Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case

Lucile Mégret; Jacques Demongeot

doi:10.3934/dcdss.2020183

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2020, Volume 13, Issue 8: 2145-2163. Doi: 10.3934/dcdss.2020183

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Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case

Lucile Mégret^1, , and
Jacques Demongeot^2,

1.
Université Pierre et Marie Curie, UMR 8256 - Adaptation Biologique et Vieillissement, 7 quai Saint Bernard, 75 252 PARIS CEDEX, France
2.
Université Grenoble Alpes, AGEIS, Team Tools for e-Gnosis Medical, Faculté de Médecine, Domaine de la Merci 38706 La Tronche, France

^* Corresponding author: Lucile Mégret
^* Corresponding author: Lucile Mégret

Received: December 31, 2018

Early access: November 2019

Published: May 2020

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Abstract

We generalize the results on the existence of an over-stable solution of singularly perturbed differential equations to the equations of the form $ \varepsilon\ddot{x}-F(x,t,\dot{x},k(t), \varepsilon) = 0 $. In this equation, the time dependence prevents from returning to the well known case of an equation of the form $ \varepsilon dy/dx = F(x,y,a, \varepsilon) $ where $ a $ is a parameter. This can have important physiological applications. Indeed, the coupling between the cardiac and the respiratory activity can be modeled with two coupled van der Pol equations. But this coupling vanishes during the sleep or the anesthesia. Thus, in a perspective of an application to optimal awake, we are led to consider a non autonomous differential equation.

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Mathematics Subject Classification: 34M25, 34M30, 34M35, 35B38.

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