Advanced Studies in Pure Mathematics

Centers and limit cycles in polynomial systems of ordinary differential equations

Valery G. Romanovski and Douglas S. Shafer

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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

Source
School on Real and Complex Singularities in São Carlos, 2012, R. N. Araújo dos Santos, V. H. Jorge Pérez, T. Nishimura and O. Saeki, eds. (Tokyo: Mathematical Society of Japan, 2016), 267-373

Dates
Received: 1 November 2013
Revised: 17 April 2014
First available in Project Euclid: 4 October 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1538621984

Digital Object Identifier
doi:10.2969/aspm/06810267

Mathematical Reviews number (MathSciNet)
MR3585783

Zentralblatt MATH identifier
1383.34053

Subjects
Primary: 34C05: Location of integral curves, singular points, limit cycles 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) 34C20: Transformation and reduction of equations and systems, normal forms 34C23: Bifurcation [See also 37Gxx] 34C45: Invariant manifolds 37G05: Normal forms 37G10: Bifurcations of singular points

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


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