A note on the period function for certain planar vector fields

Antonio Garijo; Armengol Gasull; Xavier Jarque

doi:10.1080/10236190903241689

A note on the period function for certain planar vector fields

Antonio Garijo, Armengol Gasull, Xavier Jarque

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

3 Citations (Scopus)

Abstract

Consider a smooth planar autonomous differential equation having a period annulus, P. We present a new criterion to ensure that the period function has at most one critical period on P. Our result has a compact form when the differential equation is written as ż = F(z, z̄). It is based on a suitable representation formula of the derivative of the period function which uses the infinitesimal generator associated to the continua of periodic orbits. We apply the criterion to several particular cases of the equation ż = f(z)g(z̄)/h(z, z̄), where f(z) and are holomorphic functions and h is a C2 smooth real valuated function. © 2010 Taylor & Francis.

Original language	English
Pages (from-to)	631-645
Journal	Journal of Difference Equations and Applications
Volume	16
Issue number	5
DOIs	https://doi.org/10.1080/10236190903241689
Publication status	Published - 1 May 2010

Keywords

Critical period
Meromorphic vector fields
Period function
Periodic orbit

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title = "A note on the period function for certain planar vector fields",

abstract = "Consider a smooth planar autonomous differential equation having a period annulus, P. We present a new criterion to ensure that the period function has at most one critical period on P. Our result has a compact form when the differential equation is written as {\.z} = F(z, {\=z}). It is based on a suitable representation formula of the derivative of the period function which uses the infinitesimal generator associated to the continua of periodic orbits. We apply the criterion to several particular cases of the equation {\.z} = f(z)g({\=z})/h(z, {\=z}), where f(z) and are holomorphic functions and h is a C2 smooth real valuated function. {\textcopyright} 2010 Taylor & Francis.",

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T1 - A note on the period function for certain planar vector fields

AU - Garijo, Antonio

AU - Gasull, Armengol

AU - Jarque, Xavier

PY - 2010/5/1

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N2 - Consider a smooth planar autonomous differential equation having a period annulus, P. We present a new criterion to ensure that the period function has at most one critical period on P. Our result has a compact form when the differential equation is written as ż = F(z, z̄). It is based on a suitable representation formula of the derivative of the period function which uses the infinitesimal generator associated to the continua of periodic orbits. We apply the criterion to several particular cases of the equation ż = f(z)g(z̄)/h(z, z̄), where f(z) and are holomorphic functions and h is a C2 smooth real valuated function. © 2010 Taylor & Francis.

AB - Consider a smooth planar autonomous differential equation having a period annulus, P. We present a new criterion to ensure that the period function has at most one critical period on P. Our result has a compact form when the differential equation is written as ż = F(z, z̄). It is based on a suitable representation formula of the derivative of the period function which uses the infinitesimal generator associated to the continua of periodic orbits. We apply the criterion to several particular cases of the equation ż = f(z)g(z̄)/h(z, z̄), where f(z) and are holomorphic functions and h is a C2 smooth real valuated function. © 2010 Taylor & Francis.

KW - Critical period

KW - Meromorphic vector fields

KW - Period function

KW - Periodic orbit

U2 - 10.1080/10236190903241689

DO - 10.1080/10236190903241689

M3 - Article

VL - 16

SP - 631

EP - 645

JO - Journal of Difference Equations and Applications

JF - Journal of Difference Equations and Applications

SN - 1023-6198

IS - 5

ER -