1.1 Recent work on the stability of REs has included the `block-diagonalization' technique for REs of simple mechanical systems, results on linear stability and `drift' along group orbits and Nekhoroshev stability estimates for equilibria and REs, all at non-singular points of momentum maps. One of the key theoretical aims of this project is to extend these techniques to singular points of momentum maps and to investigate phenomena associated to discrete (possibly time-reversing) symmetries and non-compact symmetry groups. We will also develop analogous theories for RPOs. Several results on bifurcations from REs have been obtained recently, but there is no satisfactory general framework. We aim to develop such frameworks for bifurcations from both REs and RPOs together with a normal form theory for nearby dynamics. For the latter analogous non-Hamiltonian techniques will provide a starting point.
1.2 The existence of classical monodromy in singular torus fibrations defined by Liouville integrable systems was first discovered in 1980 and has since been investigated in a number of concrete systems. A proof that homoclinic trajectories in the form of pinched tori imply monodromy in two degree of freedom integrable systems has also been given. The realization has grown that monodromy is not an exceptional phenomenon in classical mechanics, but very little general theory exists. The aims of this project include developing theories for classifying singularities of Liouville integrable systems and describing their associated monodromy and its persistence for non-integrable perturbations.
1.3 Criteria for the `attractiveness' of families of REs in simple mechanical systems and fluid dynamics with dissipation have been obtained. It has also been shown that individual REs can become unstable under dissipation and preliminary work indicates that the same can be true for families of REs. This project will develop these ideas and apply them to continuum systems. The effects of dissipation on heteroclinic connections will also be studied.
1.4 There has been virtually no work specifically on heteroclinic connections and chaos in symmetric Hamiltonian systems, but relevant ideas and techniques have been developed in the non-Hamiltonian context. One of the aims of the network is to adapt and combine these with standard non-symmetric Hamiltonian methods to initiate a theory of Hamiltonian symmetric chaos.
1.5 Integrability of non-holonomic systems and its connections with symmetries has not yet been the subject of extensive studies. This project aims to understand and characterize at least some basic cases and then proceed to study the effects of small perturbations on such systems.