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VOL. 68 | 2016 Centers and limit cycles in polynomial systems of ordinary differential equations
Valery G. Romanovski, Douglas S. Shafer

Editor(s) Raimundo Nonato Araújo dos Santos, Victor Hugo Jorge Pérez, Takashi Nishimura, Osamu Saeki


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.


Published: 1 January 2016
First available in Project Euclid: 4 October 2018

zbMATH: 1383.34053
MathSciNet: MR3585783

Digital Object Identifier: 10.2969/aspm/06810267

Primary: 34C05 , 34C07 , 34C20 , 34C23 , 34C45 , 37G05 , 37G10

Keywords: center , cyclicity , focus quantity

Rights: Copyright © 2016 Mathematical Society of Japan


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