August  2020, 13(8): 2145-2163. doi: 10.3934/dcdss.2020183

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  August 2020 Early access  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 and Continuous Dynamical Systems - S, 2020, 13 (8) : 2145-2163. doi: 10.3934/dcdss.2020183
References:
[1]

E. BenoîtJ.-L. CallotF. Diener and M. Diener, Chasse au canard. Ⅰ: Les canards, Collect. Math., 32 (1981), 37-74. 

[2]

K. BoldC. EdwardsJ. GuckenheimerS. GuharayK. HoffmanJ. HubbardR. 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.

[3]

M. Brøns, Bifurcations and instabilities in the Greitzer model for compressor system surge, Mathematical Engineering in Industry, 2 (1988), 51-63. 

[4]

J. BurkeM. DesrochesA. GranadosT. J. KaperM. 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.

[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.

[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.

[7]

C. Canalis-Durand, Solution formelle Gevrey d'une équation singulièrement prturbée, Asymptotic Anal., 8 (1994), 185-216. 

[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.

[9]

T. Pham DinhJ. DemongeotP. 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.

[10]

F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996). doi: 10.1090/memo/0577.

[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.

[12]

J. GrasmanH. 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.

[13]

J. GuckenheimerK. 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.

[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.

[15]

M. Nagumo, Über das anfangeswertproblem partialler Diferentialgleichungen, Jap. J. Math., 18 (1942), 41-47.  doi: 10.4099/jjm1924.18.0_41.

[16]

J. P. Ramis, Dévissage Gevrey, in Journées Singulières de Dijon, Astérisque, 59–60, Soc. Math. France, Paris, 1978,173–204.

[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.

[18]

J. P. Ramis, Théorèmes d'indice Gevrey pour les équations différentielles ordinaires, Mem. Amer. Math. Soc., 48 (1984). doi: 10.1090/memo/0296.

[19]

R. Schäfke, On a theorem of Sibuya, Letter to Sibuya, 1991.

[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.

[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.

[22]

W. Walter, An elementary proof of the Cauchy-Kowalewsky theorem, Amer. Math. Monthly, 92 (1985), 115-126. doi: 10.1080/00029890.1985.11971551.

[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.

[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.

show all references

References:
[1]

E. BenoîtJ.-L. CallotF. Diener and M. Diener, Chasse au canard. Ⅰ: Les canards, Collect. Math., 32 (1981), 37-74. 

[2]

K. BoldC. EdwardsJ. GuckenheimerS. GuharayK. HoffmanJ. HubbardR. 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.

[3]

M. Brøns, Bifurcations and instabilities in the Greitzer model for compressor system surge, Mathematical Engineering in Industry, 2 (1988), 51-63. 

[4]

J. BurkeM. DesrochesA. GranadosT. J. KaperM. 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.

[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.

[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.

[7]

C. Canalis-Durand, Solution formelle Gevrey d'une équation singulièrement prturbée, Asymptotic Anal., 8 (1994), 185-216. 

[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.

[9]

T. Pham DinhJ. DemongeotP. 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.

[10]

F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996). doi: 10.1090/memo/0577.

[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.

[12]

J. GrasmanH. 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.

[13]

J. GuckenheimerK. 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.

[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.

[15]

M. Nagumo, Über das anfangeswertproblem partialler Diferentialgleichungen, Jap. J. Math., 18 (1942), 41-47.  doi: 10.4099/jjm1924.18.0_41.

[16]

J. P. Ramis, Dévissage Gevrey, in Journées Singulières de Dijon, Astérisque, 59–60, Soc. Math. France, Paris, 1978,173–204.

[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.

[18]

J. P. Ramis, Théorèmes d'indice Gevrey pour les équations différentielles ordinaires, Mem. Amer. Math. Soc., 48 (1984). doi: 10.1090/memo/0296.

[19]

R. Schäfke, On a theorem of Sibuya, Letter to Sibuya, 1991.

[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.

[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.

[22]

W. Walter, An elementary proof of the Cauchy-Kowalewsky theorem, Amer. Math. Monthly, 92 (1985), 115-126. doi: 10.1080/00029890.1985.11971551.

[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.

[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.

[1]

Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure and Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211

[2]

Oktay Veliev. Spectral expansion series with parenthesis for the nonself-adjoint periodic differential operators. Communications on Pure and Applied Analysis, 2019, 18 (1) : 397-424. doi: 10.3934/cpaa.2019020

[3]

Hermann Brunner, Stefano Maset. Time transformations for delay differential equations. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 751-775. doi: 10.3934/dcds.2009.25.751

[4]

Nicolas Fournier. A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff. Kinetic and Related Models, 2019, 12 (3) : 483-505. doi: 10.3934/krm.2019020

[5]

Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543

[6]

N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711

[7]

David W. Pravica, Michael J. Spurr. Unique summing of formal power series solutions to advanced and delayed differential equations. Conference Publications, 2005, 2005 (Special) : 730-737. doi: 10.3934/proc.2005.2005.730

[8]

Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118

[9]

Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281

[10]

Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure and Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23

[11]

Huaiyu Jian, Xiaolin Liu, Hongjie Ju. The regularity for a class of singular differential equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1307-1319. doi: 10.3934/cpaa.2013.12.1307

[12]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991

[13]

Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027

[14]

Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39

[15]

Asif Yokus, Mehmet Yavuz. Novel comparison of numerical and analytical methods for fractional Burger–Fisher equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2591-2606. doi: 10.3934/dcdss.2020258

[16]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[17]

Katharina Schratz, Xiaofei Zhao. On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2841-2865. doi: 10.3934/dcdsb.2020043

[18]

N. I. Karachalios, H. E. Nistazakis, A. N. Yannacopoulos. Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations. Conference Publications, 2005, 2005 (Special) : 476-486. doi: 10.3934/proc.2005.2005.476

[19]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021051

[20]

Quan Zhou, Yabing Sun. High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021233

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (312)
  • HTML views (269)
  • Cited by (0)

Other articles
by authors

[Back to Top]