Skip to main content
SpringerLink
Log in

Center Manifolds, Hopf Bifurcation, and Normal Forms

  • Chapter
  • First Online:
Theory and Applications of Abstract Semilinear Cauchy Problems

Part of the book series: Applied Mathematical Sciences ((AMS,volume 201))

  • 1155 Accesses

Abstract

The purpose of this chapter is to develop the center manifold theory, Hopf bifurcation theorem, and normal form theory for abstract semilinear Cauchy problems with non-dense domain.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. H. Amann, Hopf bifurcation in quasilinear reaction-diffusion systems, in “Delay Differential Equations and Dynamical Systems”, S. N. Busenberg and M. Martelli (eds.), Lect. Notes Math. 1475, Springer, Berlin, 1991, pp. 53–63.

    Google Scholar 

  2. A. A. Andronov, A. A. Vitt and S. E. Khaikin, Theory of Oscillators (in Russian), 2nd Ed., Fizmatgiz, Moscow, 1959. English Edition, Pergamon Press, Oxford-New York-Toronto, 1966.

    MATH  Google Scholar 

  3. D. Armbruster, J. Guckenheimer, and P. Holmes, Kuramoto-Sivashinsky dynamics on the center-unstable manifold, SIAM J. Appl. Math. 49 (1989), 676–691.

    MathSciNet  MATH  Google Scholar 

  4. V. I. Arnold, Geometric Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983.

    Book  Google Scholar 

  5. P. Ashwin and Z. Mei, Normal form for Hopf bifurcation of partial differential equations on the square, Nonlinearity 8 (1995), 715–734.

    Google Scholar 

  6. A. Avez, Calcul différentiel, Masson, Paris, 1983.

    MATH  Google Scholar 

  7. J. M. Ball, Saddle point analysis for an ordinary differential equation in a Banach space and an application to dynamic buckling of a beam, in “Nonlinear Elasticity”, R. W. Dickey (ed.), Academic Press, New York, 1973, pp. 93–160.

    Google Scholar 

  8. L. Barreira and C. Vails, Center manifolds for nonuniformly partially hyperbolic diffeomorphisms, J. Math. Pures Appl. 84 (2005), 1693–1715.

    MathSciNet  MATH  Google Scholar 

  9. L. Barreira and C. Vails, Center manifolds for infinite delay, J. Differential Equations 247 (2009), 1297–1310.

    MathSciNet  MATH  Google Scholar 

  10. P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Reported 2, U. Kirchgraber and H. O. Walther (eds.), John Wiley & Sons, 1989, pp. 1–38.

    Google Scholar 

  11. P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semi flows in Banach space, Mem. Amer. Math. Soc. 135 (1998), No. 645.

    MathSciNet  MATH  Google Scholar 

  12. C. Bonatti and S. Crovisier, Center manifolds for partially hyperbolic sets without strong unstable connections, J. Inst. Math. Jussieu 15 (2016), 785–828.

    MathSciNet  MATH  Google Scholar 

  13. J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981.

    Book  MATH  Google Scholar 

  14. S. Chen, Y. Lou and J. Wei, Hopf bifurcation in a delayed reaction-diffusion-advection population model, J. Differential Equations 264 (2018), 5333–5359.

    MathSciNet  MATH  Google Scholar 

  15. S. Chen, J. Shi and J. Wei, Time delay-induced instabilities and Hopf bifurcations in general reaction-diffusion systems, J. Nonlinear Sci. 23 (2013), 1–38.

    MathSciNet  MATH  Google Scholar 

  16. C. Chicone and Y. Latushkin, Center manifolds for infinite dimensional nonautonomous differential equations, J. Differential Equations 141 (1997), 356–399.

    MathSciNet  MATH  Google Scholar 

  17. S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.

    Book  MATH  Google Scholar 

  18. C.-N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.

    Book  MATH  Google Scholar 

  19. C.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces, J. Differential Equations 94 (1991), 266–291.

    Article  MathSciNet  MATH  Google Scholar 

  20. S.-N. Chow, W. Liu, and Y. Yi, Center manifolds for smooth invariant manifolds, Trans. Amer. Math. Soc. 352 (2000), 5179–5211.

