Periodic solutions of forced isochronous oscillators at resonance

D. Bonheure; C. Fabry; D. Smets

doi:10.3934/dcds.2002.8.907

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2002, Volume 8, Issue 4: 907-930. Doi: 10.3934/dcds.2002.8.907

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Periodic solutions of forced isochronous oscillators at resonance

1.
Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, Chemin du Cyclotron, 2 , B-1348 Louvain-la-Neuve
2.
Laboratoire Jacques-Louis Lions, Université de Paris 6, 4 place Jussieu BC 187, 75252, Paris

Received: March 2001

Revised: April 2002

Published: October 2002

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Abstract

We study the existence of $2\pi$-periodic solutions for forced nonlinear oscillators at resonance, the nonlinearity being a bounded perturbation of a function deriving from an isochronous potential, i.e. a potential leading to free oscillations that all have the same period. The family of isochronous oscillators considered here includes oscillators with jumping nonlinearities, as well as oscillators with a repulsive singularity, to which a particular attention is paid. The existence results contain, as particular cases, conditions of Landesman-Lazer type. Even in the case of perturbed linear oscillators, they improve earlier results. Multiplicity and non-existence results are also given.

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Mathematics Subject Classification: 34C15, 34C25, 70K30.

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