) of Compound complex mapping and Projective Synchronization of a class of chaos System. Also we will give out some important research results.Scholars have made deep researches on the Mandelbrot-Julia sets generated from the complex mapping z← z~α + c(α∈R). Entwistle had studied the J sets of compound complex mapping z←(z~2 +c)~2+c. We generalize complex mapping as z←(z~α+ c)~β+ c(α,β∈R), and research on the structure topological inflexibility and the discontinuity evolution law of the generalized J sets of this mapping. The researches as below: generalized J sets have α-fold rotation symmetry and its center is the origin when α is integer; the different choices of angle lead to the different evolution of generalized J sets; computer the Hausdorff distance between two generalized J sets.Then we give a research into the generalized M sets of z←(z~α+c)~β+ c(α,β∈R), By plotting the Quasi-M sets, we find that: when α and β are integers, the Quasi-M sets are symmetrical about x axis, when α and β are odd, the Quasi-M sets are symmetrical about x and y axis, this has been illustrated theoretically; Making using of DeMoivre theory, when α and β are decimal fractions, the different choices of angle 9 lead to the different evolution of generalized M sets. Finally, Hausdorff distance is applied to the Quasi-M sets, offer the mathing of two Quasi-M sets.Finally, we study projective sysnchronization of a new chaos system—a modified coupled dynamos system. Using techniques from the state observer design, we present a systematic, design procedure to synchronize a modified coupled dynamos system by a scaling factor., and have proved the validity of the design. Finally, feasibility of the technique is illustrated for a modified coupled dynamos system.