We plan to use a combination of methods of dynamical systems, going back to Shilnikov, and methods of the calculus of variations to construct shadowing orbits of Hamiltonian systems and symplectic maps. Among the problems we plan to pursue are estimates for the escape time in Mather's problem on the unbounded growth of energy for a time periodic perturbation of a geodesic flow, constructing of almost collision chaotic solutions in celestial mechanics, generalization of Mather's barrier method to autonomous Lagrangian systems, investigation of secondary homoclinic orbits for perturbations of integrable symplectic maps and applications to billiard problems, variational methods in the study of multidimensional separatrix maps.
The ultimate goal of the research is to advance in understanding of the problem of Arnold's diffusion, i.e. slow drift of action variables in Hamiltonian systems that are close to integrable ones. Mather made recently important advances in this problem. This problem is closely related to many astronomical and physical phenomena. The results of the project will be used in the graduate courses on calculus of variations and dynamical systems at the University of Wisconsin-Madison.
