On the center-focus problem for the equation $\protect \frac{\protect \mathrm{d}y}{\protect \mathrm{d}x} + \Sigma _{i=1}^{n} a_i(x) y^i = 0, 0\le x \le 1$ where $a_i$ are polynomials

Gavrilov, Lubomir

doi:10.5802/ahl.41

Gavrilov, Lubomir

On the center-focus problem for the equation

\frac{d y}{d x} + Σ_{i = 1}^{n} a_{i} (x) y^{i} = 0, 0 \leq x \leq 1

where

a_{i}

are polynomials

Annales Henri Lebesgue, Volume 3 (2020), pp. 615-648.

Metadata

DOI10.5802/ahl.41

Keywords center-focus problem, Abel equation, Liénard equation

Abstract

We study irreducible components of the set of polynomial plane differential systems with a center, which can be seen as a modern formulation of the classical center-focus problem. The emphasis is given on the interrelation between the geometry of the center set and the Picard–lefschetz theory of the bifurcation (or Poincaré–Pontryagin–Melnikov) functions. Our main illustrative example is the center-focus problem for the Abel equation on a segment, which is compared to the related polynomial Liénard equation.

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