## Advanced Studies in Pure Mathematics

### Centers and limit cycles in polynomial systems of ordinary differential equations

#### Abstract

A polynomial system of differential equations on the plane with a singularity at which the eigenvalues of the linear part are complex can be placed, by means of an affine transformation and a rescaling of time, in the form $\dot x = \lambda x - y + P(x, y)$, $\dot y = x + \lambda y + Q(x, y)$. The problem of determining, when $\lambda = 0$, whether the origin is a spiral focus or a center dates back to Poincaré. This is the center problem. We discuss an approach to this problem that uses methods of computational commutative algebra. We treat generalizations of the center problem to the complex setting and to higher dimensions. The theory developed also has bearing on the cyclicity problem at the origin, the problem of determining bounds on the number of isolated periodic orbits that can bifurcate from the origin under small perturbation of the coefficients of the original system. We also treat this application of the theory. Some attention is also devoted to periodic solutions on center manifolds and their bifurcations.

#### Article information

Dates
Revised: 17 April 2014
First available in Project Euclid: 4 October 2018

https://projecteuclid.org/ euclid.aspm/1538621984

Digital Object Identifier
doi:10.2969/aspm/06810267

Mathematical Reviews number (MathSciNet)
MR3585783

Zentralblatt MATH identifier
1383.34053

Keywords
center cyclicity focus quantity

#### Citation

Romanovski, Valery G.; Shafer, Douglas S. Centers and limit cycles in polynomial systems of ordinary differential equations. School on Real and Complex Singularities in São Carlos, 2012, 267--373, Mathematical Society of Japan, Tokyo, Japan, 2016. doi:10.2969/aspm/06810267. https://projecteuclid.org/euclid.aspm/1538621984