    Google Scholar 

  21. S.-N. Chow, W. Liu and Y. Yi, Center manifolds for invariant sets, J. Differential Equations 168 (2000), 355–385.

    Article  MathSciNet  MATH  Google Scholar 

  22. S.-N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations 74 (1988), 285–317.

    Article  MathSciNet  MATH  Google Scholar 

  23. S.-N. Chow and K. Lu, C k centre unstable manifolds, Proc. Roy. Soc. Edinburgh 108A (1988), 303–320.

    Google Scholar 

  24. S.-N. Chow and K. Lu, Invariant manifolds and foliations for quasiperiodic systems, J. Differential Equations 117 (1995), 1–27.

    MathSciNet  MATH  Google Scholar 

  25. S.-N. Chow, K. Lu and Y.-Q. Shen, Normal forms for quasiperiodic evolutionary equations, Discrete Contin. Dyn. Syst. 2 (1996), 65–94.

    Google Scholar 

  26. S. N. Chow and Y. Yi, Center manifold and stability for skew-product flows, J. Dyn. Diff. Equat. 6 (1994), 543–582.

    MathSciNet  MATH  Google Scholar 

  27. M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal. 67 (1977), 53–72.

    Article  MathSciNet  MATH  Google Scholar 

  28. G. Da Prato and A. Lunardi, Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach spaces, Arch. Rational Mech. Anal. 101 (1988), 115–141.

    MathSciNet  MATH  Google Scholar 

  29. G. F. Dell’Antonio and B. D’Onofrio, Construction of a center-unstable manifold for C 1−flows and an application to the Navier-Stokes equation, Arch. Rational Mech. Anal. 93 (1986), 185–201.

    MathSciNet  MATH  Google Scholar 

  30. O. Diekmann and S. A. van Gils, Invariant manifold for Volterra integral equations of convolution type, J. Differential Equations 54 (1984), 139–180. 

    Article  MathSciNet  MATH  Google Scholar 

  31. O. Diekmann and S. A. van Gils, The center manifold for delay equations in the light of suns and stars, in “Singularity Theory and its Applications,”, Lect. Notes Math. 1463, Springer, Berlin, 1991, pp. 122–141.

    Google Scholar 

  32. O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.

    Google Scholar 

  33. J.-P. Eckmann, H. Epstein, and C.E. Wayne, Normal forms for parabolic partial differential equations, Ann. Inst. Henri Poincaré Phys. Théor. 58 (1993), 287–308.

    Google Scholar 

  34. E. Faou, B. Grébert and E. Paturel, Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. I. Finite-dimensional discretization, Numer. Math. 114 (2010), 429–458.

    Google Scholar 

  35. E. Faou, B. Grébert and E. Paturel, Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. II. Abstract splitting, Numer. Math. 114 (2010), 459–490.

    Google Scholar 

  36. T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc. 352 (2000), 2217–2238.

    Google Scholar 

  37. T. Faria, Normal forms for semilinear functional differential equations in Banach spaces and applications, Part II, Discrete Contin. Dynam. Syst. 7 (2001), 155–176.

    MathSciNet  MATH  Google Scholar 

  38. T. Faria, W. Huang and J. Wu, Smoothness of center manifolds for maps and formal adjoints for semilinear FDES in general Banach spaces, SIAM J. Math. Anal. 34 (2002), 173–203.

    MathSciNet  MATH  Google Scholar 

  39. T. Faria and L.T. Magalhães, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcations, J. Differential Equations 122 (1995), 181–200.

    Article  MathSciNet  MATH  Google Scholar 

  40. N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1979), 53–98.

    MathSciNet  MATH  Google Scholar 

  41. C. Foias, L. Hoang, E. Olson, and M. Ziane, On the solutions to the normal form of the Navier-Stokes equations, Indiana Univ. Math. J. 55 (2006), 631–686.

