|Thesis abstract: |
Nonlinear systems of ODEs depending on parameters are ubiquitous in all fields of science. The discussion of the impact of parameters on the asymptotic behavior of the dynamical system is typically the major concern. This discussion requires the determination of the so-called bifurcations, namely the parameter combinations at which some degeneracies create a structural change in the dynamics of the system. The computation of the bifurcations can be rarely performed analytically, so that one must rely upon numerical techniques. The aim of this thesis is to improve the available techniques and their software implementations and to extend them to cases which have not yet been covered: particular relevance is given to bifurcations (in smooth and non smooth systems) concerning multiple degeneracies, the so-called codimension-2 bifurcaiton points. Those points assume a strategic importance in the bifurcation analysis, since more bifurcation curves depart from them, so that they become the organizing centers of the bifurcation diagrams. Most of the identified theoretical results are implemented in MatCont, a standard software used for the bifurcation analysis, and are applied to various significant problems in engineering, biology and sociology.