Research On Strange Attractors In Planar Mapping Ang Chaos Control

The chaos dynamic behavior of Lozi mapping and Lauwerier mapping is studied; and the problems of chaos control for planar mappings and high dimensional mappings are investigated in detail.In the iterative process of the Lozi mapping, the unstable manifolds of the saddle point in the first quadrant are stretched and folded constantly. The strange attractor of Lozi mapping is the closure of the unstable manifold. On the basis of two points, which are the intersection of the stable manifold of the fixed point in the third quadrant with y-axis and the first tangent point of the mapping, the attraction domain of Lozi mapping is constructed. The structure of the attraction domain is studied and the existence of the attraction domain is proved. Results derived from the theoretical analysis are in good agreement with the numerical results.On the basis of the results of attraction domain, the capture domain of Lozi attractor is constructed further. The dynamic behavior of the unstable manifold is analyzed. According to the transverse intersection between the stable and unstable manifolds and Smale-Birkhoff Theorem, the structure and the complicated dynamic properties of the strange attractor are studied. A positive measure set is found by using Milnor measure theory, which is a necessary condition for the existence of Lozi strange attractor.The dynamic properties of Lauwerier mapping are described. The topological entropy and the Lyapunov exponents of the Lauwerier strange attractor are calculated, thus its dynamic properties are characterized quantitatively.The shift mapping on the inverse limit space of the quadratic mapping is studied. It is proved that the shift mapping is topologically transitive, and its periodic points are dense. By the inverse limit theory, it is proved that Lauwerier mapping restricted on its attractor is topologically semi-conjugate to the shift mapping on the inverse limit space when the parameter, a is equal to 4. Therefore the conclusion that Lauwerier strange attractor is chaotic in the sense of Devaney is obtained.The chaotic behaviors of both Lauwerier mapping and Lozi mapping are controlled by the OGY chaos control method and the pole placement technique of the linear control theory. The chaotic motions are controlled to be the stable periodic-1 or periodic-2 orbits.In the impluse double rotor system, the dynamic equation is used to establish a four dimensional mapping. OGY method is used to control the chaos in the mapping. It is verified that the method can also be used in higher dimensional systems.It is analyzed and studied that the values of the regulator pole and the control domain which influence the length of the control time. If the modulus of the eigenvalues of the matrix A-BKT is less than 1, then the regulator pole is selected and the purpose of control can be realized. But the different regulator pole makes the chaos control time different. It is analyzed and studied that the difference of the regulator pole and the control domain influence the control time.