    MathSciNet  MATH  Google Scholar 

  42. K. O. Friedrichs, Advanced Ordinary Differential Equations, Gordon and Breach, New York, 1965.

    MATH  Google Scholar 

  43. Th. Gallay, A center-stable manifold theorem for differential equations in Banach spaces, Comm. Math. Phys. 152 (1993), 249–268.

    Google Scholar 

  44. M. Golubitsky and P. H. Rabinowitz, A sketch of the Hopf bifurcation theorem, in “Selected Works of Eberhard Hopf with Commentaries,” C. S. Morawetz, J. B. Serrin and Y. G. Sinai (eds.), Amer. Math. Soc., Providence, 2002, pp. 111–118.

    Google Scholar 

  45. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.

    Book  MATH  Google Scholar 

  46. S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Springer, New York, 2013.

    Book  MATH  Google Scholar 

  47. J. Hadamard, Sur l’iteration et les solutions asymptotiques des equations differentielles, Bull. Soc. Math. France 29 (1901), 224–228.

    Google Scholar 

  48. J. K. Hale, Integral manifolds of perturbated differential equations, Ann. Math. 73 (1961), 496–531.

    Google Scholar 

  49. J. K. Hale, Ordinary Differential Equations, 2nd Ed., Krieger Pub., Huntington, NY, 1980.

    Google Scholar 

  50. J. K. Hale, Flows on center manifolds for scalar functional differential equations, Proc. Roy. Soc. Edinburgh 101A (1985), 193–201.

    MathSciNet  MATH  Google Scholar 

  51. J. K. Hale and J. C. F. De Oliveira, Hopf bifurcation for functional equations, J. Math. Anal. Appl. 14 (1980), 41–59.

    MathSciNet  MATH  Google Scholar 

  52. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.

    Book  MATH  Google Scholar 

  53. M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Systems, Springer, New York, 2011.

    MATH  Google Scholar 

  54. B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.

    MATH  Google Scholar 

  55. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math. 840, Springer-Verlag, Berlin, 1981.

    Book  MATH  Google Scholar 

  56. M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lect. Notes Math. 583, Springer-Verlag, New York, 1976.

    Google Scholar 

  57. A. Homburg, Global aspects of homoclinic bifurcations of vector fields, Mem. Amer. Math. Soc. 121 (1996), No. 578.

    MathSciNet  MATH  Google Scholar 

  58. E. Hopf, Abzweigung einer periodischer Lösung von einer stationären Lösung eines Differentialsystems, Ber. Verh. Sächs. Akad. Wiss. Leipzig. Math-Nat. K1 95(1943), 3–22.

    MATH  Google Scholar 

  59. H. J. Hupkes and S. M. Verduyn Lunel, Center manifold theory for functional differential equations of mixed type, J. Dyn. Diff. Equat. 19 (2007), 497–560.

    Article  MathSciNet  MATH  Google Scholar 

  60. G. Iooss, Existence et stabilité de la solution périodique secondaire intervenant dans les problèmes d’évolution du type Navier-Stokes, Arch. Rational Mech. Anal. 47 (1972), 301–329.

    Google Scholar 

  61. V. I. Iudovich, The onset of auto-oscillations in a fluid, J. Appl. Math. Mech. 35 (1971), 587–603.

    MathSciNet  Google Scholar 

  62. R. Johnson, Y. Latushkin and Schnaubelt, Reduction principle and asymptotic phase for center manifolds of parabolic systems with nonlinear boundary conditions, J. Dyn. Diff. Equat. 26 (2014), 243–266.

    MathSciNet  MATH  Google Scholar 

  63. D. D. Joseph and D. H. Sattinger, Bifurcating time periodic solutions and their stability, Arch. Rational Mech. Anal. 45 (1972), 79–109.

    MathSciNet  MATH  Google Scholar 

  64. A. Kelley, The stable, center-stable, center, center-unstable, unstable manifolds. J. Differential Equations 3 (1967), 546–570.

