The 0-Parameter Case

Abstract

As an introduction to the theory of bifurcations, in this chapter we want to consider individual vector fields, i.e., families of vector fields with a 0-dimensional parameter space. We will present two fundamentals tools: the desingularization and the asymptotic expansion of the return map along a limit periodic set. In the particular case of an individual vector field these techniques give the desired final result: the desingularization theorem says that any algebraically isolated singular point may be reduced to a finite number of elementary singularities by a finite sequence of blow-ups. If X is an analytic vector field on S ², then the return map of any elementary graphic has an isolated fixed point. As a consequence, in this special case there is no accumulation of limit cycles in the phase space. In other words, the cyclicity of each limit periodic set is less than one and any analytic vector field on the sphere has only a finite number of limit cycles.