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

 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

Received  January 2019 Published  November 2019

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.

Citation: Lucile Mégret, Jacques Demongeot. Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020183
##### References:
 [1] E. Benoît, J.-L. Callot, F. Diener and M. Diener, Chasse au canard. Ⅰ: Les canards, Collect. Math., 32 (1981), 37-74.   Google Scholar [2] K. Bold, C. Edwards, J. Guckenheimer, S. Guharay, K. Hoffman, J. Hubbard, R. Oliva and W. Weckesser, The forced van der Pol equation. Ⅱ: Canards in the reduced system, SIAM J. Appl. Dyn. Syst., 2 (2003), 570-608.  doi: 10.1137/S1111111102419130.  Google Scholar [3] M. Brøns, Bifurcations and instabilities in the Greitzer model for compressor system surge, Mathematical Engineering in Industry, 2 (1988), 51-63.   Google Scholar [4] J. Burke, M. Desroches, A. Granados, T. J. Kaper, M. Krupa and T. Vo, From canards of folded singularities to torus canards in a forced van der Pol equation, J. Nonlinear Sci., 26 (2016), 405-451.  doi: 10.1007/s00332-015-9279-0.  Google Scholar [5] C. Canalis-Durand, Formal expansions of van der Pol equation canard solutions are Gevrey, in Dynamic Bifurcations, Lecture Notes in Mathematics, 1493, Springer, Berlin, 1991, 29–39. doi: 10.1007/BFb0085022.  Google Scholar [6] C. Canalis-Durand, Solution formelle Gevrey d'une équation singulièrement prturbée : Le cas multidimensionnel, Ann. Inst. Fourier (Grenoble), 43 (1993), 469-483.  doi: 10.5802/aif.1341.  Google Scholar [7] C. Canalis-Durand, Solution formelle Gevrey d'une équation singulièrement prturbée, Asymptotic Anal., 8 (1994), 185-216.   Google Scholar [8] M. Canalis-Durand, J. P. Ramis, R. Schäfke and Y. Sibuya, Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math., 518 (2000), 95–129. doi: 10.1515/crll.2000.008.  Google Scholar [9] T. Pham Dinh, J. Demongeot, P. Baconnier and G. Benchetrit, Simulation of a biological oscillator: The respiratory rhythm, J. Theor. Biol., 103 (1983), 113-132.  doi: 10.1016/0022-5193(83)90202-3.  Google Scholar [10] F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996). doi: 10.1090/memo/0577.  Google Scholar [11] W. Eckhaus, Relaxation oscillations including a standard chase on French ducks, in Asymptotic Analysis II, Lecture Notes in Math., 985, Springer, Berlin, 1983,449–494. doi: 10.1007/BFb0062381.  Google Scholar [12] J. Grasman, H. Nijmeijer and E. J. M. Veling, Singular perturbations and a mapping on an interval for the forced van der Pol relaxation oscillator, Phys. D, 13 (1984), 195-210.  doi: 10.1016/0167-2789(84)90277-X.  Google Scholar [13] J. Guckenheimer, K. Hoffman and W. Weckesser, The forced van der Pol equation. Ⅰ: The slow flow and its bifurcations, SIAM J. Appl. Dyn. Syst., 2 (2003), 1-35.  doi: 10.1137/S1111111102404738.  Google Scholar [14] E. Matzinger, Etude des solutions surstables de l'équation de van der Pol, Ann. Fac. Sci. Toulouse Math. (6), 10 (2001), 713-744.  doi: 10.5802/afst.1010.  Google Scholar [15] M. Nagumo, Über das anfangeswertproblem partialler Diferentialgleichungen, Jap. J. Math., 18 (1942), 41-47.  doi: 10.4099/jjm1924.18.0_41.  Google Scholar [16] J. P. Ramis, Dévissage Gevrey, in Journées Singulières de Dijon, Astérisque, 59–60, Soc. Math. France, Paris, 1978,173–204.  Google Scholar [17] J. P. Ramis, Les séries k-sommables et leurs applications, in Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lecture Notes in Phys., 126, Springer, Berlin-New York, 1980,178–199.  Google Scholar [18] J. P. Ramis, Théorèmes d'indice Gevrey pour les équations différentielles ordinaires, Mem. Amer. Math. Soc., 48 (1984). doi: 10.1090/memo/0296.  Google Scholar [19] R. Schäfke, On a theorem of Sibuya, Letter to Sibuya, 1991. Google Scholar [20] Y. Sibuya, Gevrey property of formal solution in a parameter, in Asymptotic and Computational Analysis, Lecture Notes in Pure and Appl. Math., 124, Dekker, New York, 1990,393–401.  Google Scholar [21] P. Szmolyan and M. Wechselberger, Relaxation oscillations in $\mathbb{R}^3$, J. Differential Equations, 200 (2004), 69-104.  doi: 10.1016/j.jde.2003.09.010.  Google Scholar [22] W. Walter, An elementary proof of the Cauchy-Kowalewsky theorem, Amer. Math. Monthly, 92 (1985), 115-126. doi: 10.1080/00029890.1985.11971551.  Google Scholar [23] I. M. Winter-Arboleda, W. S. Gray and L. A. D. Espinosa, Fractional Fliess operators: Two approaches, 49th Annual Conference on Information Sciences and Systems, (CISS) IEEE Pres, Baltimore, MD, 2015. doi: 10.1109/CISS.2015.7086831.  Google Scholar [24] I. M. Winter-Arboleda, W. S. Gray and L. A. D. Espinosa, On global convergence of fractional Fliess operators with applications to bilinear systems, 51st Annual Conference on Information Sciences and Systems (CISS), IEEE Press, Baltimore, MD, 2017, 1-6. doi: 10.1109/CISS.2017.7926119.  Google Scholar