    Article  MathSciNet  MATH  Google Scholar 

  65. H. Kielhőfer, Bifurcation Theory: An Introduction with Applications to PDEs, Springer, New York, 2004.

    MATH  Google Scholar 

  66. I. Kmit and L. Recke, Hopf bifurcation for semilinear dissipative hyperbolic systems, J. Differential Equations 257 (2014), 264–309.

    MathSciNet  MATH  Google Scholar 

  67. H. Kokubu, Normal forms for parametrized vector fields and its application to bifurcations of some reaction diffusion equations, Japan J. Appl. Math. 1 (1984), 273–297.

    MathSciNet  MATH  Google Scholar 

  68. T. Krisztin, Invariance and noninvarince of center manifolds of time-t maps with respect to the semiflow, SIAM J. Math. Anal. 36 (2004), 717–739.

    MathSciNet  MATH  Google Scholar 

  69. N. Krylov and N. N. Bogoliubov, The Application of Methods of Nonlinear Mechanics to the Theory of Stationary Oscillations, Pub. 8 Ukrainian Acad. Sci., Kiev, 1934.

    Google Scholar 

  70. Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd Ed., Springer-Verlag, New York, 2004.

    Book  MATH  Google Scholar 

  71. S. Lang, Real Analysis, 2nd Ed., Addison-Wesley, Reading, MA, 1983.

    MATH  Google Scholar 

  72. B. Lani-Wayda, Hopf bifurcation for retarded functional differential equations and for semiflows in Banach spaces, J. Dyn. Diff. Equat. 25 (2013), 1159–1199.

    MathSciNet  MATH  Google Scholar 

  73. Y. Latushkin, J. Prüss and R. Schnaubelt, Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions, Discrete Contin. Dyn. Syst. Ser. B 9 (2008), 595–633.

    Google Scholar 

  74. A. M. Liapunov, Probléme génerale de la stabilité du mouvement, Ann. Fac. Sci. Toulouse 2 (1907), 203–474.

    Google Scholar 

  75. X.-B. Lin, Homoclinic bifurcations with weakly expanding center manifolds Dynamics Reported (New Series) 5, C. K. R. T. Jones, U. Kirchgraber and H. O. Walther (eds.), Springer-Verlag, Berlin, 1996, pp. 99–189.

    Google Scholar 

  76. X. Lin, J. So and J. Wu, Center manifolds for partial differential equations with delays, Proc. Roy. Soc. Edinburgh 122A (1992), 237–254.

    MathSciNet  MATH  Google Scholar 

  77. Z. Liu, P. Magal and S. Ruan, Hopf Bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys. 62 (2011), 191–222.

    Article  MathSciNet  MATH  Google Scholar 

  78. Z. Liu, P. Magal and S. Ruan, Center-unstable manifolds for non-densely defined Cauchy problems and applications to stability of Hopf bifurcation, Can. Appl. Math. Quart. 20 (2012), 135–178.

    Google Scholar 

  79. Z. Liu, P. Magal and S. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models, J. Differential Equations 257 (2014), 921–1011.

    Article  MathSciNet  MATH  Google Scholar 

  80. P. Magal and S. Ruan, Center Manifolds for Semilinear Equations with Non-dense Domain and Applications on Hopf Bifurcation in Age Structured Models, Mem. Amer. Math. Soc. 202 (2009), No. 951.

    Article  MathSciNet  MATH  Google Scholar 

  81. J. Marsden, The Hopf bifurcation for nonlinear semigroups, Bull. Amer. Math. Soc. 79 (1973), 537–541.

    MathSciNet  MATH  Google Scholar 

  82. J. Marsden and M. McCraken, The Hopf Bifurcation and its Applications, Springer-Verlag, New York, 1976.

    Book  Google Scholar 

  83. H. Matsunaga, S. Murakami, Y. Nagabuchi and V. M. Nguyen, Center manifold theorem and stability for integral equations with infinite delay, Funkcial. Ekvac. 58 (2015), 87–134.

    MathSciNet  MATH  Google Scholar 

  84. H. P. McKean and J. Shatah, The nonlinear Schrödinger equation and the nonlinear heat equation - Reduction to linear form, Comm. Pure Appl. Math. 44 (1991), 1067–1080.

