Advertisement

General Fenichel Theory

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

Abstract

Important Remark: This chapter provides a proof of the perturbation of normally hyperbolic invariant manifolds due to Fenichel. Chapter 3 specializes the general case to fast–slow systems, so that the current chapter is necessary only for readers interested in the proof and its strategy. It is possible to skip this chapter at first reading, get an idea how the theorem is used, and then return to the proof.

Bibliography

  1. [AMR88]
    R. Abraham, J.E. Marsden, and T. Ratiu. Manifolds, Tensor Analysis and Applications. Springer, 1988.Google Scholar
  2. [BB13]
    P. Berger and A. Bounemoura. A geometrical proof of the persistence of normally hyperbolic submanifolds. Dynamical Systems, pages 1–15, 2013. to appear.Google Scholar
  3. [BCHV93]
    H.W. Broer, S.N. Chow, A. Hagen, and G. Vegter. A normally elliptic Hamiltonian bifurcation. Z. Angewand. Math. Phys., 44(3):389–432, 1993.CrossRefzbMATHGoogle Scholar
  4. [BF96]
    F. Battelli and M. Fečkan. Global centre manifolds in singular systems. Nonlinear Different. Equat. Appl., 3:19–34, 1996.CrossRefGoogle Scholar
  5. [BK94]
    I.U. Bronstein and A.Ya. Kopanskii. Smooth invariant manifolds and normal forms. World Scientific, 1994.Google Scholar
  6. [BLZ98]
    P.W. Bates, K. Lu, and C. Zeng. Existence and persistence of invariant manifolds for semiflows in Banach spaces. Mem. Amer. Math. Soc., 135, 1998.Google Scholar
  7. [BS11b]
    S. Bianchini and L. Spinolo. Invariant manifolds for a singular ordinary differential equation. J. Differential Equat., 250(4):1788–1827, 2011.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [Car81]
    J. Carr. Applications of Centre Manifold Theory. Springer, 1981.Google Scholar
  9. [CL00]
    C. Chicone and W. Liu. On the continuation of an invariant torus in a family with rapid oscillations. SIAM J. Math. Anal., 31(2):386–415, 2000.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [DGR13]
    A. Delshams, M. Gidea, and P. Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete Contin. Dyn. Syst., 33(3):1089–1112, 2013.zbMATHMathSciNetGoogle Scholar
  11. [DLS00]
    A. Delshams, R. De La Llave, and T.M. Seara. A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of \(\mathbb{T}^{2}\). Comm. Math. Phys., 209(2):353–392, 2000.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [DLS08]
    A. Delshams, R. De La Llave, and T.M. Seara. Geometric properties of the scattering map of a normally hyperbolic invariant manifold. Adv. Math., 217(3):1096–1153, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [Eld12]
    J. Eldering. Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry. Comptes Rendus Mathematique, 350(11):617–620, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [Eld13]
    J. Eldering. Normally Hyperbolic Invariant Manifolds: The Noncompact Case. Atlantis Press, 2013.Google Scholar
  15. [Fen71]
    N. Fenichel. Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J., 21:193–225, 1971.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [Fen74]
    N. Fenichel. Asymptotic stability with rate conditions. Indiana Univ. Math. J., 23:1109–1137, 1974.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [Fen77]
    N. Fenichel. Asymptotic stability with rate conditions II. Indiana Univ. Math. J., 26:81–93, 1977.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [Fen79]
    N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Differential Equat., 31:53–98, 1979.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [Fis06]
    M.E. Fisher. Hyperbolic sets that are not locally maximal. Ergodic Theory Dyn. Syst., 26(5):1491–1510, 2006.CrossRefzbMATHGoogle Scholar
  20. [Fol99]
    G. Folland. Real Analysis - Modern Techniques and Their Applications. Wiley, 1999.Google Scholar
  21. [Fra11]
    T. Frankel. The Geometry of Physics. CUP, 2011.Google Scholar
  22. [Gar93]
    R.A. Gardner. An invariant-manifold analysis of electrophoretic traveling waves. J. Dyn. Diff. Eq., 5(4):599–606, 1993.CrossRefzbMATHGoogle Scholar
