Some remarks on inverse Jacobi multipliers around Hopf singularities
type: Item
instance of: scholarly article
title: Some remarks on inverse Jacobi multipliers around Hopf singularities
zbMATH Open document ID: 1316.37028
author: Adriana Buică
author: Isaac A. García
author: Susanna Maza
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-01-26T00:00:00Z
review text: The authors consider systems of real ordinary differential equations of the form \(\dot{x}=-y+F_1(x,y,z)\), \(\dot{y}=x+F_2(x,y,z)\), \(\dot{z}=\lambda z+F_3(x,y,z)\), where \(\lambda \neq 0\), defined in a neighbourhood of \(0\in \mathbb{R}^3\). The origin is a Hopf singularity for the corresponding vector field \(X\). A function \(V\) is an ``inverse Jacobi multiplier'' (IJM) if it is not locally null and satisfies the linear first-order partial differential equation \(XV=V\)div\(X\). The authors study several properties of IJMs which are relevant to the study of local bifurcations of periodic orbits. When the singularity is a saddle-focus they show that: (i) any two locally smooth and non-flat linearly independent IJMs have the same Taylor expansion; (ii) to any smooth and non-flat \(V\) can be associated exactly one smooth center manifold \(W_c\) of \(X\) such that \(W_c\subset V^{-1}(0)\). They consider also the question whether the properties of the vanishing set \(V^{-1}(0)\) proved in the \(3\)-dimensional case remain valid in higher dimension.
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 6390545
zbMATH Keywords: Jacobi last multipliers
zbMATH Keywords: generalized Hopf bifurcation
zbMATH Keywords: Poincaré map
zbMATH Keywords: limit cycle
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.04.034
OpenAlex ID: W2086106186
cites work: Inverse Jacobi multipliers
cites work: Local theory of nonlinear analytic ordinary differential equations
cites work: Existence of inverse Jacobi multipliers around Hopf points in \(\mathbb R^3\): emphasis on the center problem
cites work: Inverse Jacobi Multipliers: Recent Applications in Dynamical Systems
cites work: Multiple Hopf bifurcation in \(\mathbb R^3\) and inverse Jacobi multipliers
cites work: Existence and vanishing set of inverse integrating factors for analytic vector fields
cites work: The inverse integrating factor and the Poincaré map
cites work: Generalized Hopf bifurcation for planar vector fields via the inverse integrating factor
cites work: A survey on the inverse integrating factor
cites work: Properties of monodromic points on center manifolds in \(\mathbb R^3\) via Lie symmetries
cites work: Integral invariants and limit sets of planar vector fields
cites work: On the nonexistence, existence and uniqueness of limit cycles
cites work: Q1233671
DOI: 10.1016/J.JMAA.2014.04.034
description: scientific article; zbMATH DE number 6390545