    MathSciNet  MATH  Google Scholar 

  85. K. R. Meyer, The implicit function theorem and analytic differential equations, in “Dynamical Systems – Warwick 1974”, A. Manning (ed.), Lect. Notes Math. 468, Springer-Verlag, New York, 1975, pp. 191–208.

    Google Scholar 

  86. K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 2nd Ed., Springer-Verlag, New York, 2009.

    MATH  Google Scholar 

  87. A. Mielke, A reduction principle for nonautonomous systems in infinite-dimensional spaces, J. Differential Equations 65 (1986), 68–88.

    MathSciNet  MATH  Google Scholar 

  88. A. Mielke, Normal hyperbolicity of center manifolds and Saint-Vernant’s principle, Arch. Rational Mech. Anal. 110 (1990), 353–372.

    MathSciNet  MATH  Google Scholar 

  89. J. Moser, A rapidly convergent iteration method and nonlinear differential equations II, Ann. Scuo. Norm. Sup. Pisa 20 (1966), 499–535.

    Google Scholar 

  90. K. Nakanishi and W. Schlag, Invariant manifolds around soliton manifolds for the nonlinear Klein-Gordon equation, SIAM J. Math. Anal. 44 (2012), 1175–1210.

    MathSciNet  MATH  Google Scholar 

  91. Nguyen Van Minh and J. Wu, Invariant manifolds of partial functional differential equations, J. Differential Equations 198 (2004), 381–421.

    Google Scholar 

  92. N. V. Nikolenko, The method of Poincaré normal forms in problems of integrability of equations of evolution type, Russ. Math. Surv. 41 (1986), 63–114.

    MATH  Google Scholar 

  93. O. Perron, Über stabilität und asymptotische verhalten der integrale von differentialgleichungssystemen, Math. Z. 29 (1928), 129–160.

    Google Scholar 

  94. V. A. Pliss, Principal reduction in the theory of stability of motion, Izv. Akad. Nauk. SSSR Mat. Ser. 28 (1964), 1297–1324.

    Google Scholar 

  95. H. Poincaré, Sur le problème des trois corps et les équations de la dynamique, Acta Math. 13 (1890), 1–270.

    Google Scholar 

  96. M. Renardy, A centre manifold theorem for hyperbolic PDEs, Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), 363–377.

    MathSciNet  MATH  Google Scholar 

  97. T. O. Sakamoto, Hopf bifurcation in a reaction-diffusion system with conservation of mass, Nonlinearity 26 (2013), 2027–2049.

    Google Scholar 

  98. B. Sandstede, Center manifolds for homoclinic solutions, J. Dyn. Diff. Equat. 12 (2000), 449–510.

    Google Scholar 

  99. B. Sandstede and T. Theerakarn, Regularity of center manifolds via the graph transform, J. Dyn. Diff. Equat. 27 (2015), 989–1006.

    MathSciNet  MATH  Google Scholar 

  100. D. H. Sattinger, Bifurcation of periodic solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 41 (1971), 66–80.

    MathSciNet  MATH  Google Scholar 

  101. B. Scarpellini, Center manifolds of infinite dimensions I: Main results and applications, Z. Angew. Math. Phys. 42 (1991), 1–32.

    Google Scholar 

  102. A. Scheel, Radially symmetric patterns of reaction-diffusion systems, Mem. Amer. Math. Soc. 165 (2003), No. 786.

    MathSciNet  MATH  Google Scholar 

  103. R. Schnaubelt, Center manifolds and attractivity for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions, Discrete Contin. Dyn. Syst. 35 (2015), 1193–1230.

    MathSciNet  MATH  Google Scholar 

  104. G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002.

    Book  MATH  Google Scholar 

  105. J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), 685–696.

    MathSciNet  MATH  Google Scholar 

  106. C. L. Siegel, Ober die Normalform analytischer Differentialgleichungen in der Nahe einer Gleichgewichtslosung, Nachr. Akad. Wiss. Gottingen, Math.-Phys. (1952), 21–30.

