Several Problems In Applied Nonlinear Dynamical Systems

In this thesis, the following several problems in applied nonlinear dynamical systems are studied.1. The persistence of lower dimensional invariant hyperbolic tori and their stable and unstable manifolds for the smooth perturbed Hamiltonian systems is shown by using the theory of normally hyperbolic invariant manifold and a finitely smooth version of the KAM theorem. Our proof is much more succinct and clear than the previous proof on the similar problem of analytical Hamiltonian systems.2. Some dynamics in the famous ABC (Arnold-Beltrami-Childress) flow are considered. By a new like-KAM theorem and the high dimensional Melnikov method, the conditions of existence of its invariant tori and chaotic streamlines are obtained. These results negate directly a universal guess put forward by Poincare, Birkhoff, et al, and assert Arnold's original motivation for introducing this model.3. A problem on numerics and dynamics is discussed. Particularly, the damped and driven periodically sine-Gordon equation is considered, and the existence and convergence of attractors under its spectral approximation are proved. This result supplies reasonability for the previous simple Galerkin modal truncation of this equation from the viewpoint of dynamics.4. By applying the singular perturbation theory, an approximated ODE (ordinary differential equation), which is obtained by restricting the damped driven sine-Gordon equation to its GA1M (generalized asymptotic inertial manifold), is studied qualitatively. And somehomoclinic orbits and pulse orbits which explain directly chaotic jumping behavior observed in the previous numerical experiments for this equation are found.5. By the AKNS system and introducing the wave function for the integrable equations, a method to find new exact solutions from known stationary solutions of the evolution equations is proposed. Especially, a family of interesting new exact solutions of the integrable sine-Gordon equation are obtained by a similar method and these solutions contain some like-kink, like-anti-kink and like-soliton solutions, which are very significant for the further dynamics study.