1.1 The starting point for work on both REs and RPOs will be `normal forms' for neighbourhoods of the RE/RPOs in phase space. For REs these are provided by existing local descriptions of Hamiltonian group actions while for RPOs these will be combined with bundle techniques already developed in the analogous non-Hamiltonian context. Liapounov stability criteria will be obtained by applying the energy-momentum method to the resulting equations of motion. The linear stability analysis will combine methods used to describe drift in the non-singular case with methods used in the non-Hamiltonian context and with the general theory of time-reversible, equivariant, linear Hamiltonian systems. Techniques used to obtain Nekhoroshev stability estimates for REs at non-singular points will be extended to singular points. Bifurcation theory and normal form results will be obtained by combining techniques from both Hamiltonian and non-Hamiltonian theory. Finally, a combination of singular reduction, normal forms, reconstruction equations, blowing up techniques and results on perturbations of periodic manifolds will be used to study the bifurcations of RPOs from singular REs.
1.2 Singularities of torus fibrations coming from Liouville integrable system will be studied by first performing singular reduction to obtain singular one degree of freedom systems and then using topological and algebraic methods to classify their singularities. The relation between torus singularities of energy momentum maps and singularities of their complexifications is still unknown and will be investigated. Perturbations of these systems will be treated using extensions of KAM theory to integrable systems with singular action-angle coordinates.
1.3 A variety of perturbation theory methods will be used to study the effects of non-Hamiltonian perturbations on (families of) REs and RPOs including linear stability theory, reduced energy methods, the method of `shadowing curves' and Liapounov-type instability theorems.
1.4 Equivariant symbolic descriptions of dynamics near symmetric heteroclinic cycles will be obtained using symmetry adapted extensions of the Markov decomposition techniques used to prove the existence of horseshoes near homoclinic cycles. Return map techniques will be used to investigate the effects of monodromy on dynamics near perturbed pinched tori and more general heteroclinic cycles. Descriptions of lifts of `reduced chaos' to the full phase space will be obtained by combining `skew-product' ideas that have been developed recently for dissipative systems with Hamiltonian phase space reconstruction theory.
1.5 The integrability of nonholonomic systems will be studied using suitable extensions of the concept of Liouville integrability and a noncanonical perturbation theory will be developed to study near-integrable systems.