Advertisement

The Blowup Method

  • Christian Kuehn
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 191)

Abstract

This chapter deals with geometric desingularization of nonhyperbolic equilibrium points using the so-called blowup method. The main insight, due to Dumortier and Roussarie, is that singularities at which fast and slow directions interact may be converted into partially hyperbolic problems using the blowup method. The method inserts a suitable manifold, e.g., a sphere, at the singularity.

Bibliography

  1. [AG13]
    G.G. Avalos and N.B. Gallegos. Quasi-steady state model determination for systems with singular perturbations modelled by bond graphs. Math. Computer Mod. Dyn. Syst., pages 1–21, 2013. to appear.Google Scholar
  2. [Bal77]
    J.M. Ball. Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Quart. J. Math.., 28(4):473–486, 1977.CrossRefzbMATHGoogle Scholar
  3. [BFM06]
    P. Breitenlohner, P. Forgács, and D. Maison. Classification of static, spherically symmetric solutions of the Einstein-Yang-Mills theory with positive cosmological constant. Commun. Math. Phys., 261(3):569–611, 2006.CrossRefzbMATHGoogle Scholar
  4. [BM90]
    M. Brunella and M. Miari. Topological equivalence of a plane vector field with its principal part defined through newton polyhedra. J. Differential Equat., 85:338–366, 1990.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [BMD11]
    P. Bonckaert, P. De Maesschalck, and F. Dumortier. Well adapted normal linearization in singular perturbation problems. J. Dyn. Diff. Eq., 23(1):115–139, 2011.CrossRefzbMATHGoogle Scholar
  6. [BMS08]
    A. Braides, M. Maslennikov, and L. Sigalotti. Homogenization by blow-up. Applicable Analysis, 87(12):1341–1356, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [Bon87]
    C. Bonet. Singular perturbation of relaxed periodic orbits. J. Differential Equat., 66(3):301–339, 1987.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [Bru89]
    Alexander D. Bruno. Local Methods in Nonlinear Differential Equations. Springer, 1989.Google Scholar
  9. [Car52]
    M.L. Cartwright. Van der Pol’s equation for relaxation oscillations. In Contributions to the Theory of Nonlinear Oscillations II, pages 3–18. Princeton University Press, 1952.Google Scholar
  10. [Car81]
    J. Carr. Applications of Centre Manifold Theory. Springer, 1981.Google Scholar
  11. [Chi10]
    C. Chicone. Ordinary Differential Equations with Applications. Texts in Applied Mathematics. Springer, 2nd edition, 2010.Google Scholar
  12. [DH99]
    F. Dumortier and C. Herssens. Polynomial Liénard equations near infinity. J. Differential Equat., 153(1):1–29, 1999.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [DLZ97]
    F. Dumortier, C. Li, and Z. Zhang. Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops. J. Differential Equat., 139(1):146–193, 1997.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [DPK07b]
    F. Dumortier, N. Popovic, and T.J. Kaper. The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off. Nonlinearity, 20(4):855–877, 2007.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [DR96b]
    F. Dumortier and R. Roussarie. Canard Cycles and Center Manifolds, volume 121 of Memoirs Amer. Math. Soc. AMS, 1996.Google Scholar
  16. [DRS97]
    F. Dumortier, R. Roussarie, and J. Sotomayor. Bifurcations of cuspidal loops. Nonlinearity, 10(6):1369–1408, 1997.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [Dum77]
    F. Dumortier. Singularities of vector fields on the plane. J. Differential Equat., 23(1):53–106, 1977.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [Dum78]
    F. Dumortier. Singularities of Vector Fields. IMPA, Rio de Janeiro, Brazil, 1978.Google Scholar
  19. [Dum93]
    F. Dumortier. Techniques in the theory of local bifurcations: Blow-up, normal forms, nilpotent bifurcations, singular perturbations. In D. Schlomiuk, editor, Bifurcations and Periodic Orbits of Vector Fields, pages 19–73. Kluwer, Dortrecht, The Netherlands, 1993.Google Scholar
  20. [FM92]
    I. Fonseca and S. Müller. Quasiconvex integrands and lower semicontinuity in L 1. Bull. Belg. Math. Soc, 23:1081–1098, 1992.zbMATHGoogle Scholar
  21. [GH83]
    J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York, NY, 1983.CrossRefzbMATHGoogle Scholar
  22. [GKR08]
    S.D. Glyzin, A.Yu. Kolesov, and N.Kh. Rozov. Blue sky catastrophe in relaxation systems with one fast and two slow variables. Differential Equat., 44(2):161–175, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  23. [Gla77]
    R.T. Glassey. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys., 18:1794–1797, 1977.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [GLMM00]
    A. Gasull, J. Llibre, V. Maosa, and F. Maosas. The focus-centre problem for a type of degenerate system. Nonlinearity, 13(3):699–729, 2000.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [GPCS06]
