Criticality via first order development of the period constants
type: Item
instance of: scholarly article
title: Criticality via first order development of the period constants
zbMATH Open document ID: 1466.34037
author: Iván Sánchez-Sánchez
author: Joan Torregrosa
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2021-02-18T00:00:00Z
review text: Consider a two-dimensional analytic real differential system \[\dot{x}=P(x,y),\quad \dot{y}=Q(x,y)\tag{\(\ast\)}\] which has a nondegenerate center-focus at the origin. One interesting problem is to study the number of oscillations of the periodic function, also known as critical periods, which is related to subharmonic bifurcations. If the origin is a center of system \((\ast)\) and if \(T(\rho)\) is the period of the periodic orbit passing through the point \((\rho,0)\), then \(T(\rho)\) is called the period function. The value \(T(\rho^{*})\) is called a critical period of the system if \(T^{\prime}(\rho^*)=0\). In this paper, the authors analyze the bifurcation of local critical periods in the class of reversible polynomial vector fields. They denoted the maximum number of local critical periods that can bifurcate from the origin of an \(n\)th degree reversible planar system, by \(\mathcal{C}_{\textit{l}}(n)\). The technique used in this paper is based on the equivalence of the first Melnikov function for the period of perturbation of an isochronous system and the linear developments with respect to the perturbation parameters of the period constant also near the same isochronous system. With this method, the authors provide the lower bound of \(\mathcal{C}_{\textit{l}}(n)\) for some even values of the degree \(n\), more preciselly they prove that the number of local critical periods in the family of polynomial time-reversible systems of degree \(n\) is at least \(\kappa(n)\), i.e., \(\kappa(n)\leq \mathcal{C}_{\textit{l}}(n)\), where \(\kappa(4)=10\), \(\kappa(6)=22\), \(\kappa(8)=37\), \(\kappa(10)=57\), \(\kappa(12)=80\), \(\kappa(14)=106\) and \(\kappa(16)=136\).
reviewed by: Jeidy Johana Jimenez
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 7312790
zbMATH Keywords: bifurcation of critical periods
zbMATH Keywords: criticality
zbMATH Keywords: period constants
zbMATH Keywords: period function
zbMATH Keywords: time-reversible centers
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: j.na.2020.112164
OpenAlex ID: W3092975551
cites work: Isochronicity via normal form
cites work: Isochronous centres and foci via commutators and normal forms
cites work: Q4692311
cites work: On the critical points of the flight return time function of perturbed closed orbits
cites work: New lower bound for the number of critical periods for planar polynomial systems
cites work: Critical periods of perturbations of reversible rigidly isochronous centers
cites work: The monotonicity of the period function for planar Hamiltonian vector fields
cites work: Bifurcation of Critical Periods for Plane Vector Fields
cites work: Algebraic properties of the Lyapunov and period constants
cites work: On the number of critical periods for planar polynomial systems
cites work: Isochronicity for several classes of Hamiltonian systems
cites work: Qualitative theory of planar differential systems
cites work: On the period function for a family of complex differential equations
cites work: An Analytic-Numerical Method for Computation of the Liapunov and Period Constants Derived from Their Algebraic Structure
cites work: On the number of critical periods for planar polynomial systems of arbitrary degree
cites work: Isochronous Foci for Analytic Differential Systems
cites work: The nondegenerate center problem and the inverse integrating factor
cites work: On some open problems in planar differential systems and Hilbert's 16th problem
cites work: Higher order limit cycle bifurcations from non-degenerate centers
cites work: Limit cycle bifurcations from a non-degenerate center
cites work: Characterization of isochronous foci for planar analytic differential systems
cites work: Bifurcation of critical periods from Pleshkan's isochrones
cites work: HILBERT'S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS
cites work: Bifurcation of critical periods from the reversible rigidly isochronous centers
cites work: The criticality of centers of potential systems at the outer boundary
cites work: Darboux linearization and isochronous centers with a rational first integral
cites work: The period function of generalized Loud's centers
cites work: Q3715336
cites work: A Higher-Order Period Function and Its Application
cites work: Perturbed rigidly isochronous centers and their critical periods
cites work: Q3997537
cites work: A criticality result for polycycles in a family of quadratic reversible centers
cites work: The Center and Cyclicity Problems
cites work: Q3831770
cites work: Local Bifurcation of Critical Periods in Vector Fields With Homogeneous Nonlinearities of the Third Degree
cites work: Characterizing isochronous centres by Lie brackets
cites work: The period functions' higher order derivatives
cites work: The period function of Hamiltonian systems with separable variables
cites work: Q4829658
cites work: CRITICAL PERIODS OF PLANAR REVERTIBLE VECTOR FIELD WITH THIRD-DEGREE POLYNOMIAL FUNCTIONS
cites work: CRITICAL PERIODS OF THIRD-ORDER PLANAR HAMILTONIAN SYSTEMS
DOI: 10.1016/J.NA.2020.112164
description: scientific article; zbMATH DE number 7312790