  23. [GH83]
    J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York, NY, 1983.CrossRefzbMATHGoogle Scholar
  24. [Hal09]
    J.K. Hale. Ordinary Differential Equations. Dover, New York, NY, 2009.Google Scholar
  25. [Han98]
    H. Hanßmann. The quasi-periodic centre-saddle bifurcation. J. Differential Equat., 142(2):305–370, 1998.CrossRefzbMATHGoogle Scholar
  26. [Hop66]
    F.C. Hoppenstaedt. Singular perturbations on the infinite interval. Trans. Amer. Math. Soc., 123(2):521–535, 1966.CrossRefMathSciNetGoogle Scholar
  27. [HP70]
    M.W. Hirsch and C.C. Pugh. Stable manifolds and hyperbolic sets. In S.-S. Chern and S. Smale, editors, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), pages 133–163. AMS, 1970.Google Scholar
  28. [HP05]
    B. Hasselblatt and Y. Pesin. Partially hyperbolic dynamical systems. In Handbook of Dynamical Systems 1B, pages 1–55. Elsevier, 2005.Google Scholar
  29. [HPS77]
    M.W. Hirsch, C.C. Pugh, and M. Shub. Invariant Manifolds. Springer, 1977.Google Scholar
  30. [HSD03]
    M.W. Hirsch, S. Smale, and R. Devaney. Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, 2nd edition, 2003.Google Scholar
  31. [Irw70]
    M.C. Irwin. On the stable manifold theorem. Bull. London Math. Soc., 2(2):196–198, 1970.CrossRefzbMATHMathSciNetGoogle Scholar
  32. [Irw80]
    M.C. Irwin. A new proof of the pseudostable manifold theorem. J. London Math. Soc., 2(3):557–566, 1980.CrossRefMathSciNetGoogle Scholar
  33. [Jän01]
    K. Jänich. Vector Analysis. Springer, 2001.Google Scholar
  34. [Jon95]
    C.K.R.T. Jones. Geometric singular perturbation theory. In Dynamical Systems (Montecatini Terme, 1994), volume 1609 of Lect. Notes Math., pages 44–118. Springer, 1995.Google Scholar
  35. [Kel67]
    A. Kelley. The stable, center-stable, center, center-unstable, unstable manifolds. J. Differential Equat., 3(4):546–570, 1967.CrossRefzbMATHMathSciNetGoogle Scholar
  36. [KH95]
    A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. CUP, 1995.Google Scholar
  37. [KK91]
    H.K. Khalil and P.V. Kokotovic. On stability properties of nonlinear systems with slowly varying inputs. IEEE Trans. Aut. Contr., 36(2):229, 1991.Google Scholar
  38. [Kop85]
    N. Kopell. Invariant manifolds and the initialization problem for some atmospheric equations. Physica D, 14(2):203–215, 1985.CrossRefzbMATHMathSciNetGoogle Scholar
  39. [Kör04]
    T.W. Körner. A Companion to Analysis: A Second First and First Second Course in Analysis. AMS, 2004.Google Scholar
  40. [KPB00]
    L. Kocarev, U. Parlitz, and R. Brown. Robust synchronization of chaotic systems. Phys. Rev. E, 61(4):3716–3720, 2000.CrossRefGoogle Scholar
  41. [KS86]
    A. Katok and J.-M. Strelcyn. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities, volume 1222 of Springer Lecture Notes in Math. Springer, 1986.Google Scholar
  42. [Kue02]
    W. Kuehnel. Differential Geometry - Curves, Surfaces, Manifolds. AMS, 2002.Google Scholar
  43. [Lee06]
    J.M. Lee. Introduction to Smooth Manifolds. Springer, 2006.Google Scholar
  44. [Lya47]
    A.M. Lyapunov. Problème géneral de la stabilité du mouvement. Princeton University Press, 1947.Google Scholar
  45. [Man78]
    R. Mané. Persistent manifolds are normally hyperbolic. Trans. Amer. Math. Soc., 246:261–283, 1978.CrossRefzbMATHMathSciNetGoogle Scholar
  46. [Mic08]
    P.W. Michor. Topics in Differential Geometry. AMS, 2008.Google Scholar
  47. [Mie90]
    A. Mielke. Normal hyperbolicity of center manifolds and Saint-Venant’s principle. Arch. Rat. Mech. Anal., 110(4):353–372, 1990.CrossRefzbMATHMathSciNetGoogle Scholar
  48. [Mil97b]
    J.W. Milnor. Topology from the Differentiable Viewpoint. Princeton University Press, 1997.Google Scholar
  49. [MPSS93]
    J. Mallet-Paret, G.R. Sell, and Z.D. Shao. Obstructions to the existence of normally hyperbolic inertial manifolds. Indiana Univ. Math. J., 42(3):1027–1055, 1993.CrossRefzbMATHMathSciNetGoogle Scholar
  50. [MS87]
