10.25560/1265
Jukes, Alice Claire
Imperial College London
Mathematics
On homoclinic bifurcations with symmetry
Imperial College London
2007
Lamb, Jeroen
2007-04-19
2007-04-19
Doctor of Philosophy (PhD)
10044/1/1265
This thesis deals with the description of solutions of symmetric dynamical systems that lie in the neighbourhood of a homoclinic or heteroclinic cycle. Homoclinic and heteroclinic cycles are the main mechanism by which complicated behaviour is known to arise in dynamical systems. A starting point for studying the consequences of the existence of homoclinic and heteroclinic cycles is to focus on non-wandering dynamics, that is, solutions that remain in the neighbourhood of such cycles, both in the phase space and in parameter space. For general vector fields, such studies have been carried out extensively. In the context of applications differential equations often possess some additional structure. In order to describe generic phenomena in such systems, it is important to take into account this structure. Symmetry is an example of such a structure, which naturally arises in many applications. The methodology used in this thesis, based on a technique introduced by X.B. Lin in 1990, yields an effective description of the dynamics near homoclinic and heteroclinic cycles in terms of solutions of a low dimensional bifurcation equation. In the presence of symmetry, the bifurcation equation naturally inherits a structure that is directly related to the symmetry properties of the original vector field. Within this framework, robust networks with time-preserving and time-reversing symmetries are described and it is shown what the non-wandering dynamics are. For generic codimension-one equivariant systems with real leading eigenvalues the non-wandering dynamics is shown to be conjugate to a topological Markov chain. The situation is more complicated in codimension-two and a novel example is given to illustrate the difficulties and interesting phenomena that occur.