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Degenerate Hopf bifurcation formulas and Hilbert’s 16th problem. (English) Zbl 0682.58035

Summary: This paper presents explicit formulas for the solution of degenerate Hopf bifurcation problems for general systems of differential equations of dimension \(n\geq 2\), 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 \(\leq 3\). 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.

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
34C25 Periodic solutions to ordinary differential equations
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