GEOMETRY AND DYNAMICS

This part of the project will be concerned with the development of geometric and topological methods for describing aspects of the dynamics of symmetric Hamiltonian systems, including general stability and bifurcation theories for relative equilibria (REs) and periodic orbits (RPOs), the existence of, and dynamics near, (relative) heteroclinic cycles (HCs), and KAM theory for singular integrable systems and for non-holonomic systems. Specific objectives include:

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Liapounov, linear and Nekhoroshev stability criteria, bifurcation theorems and normal form techniques for REs and RPOs of symmetric Hamiltonian system.

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A classification of the heteroclinic cycles (including `pinched tori') that can occur as singular fibres of energy-momentum maps of Liouville integrable systems, and of the corresponding torus fibrations (with monodromy) over the complements of their discriminants. Development of a KAM theory for families of invariant tori with monodromy in perturbations of Liouville integrable systems with singular action-angle coordinates.

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Studies of the effects of dissipation and other non-Hamiltonian perturbations on the existence and stability of REs and RPOs and on heteroclinic cycles.

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Studies of symmetric chaos in Hamiltonian systems, including: (a) an equivariant symbolic description of the chaotic dynamics associated to perturbed heteroclinic cycle which are invariant under discrete group actions; (b) the effects of monodromy on the chaos resulting from perturbed pinched tori and more general heteroclinic cycles in non-integrable systems; (c) a description of the dynamical behaviour of the lifts of chaotic invariant sets of reduced mechanical systems to unreduced phase spaces.

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New theories for integrable and near-integrable non-holonomic mechanical systems.

Major breakthroughs are likely to include: (a) the extension of existing stability and bifurcation results to REs and RPOs with nontrivial isotropy subgroups in systems with non-compact symmetry groups and time-reversing symmetries; (b) a much improved understanding of the role that monodromy plays in symmetric Hamiltonian systems; (c) the initiation of a theory of symmetric chaos for Hamiltonian systems, including the implications for mechanical systems; (d) KAM theory for non-holonomic systems.