Bifurcations of planar vector fields and hilbert’s

In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets. The methods can be applied to theoretical problems such as Hilbert's 16th problem, but can also be used for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations. Series: Progress in Mathematics, Vol. 164 Contents: 1 Families of Two-Dimensional Vector Fields 1.1 Vector fields on surfaces of genus 0 1.2 A first approach to Bifurcation Theory 2 Limit Periodic Sets 2.1 Organizing centers for bifurcations of limit cycles 2.2 The cyclicity property 3 The 0-Parameter Case 3.1 Blowing up of singularities of vector fields 3.2 The finiteness result for analytic vector fields on S 2 3.3 The Dulac problem 4 Bifurcations of Regular Limit Periodic Sets 4.1 The return map 4.2 Regular limit periodic sets of finite codimension 4.3 Regular limit periodic set of infinite codimension 5 Bifurcations of Elementary graphics 5.1 Transition map near a hyperbolic saddle point 5.2 Unfoldings of saddle connections in the finite codimension case 5.3 Unfoldings of saddle connections of infinite codimension 5.4 Unfoldings of elementary graphics 6 Desingularization Theory and Bifurcation of Non-elementary Limit Periodic Sets 6.1 The use of rescaling formulas 6.2 Desingularization of unfoldings of cuspidal loops 6.3 A method of desingularization for analytic vector fields