    J. Marsden and J. Scheurle. The construction and smoothness of invariant manifolds by the deformation method. SIAM J. Math. Anal., 18(5):1261–1274, 1987.CrossRefzbMATHMathSciNetGoogle Scholar
  51. [MS96c]
    R. McGehee and E. Sander. A new proof of the stable manifold theorem. Z. Angew. Math. Phys., 47(4):497–513, 1996.CrossRefzbMATHMathSciNetGoogle Scholar
  52. [Mun97]
    J.R. Munkres. Analysis on Manifolds. Westview Press, 1997.Google Scholar
  53. [Nak03]
    M. Nakahara. Geometry, Topology and Physics. Taylor & Francis, 2003.Google Scholar
  54. [NS13b]
    K. Nipp and D. Stoffer. Invariant Manifolds in Discrete and Continuous Dynamical Systems. EMS, 2013.Google Scholar
  55. [Osi97]
    G. Osipenko. Linearization near a locally nonunique invariant manifold. Discr. Cont. Dyn. Syst., 3: 189–205, 1997.CrossRefzbMATHMathSciNetGoogle Scholar
  56. [Per29]
    O. Perron. Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen. Math. Zeitschr., 29(1):129–160, 1929.CrossRefMathSciNetGoogle Scholar
  57. [Pes04]
    Ya.B. Pesin. Lectures on Partial Hyperbolicity and Stable Ergodicity. EMS, 2004.Google Scholar
  58. [Poe03]
    C. Poetzsche. Slow and fast variables in non-autonomous difference equations. J. Difference Equ. Appl., 9(5):473–487, 2003.CrossRefzbMATHMathSciNetGoogle Scholar
  59. [PS70]
    C. Pugh and M. Shub. Linearization of normally hyperbolic diffeomorphisms and flows. Invent. Math., 10(3):187–198, 1970.CrossRefzbMATHMathSciNetGoogle Scholar
  60. [PS75c]
    C. Pugh and M. Shub. Axiom A actions. Invent. Math., 29(1):7–38, 1975.CrossRefzbMATHMathSciNetGoogle Scholar
  61. [PT77]
    J. Palis and F. Takens. Topological equivalence of normally hyperbolic dynamical systems. Topology, 16(4):335–345, 1977.CrossRefzbMATHMathSciNetGoogle Scholar
  62. [Rat06]
    J.G. Ratcliffe. Foundations of Hyperbolic Manifolds. Springer, 2006.Google Scholar
  63. [Rud76]
    W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, 1976.Google Scholar
  64. [Rud91]
    W. Rudin. Functional Analysis. McGraw-Hill, 1991.Google Scholar
  65. [Sac69]
    R. Sacker. A perturbation theorem for invariant manifolds and Hölder continuity. J. Math. Mech., 18:705–762, 1969.zbMATHMathSciNetGoogle Scholar
  66. [Sak90]
    K. Sakamoto. Invariant manifolds in singular perturbation problems for ordinary differential equations. Proc. Royal Soc. Ed., 116:45–78, 1990.CrossRefzbMATHMathSciNetGoogle Scholar
  67. [Sij85]
    J. Sijbrand. Properties of center manifolds. Trans. Amer. Math. Soc., 289:431–469, 1985.CrossRefzbMATHMathSciNetGoogle Scholar
  68. [Sma67]
    S. Smale. Differentiable dynamical systems. Bull. Amer. Math. Soc., 289:747–817, 1967.CrossRefMathSciNetGoogle Scholar
  69. [Spi99]
    M. Spivak. A Comprehensive Introduction to Differential Geometry. Publish or Perish, 1999. Volumes 1–5.Google Scholar
  70. [Swa83]
    R. Swanson. The spectral characterization of normal hyperbolicity. Proc. Amer. Math. Soc., 89(3): 503–509, 1983.CrossRefzbMATHMathSciNetGoogle Scholar
  71. [Tes12]
    G. Teschl. Ordinary Differential Equations and Dynamical Systems. AMS, 2012.Google Scholar
  72. [Tik52]
    A.N. Tikhonov. Systems of differential equations containing small parameters in the derivatives. Mat. Sbornik N. S., 31:575–586, 1952.MathSciNetGoogle Scholar
  73. [vS87]
    S.J. van Strien. Normal hyperbolicity and linearisability. Invent. Math., 87(2):377–384, 1987.CrossRefzbMATHMathSciNetGoogle Scholar
  74. [Wel76]
    J.C. Wells. Invariant manifolds of non-linear operators. Pac. J. Math., 62(1):285–293, 1976.CrossRefzbMATHGoogle Scholar
  75. [Whi36]
    H. Whitney. Differentiable manifolds. Ann. Math., 37(3):645–680, 1936.CrossRefMathSciNetGoogle Scholar
  76. [Wig88]
    S. Wiggins. On the detection and dynamical consequences of orbits homoclinic to hyperbolic periodic orbits and normally hyperbolic invariant tori in a class of ordinary differential equations. SIAM J. Appl. Math., 48(2):262–285, 1988.CrossRefzbMATHMathSciNetGoogle Scholar
  77. [Wig94]
    S. Wiggins. Normally Hyperbolic Invariant Manifolds in Dynamical Systems. Springer, 1994.Google 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