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##### References:
 [1] E. Benoît, J.-L. Callot, F. Diener and M. Diener, Chasse au canard. Ⅰ: Les canards, Collect. Math., 32 (1981), 37-74.   Google Scholar [2] K. Bold, C. Edwards, J. Guckenheimer, S. Guharay, K. Hoffman, J. Hubbard, R. Oliva and W. Weckesser, The forced van der Pol equation. Ⅱ: Canards in the reduced system, SIAM J. Appl. Dyn. Syst., 2 (2003), 570-608.  doi: 10.1137/S1111111102419130.  Google Scholar [3] M. Brøns, Bifurcations and instabilities in the Greitzer model for compressor system surge, Mathematical Engineering in Industry, 2 (1988), 51-63.   Google Scholar [4] J. Burke, M. Desroches, A. Granados, T. J. Kaper, M. Krupa and T. Vo, From canards of folded singularities to torus canards in a forced van der Pol equation, J. Nonlinear Sci., 26 (2016), 405-451.  doi: 10.1007/s00332-015-9279-0.  Google Scholar [5] C. Canalis-Durand, Formal expansions of van der Pol equation canard solutions are Gevrey, in Dynamic Bifurcations, Lecture Notes in Mathematics, 1493, Springer, Berlin, 1991, 29–39. doi: 10.1007/BFb0085022.  Google Scholar [6] C. Canalis-Durand, Solution formelle Gevrey d'une équation singulièrement prturbée : Le cas multidimensionnel, Ann. Inst. Fourier (Grenoble), 43 (1993), 469-483.  doi: 10.5802/aif.1341.  Google Scholar [7] C. Canalis-Durand, Solution formelle Gevrey d'une équation singulièrement prturbée, Asymptotic Anal., 8 (1994), 185-216.   Google Scholar [8] M. Canalis-Durand, J. P. Ramis, R. Schäfke and Y. Sibuya, Gevrey solutions of singularly perturbed differential equations, J. Reine Angew. Math., 518 (2000), 95–129. doi: 10.1515/crll.2000.008.  Google Scholar [9] T. Pham Dinh, J. Demongeot, P. Baconnier and G. Benchetrit, Simulation of a biological oscillator: The respiratory rhythm, J. Theor. Biol., 103 (1983), 113-132.  doi: 10.1016/0022-5193(83)90202-3.  Google Scholar [10] F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996). doi: 10.1090/memo/0577.  Google Scholar [11] W. Eckhaus, Relaxation oscillations including a standard chase on French ducks, in Asymptotic Analysis II, Lecture Notes in Math., 985, Springer, Berlin, 1983,449–494. doi: 10.1007/BFb0062381.  Google Scholar [12] J. Grasman, H. Nijmeijer and E. J. M. Veling, Singular perturbations and a mapping on an interval for the forced van der Pol relaxation oscillator, Phys. D, 13 (1984), 195-210.  doi: 10.1016/0167-2789(84)90277-X.  Google Scholar [13] J. Guckenheimer, K. Hoffman and W. Weckesser, The forced van der Pol equation. Ⅰ: The slow flow and its bifurcations, SIAM J. Appl. Dyn. Syst., 2 (2003), 1-35.  doi: 10.1137/S1111111102404738.  Google Scholar [14] E. Matzinger, Etude des solutions surstables de l'équation de van der Pol, Ann. Fac. Sci. Toulouse Math. (6), 10 (2001), 713-744.  doi: 10.5802/afst.1010.  Google Scholar [15] M. Nagumo, Über das anfangeswertproblem partialler Diferentialgleichungen, Jap. J. Math., 18 (1942), 41-47.  doi: 10.4099/jjm1924.18.0_41.  Google Scholar [16] J. P. Ramis, Dévissage Gevrey, in Journées Singulières de Dijon, Astérisque, 59–60, Soc. Math. France, Paris, 1978,173–204.  Google Scholar [17] J. P. Ramis, Les séries k-sommables et leurs applications, in Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lecture Notes in Phys., 126, Springer, Berlin-New York, 1980,178–199.  Google Scholar [18] J. P. Ramis, Théorèmes d'indice Gevrey pour les équations différentielles ordinaires, Mem. Amer. Math. Soc., 48 (1984). doi: 10.1090/memo/0296.  Google Scholar [19] R. Schäfke, On a theorem of Sibuya, Letter to Sibuya, 1991. Google Scholar [20] Y. Sibuya, Gevrey property of formal solution in a parameter, in Asymptotic and Computational Analysis, Lecture Notes in Pure and Appl. Math., 124, Dekker, New York, 1990,393–401.  Google Scholar [21] P. Szmolyan and M. Wechselberger, Relaxation oscillations in $\mathbb{R}^3$, J. Differential Equations, 200 (2004), 69-104.  doi: 10.1016/j.jde.2003.09.010.  Google Scholar [22] W. Walter, An elementary proof of the Cauchy-Kowalewsky theorem, Amer. Math. Monthly, 92 (1985), 115-126. doi: 10.1080/00029890.1985.11971551.  Google Scholar [23] I. M. Winter-Arboleda, W. S. Gray and L. A. D. Espinosa, Fractional Fliess operators: Two approaches, 49th Annual Conference on Information Sciences and Systems, (CISS) IEEE Pres, Baltimore, MD, 2015. doi: 10.1109/CISS.2015.7086831.  Google Scholar [24] I. M. Winter-Arboleda, W. S. Gray and L. A. D. Espinosa, On global convergence of fractional Fliess operators with applications to bilinear systems, 51st Annual Conference on Information Sciences and Systems (CISS), IEEE Press, Baltimore, MD, 2017, 1-6. doi: 10.1109/CISS.2017.7926119.  Google Scholar
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