Composite Expansions and Singularly Perturbed Differential Equations

  • Augustin Fruchard
  • Reinhard Schäfke
Part of the Lecture Notes in Mathematics book series (LNM, volume 2066)


We assume Φ analytic with respect to x and y in a domain \(\mathcal{D}\subset {\mathbb{C}}^{2}\) and of Gevrey order 1 with respect to \(\epsilon \) in S; this also allows to treat equations containing a control parameter. This is useful for equations where canards solutions might occur for certain values of the parameter.


  1. 1.
    Ackerberg, R.C., O’Malley, R.E.: Boundary layer problems exhibiting resonance. Stud. Appl. Math. 49, 277–295 (1970)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Balser, W.: Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. Springer, New York (2000)zbMATHGoogle Scholar
  3. 3.
    Benoît, É., Callot, J.-L., Diener, F., Diener, M.: Chasse au canard. Collect. Math. 31, 37–119 (1981)Google Scholar
  4. 4.
    Benoît, É., El Hamidi, A., Fruchard, A.: On combined asymptotic expansions in singular perturbations. Electron. J. Diff. Equat. 51, 1–27 (2002)Google Scholar
  5. 5.
    Benoît, É., Fruchard, A., Schäfke, R., Wallet, G.: Solutions surstables des équations différentielles complexes lentes-rapides à point tournant. Ann. Fac. Sci. Toulouse Math. VII 4, 627–658 (1998)CrossRefGoogle Scholar
  6. 6.
    Benoît, É., Fruchard, A., Schäfke, R., Wallet, G.: Overstability: Toward a global study. C. R. Acad. Sci. Paris I 326, 873–878 (1998)zbMATHCrossRefGoogle Scholar
  7. 7.
    Bonckaert, P., De Maesschalck, P., Gevrey normal forms of vector fields with one zero eigenvalue. J. Math. Anal. Appl. 344, 301–321 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Canalis-Durand, M., Mozo-Fernandez, J., Schäfke, R.: Monomial summability and doubly singular differential equations. J. Differ. Equat. 233, 485–511 (2007)zbMATHCrossRefGoogle Scholar
  9. 9.
    Canalis-Durand, M., Ramis, J.-P., Schäfke, R., Sibuya, Y.: Gevrey solutions of singularly perturbed differential equations. J. Reine Angew. Math. 518, 95–129 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    De Maesschalck, P.: On maximum bifurcation delay in real planar singularly perturbed vector fields. Nonlinear Anal. 68, 547–576 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    De Maesschalck, P.: Ackerberg-O’Malley resonance in boundary value problems with a turning point of any order. Commun. Pure Appl. Anal. 6, 311–333 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    De Maesschalck, P.: Gevrey properties of real planar singularly perturbed systems. J. Differ. Equat. 238, 338–365 (2007)zbMATHCrossRefGoogle Scholar
  13. 13.
    De Maesschalck, P., Dumortier, F.: Canard solutions at non-generic turning points. Trans. Am. Math. Soc. 358, 2291–2334 (2006)zbMATHCrossRefGoogle Scholar
  14. 14.
    Diener, M.: Regularizing microscopes and rivers. SIAM J. Math. Anal. 25, 148–173 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Dorodnitsyn, A.A.: Asymptotic solution of the Van der Pol equation. Priklad. Mat. Mekh. 11, 313–328 (1947) (in russian)Google Scholar
  16. 16.
    Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Memoir. Am. Math. Soc. 577, 1996Google Scholar
  17. 17.
    Eckhaus, W.: Asymptotic analysis of singular perturbations. Studies in Mathematics and its Applications, vol. 9. North-Holland, Amsterdam (1979)Google Scholar
  18. 18.
    Erdélyi, A.: Singular perturbations of boundary value problems involving ordinary differential equations. J. Soc. Indust. Appl. Math. 11, 105–116 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equat. 31, 53–98 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Forget, T.: Points tournants dégénérés, Thèse de Doctorat, Université de La Rochelle, 2007Google Scholar
  21. 21.
