Electron. J. Diff. Equ., Vol. 2012 (2012), No. 22, pp. 1-17.

Limit cycles for fourth-order autonomous differential equations

Jaume Llibre, Amar Makhlouf

Abstract:
We provide sufficient conditions for the existence of periodic solutions of the fourth-order differential equation
$$ \ddddot x -(\lambda+\mu) \dddot x +(1+\lambda \mu)\ddot x -(\lambda+\mu)\dot x+\lambda \mu x = \varepsilon F(x, \dot x, \ddot x, \dddot x), $$
where $\lambda$, $\mu$ and $\varepsilon$ are real parameters, $\varepsilon$ is small and F is a nonlinear function.

Submitted October 12, 2011. Published February 7, 2012.
Math Subject Classifications: 37G15, 37C80, 37C30.
Key Words: Periodic orbit; fourth-order differential equation; averaging theory.

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Jaume Llibre
Departament de Matematiques
Universitat Autonoma de Barcelona
08193 Bellaterra, Barcelona, Catalonia, Spain
email: jllibre@mat.uab.cat
Amar Makhlouf
Department of Mathematics, University of Annaba
BP 12, Annaba, Algeria
email: makhloufamar@yahoo.fr

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