The research deals with a qualitative study of differentiable dynamical systems and their bifurcations. Emphasis is put on the local problems, more specifically on singularities of vector fields (systems of ordinary differential equations) and fixed points of diffeomorphisms. Inspired by elementary catastrophe theory, one looks for generic and stable properties. Standard ingredients of the investigation can be outlined as follows. After reducing the dynamical system or a family of dynamical systems, to a limited number of essential degrees of freedom (reduction to an invariant manifold like the centre manifold), one of the central procedures is the concept of normal form. Using an appropriate change of coordinates, the system can be transformed into a simple form, suitable for a further detailed analysis. In this, blowing up-techniques and rescalings play an important role. They reduce the study to simple polynomial systems (mostly in the plane or in 3-space for the problems under consideration) which then need to be considered globally. This is done by means of geometric and topological methods, notions like transversality and structural stability, the method of Hamiltonian bifurcations and Abelian integrals, lyapunov functions and related techniques. The main point is the study of homoclinic and heteroclinic orbits and especially of limit cycles (isolated periodic movements). As such the research group is also interested in questions related to Hilbert's 16th problem (dealing with the number of limit cycles of polynomial vector fields on the plane). The local study, as described above, also leads in a natural way to singular perturbation problems. Their systematic investigation became one of the main themes of the group. In recent years results have been obtained on the following subjects: - Generic local 3 and 4-parameter families of planar vector fields - Local and global bifurcations for quadratic vector fields on the plane, in relation with Hilbert's 16th problem - Conjugacies and normal-linearizations of diffeomorphisms along invariant manifolds; moduli fortopological conjugacy - Relation between local diffeomorphisms and singularities of vector fields on the plane - Singularities of vector fields on 3-space and their unfoldings - Limit cycles and transit-time analysis in singular perturbation problems - Study of polynomial Liénard-equations.
BIFURCATIONS, DYNAMICAL SYSTEMS, LIMIT CYCLES, NORMAL FORMS, PERTURBATIONS, SINGULARITIES, UNFOLDINGS, VECTOR FIELDS
Peter DE MAESSCHALCK