    Forget, T.: Solutions canards en des points tournants dégénérés. Ann. Fac. Sci. Toulouse Math. 16, 799–816 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Forget, T.: Asymptotic study of planar canard solutions. Bull. Belg. Math. Soc. Simon Stevin 15, 809–824 (2008)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Fraenkel, L.E.: On the method of matched asymptotic expansions. Proc. Cambridge Philos. Soc. 65, 209–284 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Fruchard, A., Schäfke, R.: Exceptional complex solutions of the forced Van der Pol equation. Funkcialaj Ekvacioj 42, 201–223 (1999)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Fruchard, A., Schäfke, R.: Overstability and resonance. Ann. Inst. Fourier Grenoble 53, 227–264 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Fruchard, A., Schäfke, R.: A survey of some results on overstability and bifurcation delay. Discrete Cont. Dyn. Syst. S 2, 931–965 (2009)zbMATHCrossRefGoogle Scholar
  27. 27.
    Fruchard, A., Schäfke, R.: De nouveaux développements asymptotiques combinés pour la perturbation singulière. Actes du Colloque à la mémoire d’Emmanuel Isambert Publications Univ. Paris 13, 125–161 (2012)Google Scholar
  28. 28.
    Fruchard, A., Schäfke, R.: Composite asymptotic expansions and turning points of singularly perturbed ordinary differential equations. C. R. Math. Acad. Sci. 348, 1273–1277 (2010)zbMATHCrossRefGoogle Scholar
  29. 29.
    Gautheron, V., Isambert, E.: Finitely differentiable ducks and finite expansions. In: Benoît, E. (ed.) Dynamic Bifurcations, Lect. Notes Math., vol. 1493, pp. 40–56. Springer, New York (1991)CrossRefGoogle Scholar
  30. 30.
    Van Gils, S., Krupa, M., Szmolyan, P.: Asymptotic expansions using blow-up Z. Angew. Math. Phys. 56, 369–397 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Hek, G.: Geometric singular perturbation theory in biological practice. J. Math. Biol. 60, 347–386 (2010)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Isambert, E.: Nonsmooth ducks and regular perturbations of rivers, I and II. J. Math. Anal. Appl. 200, 14–33 and 289–306 (1996)Google Scholar
  33. 33.
    Jones, C.K.R.T.: Geometric singular perturbation theory. In: Dynamical Systems, Lect. Notes Math., vol. 1609. Springer, New York (1995)Google Scholar
  34. 34.
    Kaplun, S., Lagerstrom, P.A.: Asymptotic expansions of Navier-Stokes solutions for small Reynolds numbers. J. Math. Mech. 6, 585–593 (1957)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Kevorkian, J., Cole, J.D.: Perturbation methods in applied mathematics. Applied Mathematical Sciences, vol. 34. Springer, New York (1981)Google Scholar
  36. 36.
    Kopell, N.: A geometric approach to boundary layer problems exhibiting resonance. SIAM J. Appl. Math. 37, 436–458 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Kopell, N.: The singularly perturbed turning-point problem: a geometric approach. In: Singular perturbations and asymptotics, Proc. Adv. Sem., Math. Res. Center, University of Wisconsin, Madison, Wisconsin (1980) 173–190Google Scholar
  38. 38.
    Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equat. 174, 312–368 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Krupa, M., Szmolyan, P.: Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions. SIAM J. Math. Anal., 33, 286–314 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Lagerstrom, P.A.: Matched asymptotic expansions: ideas and techniques. Applied Mathematical Sciences, vol. 76. Springer, New York (1988)Google Scholar
  41. 41.
    Lobry, C.: Dynamic bifurcations. In: Benoît, E. (ed.) Dynamic Bifurcations, Lect. Notes Math., vol. 1493, pp. 1–13. Springer, New York (1991)CrossRefGoogle Scholar
  42. 42.
