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Journal of Dynamics and Differential Equations

, Volume 27, Issue 3–4, pp 989–1006 | Cite as

Regularity of Center Manifolds via the Graph Transform

  • Björn SandstedeEmail author
  • Thunwa Theerakarn
Article

Abstract

The purpose of this paper is to give a short self-contained proof of the center-manifold theorem for maps and vector fields in finite-dimensional spaces using the graph transform. In particular, regularity of the center manifold is established using a direct argument that is based on the closedness of sets of differentiable functions whose highest derivatives are Lipschitz continuous in the space of continuous functions; this argument avoids the fiber contraction theorem that is commonly used in this context.

Keywords

Center manifolds Graph transform Regularity 

Mathematics Subject Classification

37G10 34C23 37L10 

Notes

Acknowledgments

Sandstede was partially supported by the NSF through Grant DMS-0907904. Theerakarn acknowledges support by Brown University through an Undergraduate Teaching and Research Award. This paper is dedicated to the memory of Professor Klaus Kirchgässner: he was a terrific mentor and an engaged researcher who continues to be an inspiration to many of us.

References

  1. 1.
    Bates, P.W., Jones, C.K.R.T.: Invariant manifolds for semilinear partial differential equations. Dynamics Reported, pp. 1–38. Wiley, Chicester (1989)CrossRefGoogle Scholar
  2. 2.
    Bates, P.W., Lu, K., Zeng, C.: Existence and persistence of invariant manifolds for semiflows in Banach space. Mem. Am. Math. Soc. 135, viii+129 (1998)MathSciNetGoogle Scholar
  3. 3.
    Bonatti, C., Crovisier, S.: Center manifolds for partially hyperbolic sets without strong unstable connections. J. Inst. Math. Jussieu (2015) (To appear)Google Scholar
  4. 4.
    Carr, J.: Applications of Centre Manifold Theory. Applied Mathematical Sciences. Springer, New York (1981)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chicone, C.: Ordinary Differential Equations with Applications. Springer, New York (2006)zbMATHGoogle Scholar
  6. 6.
    Chow, S.-N., Lin, X.-B., Lu, K.: Smooth invariant foliations in infinite-dimensional spaces. J. Differ. Equ. 94, 266–291 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Chow, S.-N., Liu, W., Yi, Y.: Center manifolds for invariant sets. J. Differ. Equ. 168, 355–385 (2000a)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Chow, S.-N., Liu, W., Yi, Y.: Center manifolds for smooth invariant manifolds. Trans. Am. Math. Soc. 352, 5179–5211 (2000b)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Chow, S.-N., Lu, K.: Invariant manifolds for flows in Banach spaces. J. Differ. Equ. 74, 285–317 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Dangelmayr, G., Fiedler, B., Kirchgässner, K., Mielke, A.: Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability. Pitman Research Notes in Mathematics Series. Longman, Harlow (1996)zbMATHGoogle Scholar
  11. 11.
    de la Llave, R.: A smooth center manifold theorem which applies to some ill-posed partial differential equations with unbounded nonlinearities. J. Dyn. Differ. Equ. 21, 371–415 (2009)CrossRefzbMATHGoogle Scholar
  12. 12.
    Dias, F., Iooss, G.: Water-Waves as a Spatial Dynamical System. Handbook of Mathematical Fluid Dynamics II, pp. 443–499. North-Holland, Amsterdam (2003)Google Scholar
  13. 13.
    Doelman, A., Sandstede, B., Scheel, A., Schneider, G.: Propagation of hexagonal patterns near onset. Eur. J. Appl. Math. 14, 85–110 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Doelman, A., Sandstede, B., Scheel, A., Schneider, G.: The dynamics of modulated wave trains. Mem. Am. Math. Soc. 199, viii+105 (2009)MathSciNetGoogle Scholar
  15. 15.
    Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21(193—-226), 1972 (1971)MathSciNetGoogle Scholar
  16. 16.
    Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 53–98 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Fischer, G.: Zentrumsmannigfaltigkeiten bei elliptischen Differentialgleichungen. Math. Nachr. 115, 137–157 (1984)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Gallay, T.: A center-stable manifold theorem for differential equations in Banach spaces. Comm. Math. Phys. 152, 249–268 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Haragus, M., Iooss, G.: Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-dimensional Dynamical Systems. Springer, London (2011)CrossRefzbMATHGoogle Scholar
