Periodic Orbits Near Equilibria via Averaging Theory of Second Order

Luis Barreira, Jaume Llibre, Claudia Valls

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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 languageEnglish
Pages (from-to)715-731
JournalMathematical Modelling and Analysis
Issue number5
Publication statusPublished - 1 Nov 2012


  • differential equation
  • nonlinear differential equation
  • perturbation method


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