The proposed project consists of two main parts. The first part is investigation generic properties of discrete dynamical systems (diffeomorphisms) on a compact manifold. We apply Newton Interpolation technique to investigate bifurcations of homoclinic tangency for 2-dimensional diffeomorphisms and, in particular, Newhouse phenomenon of infinitely many coexisting sinks. Homoclinic tangency has proved to be the most interesting feature of dynamics in dimension 2. The second is investigation of instabilities of generic of Hamiltonian systems, primarily using Mather theory.
Enormous number of systems in the nature and technological processes are described by ordinary differential equations. Therefore, it is extremely important to understand long time behavior and stability of trajectories of in some sense generic ordinary differential equations. The first part of proposed project is an attempt to contribute to understanding of complicated behavior of chaotic low dimensional systems. Motion of plans, comets, and celestial dynamics in general are described by Hamiltonian equations. The second part of the project is devoted to investigation of such dynamics using primarily variational methods developed by many people and J. Mather in particular.
