Rank One Chaos In Henon Map And The Dynamics Of Homoclinic Tangles In Lorenz-like Equation

In this thesis, we study the strange attractors in periodically perturbed Henon map and the dynamics of homoclinic tangles in Lorenz-like equation. There are two parts:In the first part, we investigate the dynamics of discrete system which is kicked at periodic time. The theories of rank one chaos’31are based on the studies of Benedicks and Carleson on Henon maps[1,2]. Then it is applied to concrete differential equations. The first application is the dynamics of periodically kicked Hopf limit cycle. They proved the existence of the rank one chaos theoretically and observed the corresponding numerical results[3,4]. The second application is for all limit cycles, they can be turned into rank one attractors under the periodical perturbation through the definition of shear factor[5]. Motivated by the limit cycle in differential equations, we study the rank one chaos in Henon map admitting stable periodic solution. Firstly, there will be a periodic solution. Secondly, add periodically kicks to the map. Our results indicate strongly that a discrete system also has rank one chaos.The second part is about the dynamics of homoclinic tangles in three-dimensional Lorenz-like equation. There have made a systematic study on homoclinic tangles for pe-riodically perturbed two-dimensional equations by a precise derivation of the separatrix map[6]around homoclinic solutions[7-10]. There exists four types of strange attractors and they are transient tangle, sinks, Henon-like attractors and rank one attractors char-acterized by an SRB measure[11-14]. They also prove that these strange attractors occur repeatedly in a fixed pattern. So far, there is rarely results about the homoclinic tangles in three-dimensional equation or higher-dimensional equation. In this thesis, we study the dynamics of the homoclinic tangles in three-dimensional Lorenz-like equations by numer-ical simulation. And we also get three types dynamic phenomena:sinks, rank one chaos and Henon-like attractors characterized by an SRB measure. There is also a well-defined pattern for the occurrence of these three dynamical phenomena.