    Google Scholar 

  107. C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Springer-Verlag, New York, 1971.

    Book  MATH  Google Scholar 

  108. J. Sijbrand, Properties of center manifolds, Trans. Amer. Math. Soc. 289 (1985), 431–469.

    Article  MathSciNet  MATH  Google Scholar 

  109. G. Simonett, Center manifolds for quasilinear reaction-diffusion systems, Differential Integral Equations 8 (1995), 753–796.

    Google Scholar 

  110. G. Simonett, Hopf bifurcation and stability for a quasilinear reaction-diffusion system, in “Evolution Equations,” G. Ferreyra, G. Goldstein and F. Neubrander (eds.), Lect. Notes Pure Appl. Math. 168, Dekker, New York, 1995, pp. 407–418.

    Google Scholar 

  111. E. Stumpf, On a differential equation with state-dependent delay: A global center-unstable manifold bordered by a periodic orbit, Ph.D. Dissertation, University of Hamburg, 2010.

    Google Scholar 

  112. E. Stumpf, Attraction property of local center-unstable manifolds for differential equations with state-dependent delay, Electron. J. Qual. Theory Differ. Equ. (2015), No. 4, 45 pp.

    Google Scholar 

  113. E. Stumpf, A note on local center manifolds for differential equations with state-dependent delay, Differential Integral Equations 29 (2016), 1093–1106.

    Google Scholar 

  114. L. Turyn, A center-unstable manifold theorem for parametrically excited surface waves, SIAM J. Math. Anal. 27 (1996), 241–257.

    MathSciNet  MATH  Google Scholar 

  115. A. Vanderbauwhede, Invariant manifolds in infinite dimensions, in “Dynamics of Infinite Dimensional Systems”, S. N. Chow and J. K. Hale (eds.), Springer-Verlag, Berlin, 1987, pp. 409–420.

    Google Scholar 

  116. A. Vanderbauwhede, Center manifold, normal forms and elementary bifurcations, Dynamics Reported 2, U. Kirchgraber and H. O. Walther (eds.), John Wiley & Sons, 1989, pp. 89–169.

    Google Scholar 

  117. A. Vanderbauwhede and S. A. van Gils, Center manifolds and contractions on a scale of Banach spaces, J. Funct. Anal. 72 (1987), 209–224.

    Article  MathSciNet  MATH  Google Scholar 

  118. A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynamics Reported (New Series) 1, C. K. R. T. Jones, U. Kirchgraber and H. O. Walther (eds.), Springer-Verlag, Berlin, 1992, pp. 125–163.

    Google Scholar 

  119. S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Springer-Verlag, New York, 1994.

    Book  MATH  Google Scholar 

  120. J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.

    Book  MATH  Google Scholar 

  121. Y. Yi, A generalized integral manifold theorem, J. Differential Equations 102 (1993), 153–187.

    Article  MathSciNet  MATH  Google Scholar 

  122. K. Yoshida, The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology, Horoshima Math. J. 12 (1982), 321–348.

    Google Scholar 

  123. E. Zehnder, A simple proof of a generalization of a theorem by C. L. Siegel, in “Geometry and Topology”, J. Palis and M. do Carmo (eds.), Lect. Notes Math. 597, Springer-Verlag, Berlin, 1977, pp. 855–866.

    Google Scholar 

  124. E. Zehnder, C. L. Siegel’s linearization theorem in infinite dimensions, Manuscripta Math. 23 (1978), 363–371.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Bordeaux, Université de Bordeaux, Talence, France

    Pierre Magal

  2. Department of Mathematics, University of Miami, Coral Gables, FL, USA

    Shigui Ruan

Authors
  1. Pierre Magal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shigui Ruan
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Magal, P., Ruan, S. (2018). Center Manifolds, Hopf Bifurcation, and Normal Forms. In: Theory and Applications of Abstract Semilinear Cauchy Problems. Applied Mathematical Sciences, vol 201. Springer, Cham. https://doi.org/10.1007/978-3-030-01506-0_6

Download citation

Publish with us

Policies and ethics

Search

Navigation