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

From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation

“…We will give in future works: i) details of the proofs announced in this paper about the existence of "canard" solutions in the equations (5) (cf. [26]) ii) more simulations of the coupled system showing their qualitative agreement with experimental observations acquired in empirical situations of cardiorespiratory decoupling. In the real system, the respiration can be voluntarily ruled, hence an exogenous control can replace the endogenous one, e.g., for better preparing the heart to the effort.…”

“…It is worth to notice that this result can be applied to any forced van der Pol (cf. [26]). The same approach can be used for simulating the disappearance (pathologic in neurodegenerative diseases and physiologic during the sleep) of the respiratory sinus arrhythmia (RSA).…”

“…This situation corresponds to the classical problem of the van der Pol system with a periodical forcing, which has been intensively studied (see for example [5-9,16-18, 22, 25]), but only in the case of a classical forcing (e.g., constant or cosine). In [26], we prove the existence of canard solutions for a large family of nonautonomous equations, which contains in particular the equations (5). It is well known that the forced van der Pol equation:…”

“…In the case where k(t)g(t) = α is a constant parameter, there are exactly two canard solutions and E. Matzinger proved in [25] the existence of a uniform asymptotic development of a solution in sectors centred at the value u = 1 and calculated an explicit approximation of this solution. We prove in [26] the existence of formal solutions, which are Gevrey asymptotic to over-stable solutions for a large class of equation which contains the equation (6).…”

“…DEMONGEOT, ISTRATE, KHLAIFI, MÉGRET, TARAMASCO AND THOMAS by a system of two coupled Liénard equations [8,26]. In the case where these equations are of van der Pol or FitzHugh-Nagumo type, they can represent the bulbar vegetative coupling between the respiratory control system and the cardiomoderator system [2][3][4].…”