Solutions of slowly and periodically varying differential equations. Applications to some classical examples

Nadir Sari (1)

1 LaSIE - Laboratoire des Sciences de l'Ingénieur pour l'Environnement - UMR 7356

We study a singularly perturbed second-order differential equation describing a slowly and periodically varying hamil-tonian system. Typical dynamics governed by this type of system are, for example, equations of forced pendulum, of Duffing or of the "shallow water sloshing" problem. Using symmetries of this equation and singular perturbation tools, we describe dynamics, by splitting the phase space in regions where the motion is oscil-latory and others where it is unbounded, and study dynamics in each kind of regions. Finally we establish the existence periodic solutions and give the structure of these solutions in term of response curves. In particular, our results extend and complete the ones stated in [1, 4, 6] and answer to some open questions within. We also give new results about multiplicity of periodic solutions of forced pendulum equation. To illustrate our results, we conclude this work by a numerical study of these classical examples.

Mots clés: singular perturbation; fast oscillations; periodic solu- tions; nonlinear resonance

Domaines: Systèmes dynamiques [math.DS]

Nadir Sari. Solutions of slowly and periodically varying differential equations. Applications to some classical examples. 2020. ⟨hal-02506680⟩