Section 1

GEOMETRY AND DYNAMICS

Topics

Project objectives

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:

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

• 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.

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

• 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.

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

Major breakthroughs are likely to include:

1. 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;
   
2. a much improved understanding of the role that monodromy plays in symmetric Hamiltonian systems;
   
3. the initiation of a theory of symmetric chaos for Hamiltonian systems, including the implications for mechanical systems;
   
4. KAM theory for non-holonomic systems.

Scientific originality

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.

Research Method

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.