| This thesis investigates the dynamics of homoclinic tangle and rank one chaos in Duffing equation with cubic weakly damped. Following is the structure of this thesis.The first chapter is background, we first introduce the Smale horseshoe map and symbolic dynamics. Secondly, homoclinic tangle theory and rank one chaos theory which are presented by the reference [8] is introduced. These theory provide a new way for applying nonhyperbolic map theory in differential equations. The homoclinic tangle the-ory presents a more systemic and complete description for homoclinic tangle’s overall dynamic structure.In the second chapter, we investigate the parameter conditions under which Duffing equation with cubic weakly damped has one or two homoclinic(s) by Melnikov method and qualitative theory, and also has a dissipative saddle point. Based on this, we ana-lyze the dynamic of homoclinic tangle for periodically perturbed Duffing equation with cubic weakly damped. When the system has one homoclinic solution, there exist three types of strange attractors after periodically perturbed and they are sinks representing stable dynamical behavior, Henon-like attractors characterized by an SRB measure and transient tangle. These strange attractors occur repeatedly in a fixed pattern. When the system has two homoclinic solutions, there exists five types of strange attractors after periodically perturbed and they are bilateral Henon-like attractors characterized by an SRB measure, bilateral sinks, bilateral rank one attractors, unilateral sinks, unilateral Henon-like attractors characterized by an SRB measure. These strange attractors also occur repeatedly in a fixed pattern.The third chapter is about rank one dynamics of Duffing equation with cubic weakly damped. According to the rank one theory, there exists rank one chaos after periodically kicking the stable limit cycle in autonomous system. We first get the parameter conditions under which Duffing equation with weakly damped has one or two stable limit cycle(s) by Melnikov method. Based on this, we obtain rank one chaos by periodically kicking one stable limit cycle. Especially by periodically kicking two stable limit cycles, we also obtain rank one chaos. This is the latest results at present. |
