Vaillancourt, Remi,Nolte, Klaus-Georg.2009-03-252009-03-2519941994Source: Dissertation Abstracts International, Volume: 56-04, Section: B, page: 2065.9780315959637http://hdl.handle.net/10393/9695http://dx.doi.org/10.20381/ruor-7920Based upon the combination of the pseudo-arclength continuation method the Poincare map and various defining equations, several systems of equations are constructed that trace nonhyperbolic periodic solutions of autonomous ordinary differential equations. A system that equally well continues a saddle-node, a period-doubling and a secondary Hopf bifurcation across a two-dimensional parameter space is presented first. Additional formulae are provided which, along with the continuation, completely characterize these codimension-one bifurcations and therefore lead to the detection of certain codimension-two bifurcations, in particular Takens-Bogdanov bifurcations, cusps, isola formation points or perturbed bifurcation points and degenerate period-doublings and degenerate secondary Hopf bifurcations. Secondly, systems are developed which continue these codimension-two bifurcations across a three-dimensional parameter space, thereby detecting certain codimension-three bifurcations of periodic orbits. The application of these ideas to a five-dimensional system describing a two-level laser leads to a variety of interesting bifurcations. A winged cusp, a swallow tail, two kinds of degenerate Takens-Bogdanov points and isola formation points for different codimension-one loops are found. Also, maximal bounds in a three-dimensional parameter domain for the existence of certain periodic solutions are given. Moreover, the coexistence of several attractors of the same and/or different topological structure is demonstrated. Finally, attractive tori are found in a systematic way and briefly discussed.244 p.Mathematics.Thesis