Degenerate Hopf Bifurcation Formulas and Hilbert’s 16th Problem

This paper presents explicit formulas for the solution of degenerate Hopf bifurcation problems for general systems of differential equations of dimension $n \geqq 2$ , with smooth vector fields. The main new result is the general solution of the problem for a weak focus of order 3. For bifurcation problems with a distinguished parameter, we present the formulas for the defining conditions of all cases with codimension $ \leqq 3$. The formulas have been applied to Hilbert’s 16th problem, yielding a new proof of Bautin’s theorem, and correcting an error in Bautin’s formula for the third focal value. The approach used is the Lyapunov–Schmidt method. A review of five other approaches is given, along with literature references and comparisons to the present work.