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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:

34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
14P25 Topology of real algebraic varieties
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34C23 Bifurcation theory for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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