SRB Attractors With Intermingled Basins And Bifurcation Diagram Of A Class Of Plane Vector Field

The thesis consists of two main parts.The first part belongs to the smooth ergodic theory. We investigate a class of non-hyperbolic diffeomorphism defined on the product space. By using the Pesin theory combined with the general theory of differentiable dynamical systems, we prove that there are exactly two SRB attractors, and their basins cover a full measure subset of the ambient manifold. So the system is "typical" in the space of general dynamical systems in the sense of some recent conjectures. Furthermore, we prove that the basins of SRB attractors have the strange intermingled phenomenon, i.e. they are measure-theoretically dense in each other. The intermingled phenomenons have been observed in many physical systems by numerical experiment, and considered in relation to some fundamental problems in physical, biology and computer science etc. At last, we describe a concrete example for application.The second part is about the geometrical theory of the vector field. By using the qualitative method of ordinary differential equation, we research the structure of phase diagram and the bifurcation problem of a class of plane Lienard systems. On the basis of Poincare-Bendixson principle, we construct a global positive invariant set, and prove that existence, uniqueness and stability of limit cycle. Furthermore, in virtue of the ideas of general rotational vector field, we obtain the conditions and parameter region for nonexistence of limit cycle. In particular, although no using Poincare compactification, we can obtain the global structure of phase diagram. As a consequence, we get a complete description of the phase space and bifurcation. At last, our results are applied to a physiology model, and we also discuss the resonance phenomenons of the perturbed system.