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Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. (English) Zbl 1063.34026

Summary: The original Hilbert’s 16th problem can be split into four parts consisting of problems A-D. In this paper, the progress of study on Hilbert’s 16th problem is presented, and the relationship between Hilbert’s 16th problem and bifurcations of planar vector fields is discussed. The material is presented in eight sections.

Section 1: Introduction: what is Hilbert’s 16th problem?

Section 2: The first part of Hilbert’s 16th problem.

Section 3: The second part of Hilbert’s 16th problem: introduction.

Section 4: Focal values, saddle values and finite cyclicity in a fine focus, closed orbit and homoclinic loop.

Section 5: Finiteness problem.

Section 6: The weakened Hilbert’s 16th problem.

Section 7: Global and local bifurcations of Z q -equivariant vector fields.

Section 8: The rate of growth of Hilbert number H(n) with n.


MSC:
34C07Theory of limit cycles of polynomial and analytic vector fields
34-02Research monographs (ordinary differential equations)
14P25Topology of real algebraic varieties
34C05Location of integral curves, singular points, limit cycles (ODE)
34C08Connections of ODE with real algebraic geometry
34C23Bifurcation (ODE)
37G15Bifurcations of limit cycles and periodic orbits