The theory of dynamical systems aims to understand the nature of the behaviour of solutions of evolution equations, describing processes in a broad spectrum of scientific disciplines. Dynamical systems that arise in the context of applications often admit additional structure with important consequences for the dynamics. For instance, mechanical systems often possess symmetry and Hamiltonian structure. Many mechanical systems are described by Hamiltonian equations, such as the celebrated Henon-Heiles model of galactic motion, the motion of nonlinear three-dimensional vibrations of strings, localized travelling waves in Hamiltonian lattices (Fermi-Pasta-Ulam chain), vortex dynamics (related to hydrodynamics problems) and non-holonomic dynamics. The main goal of the proposed project is to develop mathematical methods of the bifurcation theory for dynamical systems with special structures. In particular, the focus will be on bifurcations involving homoclinic solutions, which lie at the basis of the understanding of complicated recurrent dynamics, better known as chaos. While homoclinic bifurcations have been extensively studied in the context of general systems (without additional structure), the problem of homoclinic bifurcation in Hamiltonian systems has received relatively little attention, despite its obvious relevance for many practical applications. This is mainly due to the fact that homoclinic bifurcations in Hamiltonian systems are often much more challenging than those in general systems. The project objectives include the study of global bifurcations in systems with different types of homoclinic and heteroclinic orbits leading to a creation of novel methods for the study of Hamiltonian systems with symmetry. An important objective of the proposed project is the application of these mathematical methods to study the dynamics of an axisymmetric rigid body in a gravity field, which is a fundamental open problem in the field of theoretical mechanics.
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