# Centennial History of Hilbert’s 16th Problem

@article{Ilyashenko2002CentennialHO, title={Centennial History of Hilbert’s 16th Problem}, author={Yulii Il'yashenko}, journal={Bulletin of the American Mathematical Society}, year={2002}, volume={39}, pages={301-354} }

The second part of Hilbert’s 16th problem deals with polynomial differential equations in the plane. It remains unsolved even for quadratic polynomials. There were several attempts to solve it that failed. Yet the problem inspired significant progress in the geometric theory of planar differential equations, as well as bifurcation theory, normal forms, foliations and some topics in algebraic geometry. The dramatic history of the problem, as well as related developments, are presented below. §1… Expand

#### 324 Citations

Extrema of the Andronov-Hopf function of a polynomial Lienard system

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- 2005

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Attractors of one-dimensional maps generated by b oundary value problems

- Mathematics
- Proceedings. 2005 International Conference Physics and Control, 2005.
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There exist boundary value problems for which the investigation is reduced to a study of one-dimensional maps given implicitly by the relationship xn+1 + ax n = F (xn + bx n+1). We discuss some… Expand

Limit cycles of quadratic systems

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Abstract In this paper, the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving Hilbert’s Sixteenth Problem in this special case… Expand

Hilbert's 16th problem

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We provide an upper bound for the number of limit cycles that planar polynomial differential systems of a given degree may have. The bound turns out to be a polynomial of degree four in the degree of… Expand

The Infinitesimal Hilbert’s 16th Problem in the Real and Complex Planes

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Abstract.In this paper we study the infinitesimal Hilbert’s 16th problem on $${\mathbb{R}}^{2}$$ and $${\mathbb{C}}^{2}$$, describing some techniques used for its research in the last few years. We… Expand

From the sixteenth Hilbert problem to tropical geometry

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- 2008

Abstract.Hilbert’s problem on the topology of algebraic curves and surfaces (the sixteenth problem from the famous list presented at the second International Congress of Mathematicians in 1900) was… Expand

Hilbert's 16th problem. I. When differential systems meet variational methods

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We provide an upper bound for the number of limit cycles that planar polynomial differential systems of a given degree may have. The bound turns out to be a polynomial of degree three in the degree… Expand

Algebraične metode v dinamičnih sistemih

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- 2018

Algebraic Methods in Dynamical Systems. In this monograph the theory of dynamical systems meets abstract algebra. From the dynamical point of view we are focused to center and cyclicity problems.… Expand

Symbolic computation of limit cycles associated with Hilbert’s 16th problem

- Computer Science
- 2009

In this paper, efficient computations for the focus values and polynomial equations are discussed, showing how to deal with the complexity in the computation of non-linear dynamical systems. Expand

Limit cycles of three-dimensional polynomial vector fields

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We consider polynomial vector fields in three-dimensional space of the form and their polynomial perturbations of degree ?n. We study limit cycles of three-dimensional systems bifurcating from closed… Expand

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