  20. 20.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics. Springer, Berlin (1981)zbMATHGoogle Scholar
  21. 21.
    Hirsch, M.W., Pugh, C.C., Shub, M.: Lecture Notes in Mathematics. Springer, Berlin (1977)Google Scholar
  22. 22.
    Homburg, A.J.: Global aspects of homoclinic bifurcations of vector fields. Mem. Am. Math. Soc 121, viii+128 (1996)MathSciNetGoogle Scholar
  23. 23.
    Iooss, G., Kirchgässner, K.: Travelling waves in a chain of coupled nonlinear oscillators. Comm. Math. Phys. 211, 439–464 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Iooss, G., Mielke, A.: Bifurcating time-periodic solutions of Navier-Stokes equations in infinite cylinders. J. Nonlinear Sci. 1, 107–146 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Johnson, M.A., Noble, P., Rodrigues, L.M., Zumbrun, K.: Nonlocalized modulation of periodic reaction diffusion waves: the Whitham equation. Arch. Ration. Mech. Anal. 207, 669–692 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Kelley, A.: The stable, center-stable, center, center-unstable, unstable manifolds. J. Differ. Equ. 3, 546–570 (1967)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Kirchgässner, K.: Wave-solutions of reversible systems and applications. J. Differ. Equ. 45, 113–127 (1982)CrossRefzbMATHGoogle Scholar
  28. 28.
    Kirchgässner, K.: Nonlinearly Resonant Surface Waves and Homoclinic Bifurcation. Advances in Applied Mechanics, pp. 135–181. Academic Press, Boston (1988)Google Scholar
  29. 29.
    Krieger, J., Nakanishi, K., Schlag, W.: Center-stable manifold of the ground state in the energy space for the critical wave equation. Math. Ann. 361, 1–50 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Marsden, J.E., McCracken, M.: The Hopf Bifurcation and Its Applications. Springer, New York (1976)CrossRefzbMATHGoogle Scholar
  31. 31.
    Mielke, A.: A reduction principle for nonautonomous systems in infinite-dimensional spaces. J. Differ. Equ. 65, 68–88 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Mielke, A.: Reduction of quasilinear elliptic equations in cylindrical domains with applications. Math. Methods Appl. Sci. 10, 51–66 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Mielke, A.: Hamiltonian and Lagrangian Flows on Center Manifolds. Lecture Notes in Mathematics. Springer, Berlin (1991). with applications to elliptic variational problemsCrossRefzbMATHGoogle Scholar
  34. 34.
    Mielke, A.: On nonlinear problems of mixed type: a qualitative theory using infinite-dimensional center manifolds. J. Dyn. Differ. Equ. 4, 419–443 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Mielke, A.: The Ginzburg-Landau Equation in Its Role as a Modulation Equation. Handbook of Dynamical Systems, pp. 759–834. North-Holland, Amsterdam (2002)Google Scholar
  36. 36.
    Pego, R.L., Quintero, J.R.: A host of traveling waves in a model of three-dimensional water-wave dynamics. J. Nonlinear Sci. 12, 59–83 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  37. 37.
    Pliss, V.A.: A reduction principle in the theory of stability of motion. Izv. Akad. Nauk SSSR Ser. Mat. 28, 1297–1324 (1964)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Sandstede, B.: Center manifolds for homoclinic solutions. J. Dyn. Differ. Equ. 12, 449–510 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  39. 39.
    Sandstede, B., Scheel, A.: Essential instability of pulses and bifurcations to modulated travelling waves. Proc. R. Soc. Edinb. A 129, 1263–1290 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  40. 40.
    Sandstede, B., Scheel, A.: Defects in oscillatory media: toward a classification. SIAM J. Appl. Dyn. Syst. 3, 1–68 (2004). (electronic)CrossRefMathSciNetzbMATHGoogle Scholar
  41. 41.
    Shub, M.: Global Stability of Dynamical Systems. Springer, New York (1987)CrossRefzbMATHGoogle Scholar
  42. 42.
    Sijbrand, J.: Properties of center manifolds. Trans. Am. Math. Soc. 289, 431–469 (1985)CrossRefMathSciNetzbMATHGoogle Scholar
  43. 43.
    Takens, F., Vanderbauwhede, A.: Local Invariant Manifolds and Normal Forms. Handbook of Dynamical Systems, pp. 89–124. Elsevier, Amsterdam (2010)Google Scholar
  44. 44.
    Vanderbauwhede, A.: Centre Manifolds, Normal Forms and Elementary Bifurcations. Dynamics Reported, pp. 89–169. Wiley, Chichester (1989)Google Scholar
  45. 45.
    Vanderbauwhede, A., Iooss, G.: Center Manifold Theory in Infinite Dimensions. Dynamics Reported, pp. 125–163. Springer, Berlin (1992)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA

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