On persistent centers
type: Item
instance of: scholarly article
title: On persistent centers
zbMATH Open document ID: 1183.34041
author: Anna Cima
author: Armengol Gasull
author: J. C. Medrado
published in: Bulletin des Sciences Mathématiques
publication date: 2009-09-29T00:00:00Z
review text: A system of analytic ordinary differential equations on the plane with a monodromic critical point at the origin at which the linear part has purely imaginary eigenvalues can be written in the form \[ \dot x = -y + P(x,y),\quad \dot y = x + Q(x,y), \] or in complex coordinates \(z = x + iy\) as \[ \dot z = iz + F(z, \bar z),\tag{1} \] Fix such a system. The authors define the origin to be a \textit{persistent center} for that system if it is a center for the system \(\dot z = iz + \lambda F(z, \bar z)\) for all complex \(\lambda\). The origin is termed a \textit{weakly persistent center} for that system if it is a center for the system \(\dot z = iz + c F(z, \bar z)\) for all real \(c\). The authors identify five broad classes of general persistent centers (e.g., the \textit{holomorphic} class \(\dot z = iz + f(z)\) with \(f(0) = f'(0) = 0\)). They also completely characterize collections of persistent centers within some general classes of systems, e.g., systems (1) in which \(F\) is a polynomial function of degree at most three.
reviewed by: Douglas S. Shafer
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 5608813
zbMATH Keywords: center-focus problem
zbMATH Keywords: Lyapunov constants
zbMATH Keywords: Melnikov functions
MaRDI profile type: Publication
full work available at URL: j.bulsci.2008.08.007
OpenAlex ID: W2090591360
cites work: Q4768040
cites work: Algebraic properties of the Lyapunov and period constants
cites work: Integrable systems in the plane with center type linear part
cites work: Q4283276
cites work: Degenerate Hopf Bifurcation Formulas and Hilbert’s 16th Problem
cites work: Successive derivatives of a first return map, application to the study of quadratic vector fields
cites work: On the period function for a family of complex differential equations
cites work: An explicit expression of the first Lyapunov and period constants with applications
cites work: An Analytic-Numerical Method for Computation of the Liapunov and Period Constants Derived from Their Algebraic Structure
cites work: A new approach to the computation of the Lyapunov constants
cites work: A relation between small amplitude and big limit cycles.
cites work: The center problem for a linear center perturbed by homogeneous polynomials
cites work: On the Poincaré–Lyapunov constants and the Poincare series
cites work: Bifurcation formulae derived from center manifold theory
cites work: On Second Order Bifurcations of Limit Cycles
cites work: Higher order bifurcations of limit cycles
cites work: Q4213962
cites work: Theory of values of singular point in complex autonomous differential systems
cites work: Algorithmic Derivation of Centre Conditions
cites work: Bifurcation of planar vector fields and Hilbert's sixteenth problem
cites work: Averaging methods in nonlinear dynamical systems
cites work: Quadratic systems with center and their perturbations
DOI: 10.1016/J.BULSCI.2008.08.007
description: scientific article; zbMATH DE number 5608813