*(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 $H\left(n\right)$ with $n$.

##### MSC:

34C07 | Theory of limit cycles of polynomial and analytic vector fields |

34-02 | Research monographs (ordinary differential equations) |

14P25 | Topology of real algebraic varieties |

34C05 | Location of integral curves, singular points, limit cycles (ODE) |

34C08 | Connections of ODE with real algebraic geometry |

34C23 | Bifurcation (ODE) |

37G15 | Bifurcations of limit cycles and periodic orbits |