Periodic Orbits Near Equilibria via Averaging Theory of Second Order

Luis Barreira; Jaume Llibre; Claudia Valls

doi:https://doi.org/10.3846/13926292.2012.736090

Periodic Orbits Near Equilibria via Averaging Theory of Second Order

Luis Barreira, Jaume Llibre, Claudia Valls

Department of Mathematics

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

Abstract

Lyapunov, Weinstein and Moser obtained remarkable theorems giving sufficient conditions for the existence of periodic orbits emanating from an equilibrium point of a differential system with a first integral. Using averaging theory of first order we established in [1] a similar result for a differential system without assuming the existence of a first integral. Now, using averaging theory of the second order, we extend our result to the case when the first order average is identically zero. Our result can be interpreted as a kind of degenerated Hopf bifurcation. © 2012 Copyright Vilnius Gediminas Technical University.

Original language	English
Pages (from-to)	715-731
Journal	Mathematical Modelling and Analysis
Volume	17
Issue number	5
DOIs	https://doi.org/10.3846/13926292.2012.736090
Publication status	Published - 1 Nov 2012

Keywords

differential equation
nonlinear differential equation
perturbation method

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abstract = "Lyapunov, Weinstein and Moser obtained remarkable theorems giving sufficient conditions for the existence of periodic orbits emanating from an equilibrium point of a differential system with a first integral. Using averaging theory of first order we established in [1] a similar result for a differential system without assuming the existence of a first integral. Now, using averaging theory of the second order, we extend our result to the case when the first order average is identically zero. Our result can be interpreted as a kind of degenerated Hopf bifurcation. {\textcopyright} 2012 Copyright Vilnius Gediminas Technical University.",

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author = "Luis Barreira and Jaume Llibre and Claudia Valls",

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T1 - Periodic Orbits Near Equilibria via Averaging Theory of Second Order

AU - Barreira, Luis

AU - Llibre, Jaume

AU - Valls, Claudia

PY - 2012/11/1

Y1 - 2012/11/1

N2 - Lyapunov, Weinstein and Moser obtained remarkable theorems giving sufficient conditions for the existence of periodic orbits emanating from an equilibrium point of a differential system with a first integral. Using averaging theory of first order we established in [1] a similar result for a differential system without assuming the existence of a first integral. Now, using averaging theory of the second order, we extend our result to the case when the first order average is identically zero. Our result can be interpreted as a kind of degenerated Hopf bifurcation. © 2012 Copyright Vilnius Gediminas Technical University.

AB - Lyapunov, Weinstein and Moser obtained remarkable theorems giving sufficient conditions for the existence of periodic orbits emanating from an equilibrium point of a differential system with a first integral. Using averaging theory of first order we established in [1] a similar result for a differential system without assuming the existence of a first integral. Now, using averaging theory of the second order, we extend our result to the case when the first order average is identically zero. Our result can be interpreted as a kind of degenerated Hopf bifurcation. © 2012 Copyright Vilnius Gediminas Technical University.

KW - differential equation

KW - nonlinear differential equation

KW - perturbation method

U2 - https://doi.org/10.3846/13926292.2012.736090

DO - https://doi.org/10.3846/13926292.2012.736090

M3 - Article

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VL - 17

SP - 715

EP - 731

JO - Mathematical Modelling and Analysis

JF - Mathematical Modelling and Analysis

IS - 5

ER -

Periodic Orbits Near Equilibria via Averaging Theory of Second Order

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