\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author: Lucile Mégret

    * Corresponding author: Lucile Mégret 
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

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

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [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.
  • 加载中
SHARE

Article Metrics

HTML views(1513) PDF downloads(336) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return