    A. Garcia, E. Pérez-Chavela, and A. Susin. A generalization of the Poincaré compactification. Arch. Rat. Mech. Anal., 179(2):285–302, 2006.CrossRefzbMATHGoogle Scholar
  26. [Guc84]
    J. Guckenheimer. Multiple bifurcation problems of codimension two. SIAM J. Math. Anal., 15(1):1–49, 1984.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [Hir64a]
    H. Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero: I. Ann. of Math., 79(1):109–203, 1964.CrossRefzbMATHMathSciNetGoogle Scholar
  28. [Hir64b]
    H. Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero: II. Ann. of Math., 79(2):205–326, 1964.CrossRefMathSciNetGoogle Scholar
  29. [IY08]
    Yu. Ilyashenko and S. Yakovenko. Lectures on Analytic Differential Equations. AMS, 2008.Google Scholar
  30. [JL92]
    W. Jäger and S. Luckhaus. On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc., 329(2):819–824, 1992.CrossRefzbMATHMathSciNetGoogle Scholar
  31. [Kal03]
    V. Kaloshin. The existential Hilbert 16-th problem and an estimate for cyclicity of elementary polycycles. Invent. Math., 151(3):451–512, 2003.CrossRefzbMATHMathSciNetGoogle Scholar
  32. [KM89]
    A.Yu. Kolesov and E.F. Mishchenko. Existence and stability of the relaxation torus. Russ. Math. Surv., 44(3):204–205, 1989.Google Scholar
  33. [Kol94]
    A.Yu. Kolesov. On the existence and stability of a two-dimensional relaxational torus. Math. Notes, 56(6):1238–1243, 1994.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [Kol07]
    J. Kollár. Lectures on Resolution of Singularities. Princeton University Press, 2007.Google Scholar
  35. [KS01b]
    M. Krupa and P. Szmolyan. Extending geometric singular perturbation theory to nonhyperbolic points - fold and canard points in two dimensions. SIAM J. Math. Anal., 33(2):286–314, 2001.CrossRefzbMATHMathSciNetGoogle Scholar
  36. [KS01d]
    M. Krupa and P. Szmolyan. Geometric analysis of the singularly perturbed fold. in: Multiple-Time-Scale Dynamical Systems, IMA Vol. 122:89–116, 2001.Google Scholar
  37. [KS01e]
    M. Krupa and P. Szmolyan. Relaxation oscillation and canard explosion. J. Differential Equat., 174: 312–368, 2001.CrossRefzbMATHMathSciNetGoogle Scholar
  38. [Kue14]
    C. Kuehn. Normal hyperbolicity and unbounded critical manifolds. Nonlinearity, 27(6):1351–1366, 2014.CrossRefzbMATHMathSciNetGoogle Scholar
  39. [Lee06]
    J.M. Lee. Introduction to Smooth Manifolds. Springer, 2006.Google Scholar
  40. [Lev90]
    H.A. Levine. The role of critical exponents in blowup theorems. SIAM Rev., 32(2):262–288, 1990.CrossRefzbMATHMathSciNetGoogle Scholar
  41. [Lu76]
    Y.-C. Lu. Singularity Theory and an Introduction to Catastrophe Theory. Springer, 1976.Google Scholar
  42. [LXY03]
    W. Liu, D. Xiao, and Y. Yi. Relaxation oscillations in a class of predator–prey systems. J. Differential Equat., 188(1):306–331, 2003.CrossRefzbMATHMathSciNetGoogle Scholar
  43. [Mae07a]
    P. De Maesschalck. Ackerberg-O’Malley resonance in boundary value problems with a turning point of any order. Comm. Pure Appl. Anal., 6(2):311–333, 2007.CrossRefzbMATHGoogle Scholar
  44. [MKKR94]
    E.F. Mishchenko, Yu.S. Kolesov, A.Yu. Kolesov, and N.Kh. Rozov. Asymptotic Methods in Singularly Perturbed Systems. Plenum Press, 1994.Google Scholar
  45. [MR80]
    E.F. Mishchenko and N.Kh. Rozov. Differential Equations with Small Parameters and Relaxation Oscillations (translated from Russian). Plenum Press, 1980.Google Scholar
  46. [Pan06]
    D. Panazzolo. Resolution of singularities of real-analytic vector fields in dimension three. Acta Math., 197:167–289, 2006.CrossRefzbMATHMathSciNetGoogle Scholar
  47. [SS04a]
    B. Sandstede and A. Scheel. Evans function and blow-up methods in critical eigenvalue problems. Discr. Cont. Dyn. Syst., 10:941–964, 2004.CrossRefzbMATHMathSciNetGoogle Scholar
  48. [SS05a]
    B. Sandstede and A. Scheel. Absolute instabilities of standing pulses. Nonlinearity, 18:331–378, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  49. [SW04]
    P. Szmolyan and M. Wechselberger. Relaxation oscillations in \(\mathbb{R}^{3}\). J. Differential Equat., 200:69–104, 2004.CrossRefzbMATHMathSciNetGoogle Scholar
  50. [Wec02]
    M. Wechselberger. Extending Melnikov-theory to invariant manifolds on non-compact domains. Dynamical Systems, 17(3):215–233, 2002.CrossRefzbMATHMathSciNetGoogle Scholar
  51. [Wlo05]
    J. Wlodarczyk. Simple Hironaka resolution in characteristic zero. J. Amer. Math. Soc., 18(4):779–822, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  52. [Zol87]
    H. Zoldek. Bifurcations of certain family of planar vector fields tangent to axes. J. Differential Equat., 67(1):1–55, 1987.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Christian Kuehn
    • 1
  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria

Personalised recommendations