    Matzinger, É.: Étude d’équations différentielles ordinaires singulièrement perturbées au voisinage d’un point tournant. Thesis, Preprint IRMA 2000/53, Strasbourg (2000)Google Scholar
  43. 43.
    Matzinger, É.: Étude des solutions surstables de l’équation de Van der Pol. Ann. Fac. Sci. Toulouse 10, 713–744 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Matzinger, É.: Asymptotic behaviour of solutions near a turning point: the example of the brusselator equation. J. Differ. Equat. 220, 478–510 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Mischenko, E.F., Rozov, N.Ch.: Differential Equations with Small Parameters and Relaxation Oscillations. Plenum Press, New York and London (1980)CrossRefGoogle Scholar
  46. 46.
    O’Malley, R.E.: Singular perturbation methods for ordinary differential equations. Applied Mathematical Sciences, vol. 89. Springer, New York (1991)Google Scholar
  47. 47.
    Panazzolo, D.: On the existence of canard solutions. Publ. Mat. 44, 503–592 (2000)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Ramis, J.-P.: Dévissage Gevrey. Astérisque 59–60, 173–204 (1978)MathSciNetGoogle Scholar
  49. 49.
    Ramis, J.-P.: Les séries k-sommables et leurs applications. In: Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lect. Notes Physics, vol. 126, pp. 178–199. Springer, New York (1980)Google Scholar
  50. 50.
    Sibuya, Y.: Gevrey property of formal solutions in a parameter. In: Asymptotic and computational analysis (Winnipeg, MB, 1989). Lecture Notes in Pure and Appl. Math., vol. 124, pp. 393–401. Dekker, New York (1990)Google Scholar
  51. 51.
    Sibuya, Y.: Linear differential equations in the complex domain. Problems of Analytic Continuation. Am. Math. Soc., Providence (RI) (1990)zbMATHGoogle Scholar
  52. 52.
    Sibuya, Y.: Uniform simplification in a full neighborhood of a transition point. Memoi. Am. Math. Soc. 149 (1974)Google Scholar
  53. 53.
    Sibuya, Y.: A theorem concerning uniform simplification at a transition point and a problem of resonance. SIAM J. Math. Anal. 12(5), 653–668 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Skinner, L.A.: Singular Perturbation Theory. Springer, New York (2011)zbMATHCrossRefGoogle Scholar
  55. 55.
    Skinner, L.A.: Uniform solution of boundary layer problems exhibiting resonance. SIAM J. Appl. Math. 47, 225–231 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Skinner, L.A.: Matched expansion solutions of the first-order turning point problem. SIAM J. Math. Anal. 25, 1402–1411 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Skinner, L.A.: A class of singularly perturbed singular Volterra integral equations. Asymptot. Anal. 22, 113–127 (2000)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Szmolyan, P., Wechselberger, M.: Canards in R3. J. Differ. Equat. 177, 419–453 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Vasil’eva, A.B., Butuzov, V.F.: Asymptotic Expansions of the Solutions of Singularly Perturbed Equations. Izdat. “Nauka”, Moscow (1973) (in Russian)Google Scholar
  60. 60.
    Wallet, G.: Surstabilité pour une équation différentielle analytique en dimension un. Ann. Inst. Fourier 40, 557–595 (1990)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Wasow, W.: Asymptotic Expansions for Ordinary Differential Equations. Interscience, New York (1965)zbMATHGoogle Scholar
  62. 62.
    Wasow, W.: Linear Turning Point Theory. Springer, New York (1985)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Augustin Fruchard
    • 1
  • Reinhard Schäfke
    • 2
  1. 1.Laboratoire de Mathématiques, Informatique et ApplicationsUniversité de Haute AlsaceMulhouseFrance
  2. 2.Institut de Recherche Mathématique AvancéeUniversité de StrasbourgStrasbourgFrance

Personalised recommendations