Advertisement

Quasi-Codimension 3 Bifurcation of Invariant T2 Tori for Maps

  • G. Iooss
  • J. E. Los
Conference paper

Abstract

Bifurcations of invariant tori of dimension higher than one for families of maps, or higher than two for families of vector fields in dissipative systems is a puzzling problem. There are experimental evidences of the existence of such tori, for instance in Bénard convection problem [Go-Be] and in Taylor Couette problem [GRS].

Keywords

Normal Form Hopf Bifurcation Implicit Function Theorem Invariant Torus Invariant Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [Bost]
    J.B. Bost.Tores invariants des systèmes dynamiques hamiltoniens. Séminaire Bourbaki 1984–1985 n° 639.Google Scholar
  2. [B.H.T.]
    H.W. Broer, G.B. Huitema, F. Takens. Unfoldings of quasi-periodic tori. Preprint (Univ. of Groningen).Google Scholar
  3. [Bra-B]
    B.L.J. Braaksma, H.W. Broer. On a quasi-periodic Hopf bifurcation. Ann. IHP. Analyse non linéaire, 4,2,p115–168, (1987).MATHMathSciNetGoogle Scholar
  4. [Ch-Io]
    A. Chenciner, G. Iooss. Bifurcation de tores invariants. Arch. Rat. Mech. Anal.69,3,109–198,(1979)MATHMathSciNetGoogle Scholar
  5. [Ch-Io]
    A. Chenciner, G. Iooss. and Persistance et bifurcation de tores invariants.Arch.Rat.Mech.Anal.71,4,301–306,(1979).MATHMathSciNetGoogle Scholar
  6. [Chen]
    A. Chenciner. Bifurcation de point fixes elliptiques I, courbes invariantes, Publ Math IHES, 61, 67–127, (1985).MATHMathSciNetGoogle Scholar
  7. [Go-Be]
    J.P. Gollub,S.V. Benson. Many routes to turbulent convection.J.Fluid. Mech. 100,3,449,(1980).CrossRefADSGoogle Scholar
  8. [GRS]
    M. Gorman,L.A. Reith,H.L. Swinney. Modulation patterns, multiple frequencies,and other phenomena in circular Couette flow. Nonlinear dynamics. Ann.N.YAcad.Sci.357.10–21.R.Helleman ed. (1980).CrossRefADSGoogle Scholar
  9. [Gu-Ho]
    J. Guckenheimer, P. Holmes. Nonlinear oscillations, Dynamical systems, and Bifurcations of vector fields. Appl.Math.Sci 42. Springer Verlag (1983).MATHGoogle Scholar
  10. [Ham]
    R.S. Hamilton. The inverse function theorem of Nash-Moser. BAMS, 7, 1,65–222, (1982).CrossRefMATHMathSciNetGoogle Scholar
  11. [Her79]
    M. Herman.Sur la conjugaison differentiable des diffeomorphismes du cercle a des rotations. Pub.Math IHES, 49, 5–233,(1979).MATHMathSciNetGoogle Scholar
  12. [Her83]
    M. Herman. Sur les courbes invariantes par les difféomorphismes de l’anneau I. Astérisque, 103–104, (1983).Google Scholar
  13. [Io-La]
    G. Iooss,W.F. Langford. Conjectures on the route to turbulence via bifurcations. Nonlinear Dynamics. Ann.N.Y.Acad.Sci, 357, 489–505, R.Helleman ed. (1980).CrossRefADSGoogle Scholar
  14. [Io81]
    G. Iooss. Bifurcations élémentaires-successions et interactions. Nonlinear phenomena in Chemical dynamics. Vidal- Pacault ed, 71–78,(1981).Google Scholar
  15. [Io-Lo]
    G. Iooss, J.E. Los. Quasi-genericity of bifurcations to invariant tori for maps. Preprint 1987.Google Scholar
  16. [Jo-Ze]
    R. Jost, E. Zehnder. A generalization of the Hopf Bifurcation Theorem. Helv. Phys. Acta, 45, 258–276, (1972).Google Scholar
  17. [Lan]
    O.E. Lanford. Bifurcation of periodic solutions into invariant tori: the work of Ruelle and Takens. Lect.Notes.Math,322, p159–192, Springer Verlag(1973).CrossRefMathSciNetGoogle Scholar
  18. [Los1]
    J.E.Los. Phénomènes de petits diviseurs dans les dédoublements de courbes invariantes. Ann. IHP, Analyse non linéaire (to appear).Google Scholar
  19. [Los2]
    J.E. Los. Doubling bifurcation for invariant curves: C invariant curve on Ck cylinder. (Submitted lo Nonlinearity). Preprint n° 124 Univ Nice (1986).Google Scholar
  20. [Mos]
    J. Moser. Convergent series expansion for quasi-periodic motion, Math. Annalen, 169, 136–176 (1967).CrossRefMATHGoogle Scholar
  21. [Ru-Ta]
    D. Ruelle, F. Takens. On the nature of turbulence. Comm. Math. Phys., 20, 167–192, (1971).CrossRefMATHADSMathSciNetGoogle Scholar
  22. [Russ]
    H. Rüssmann.Über invariante kurven differenzierbarer abbildungen eines kreisrings.Nach. Akad. Wiss. Göttingen Math. Phys. K1.II, 67–105,(1970).Google Scholar
  23. [Sac]
    R. Sacker. On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations. Thesis, New-York University, IMM-NYU 333, 1964.Google Scholar
  24. [Sch]
    J. Scheurle. Bifurcation of quasi-periodic solutions from equilibrium points of reversible dynamical systems. Arch Rat Mech Anal, 97,2,103–139,1987.CrossRefMATHMathSciNetGoogle Scholar
  25. [Sell]
    G.R. Sell. Bifurcation of higher dimensional tori. Arch.Rat.Mech. Anal. 69,3,199–230,(1979).CrossRefMATHMathSciNetGoogle Scholar
  26. [Yoc]
    J.C. Yoccoz. Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne. Ann.Sci.ENS.17, 333–359 (1984).MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • G. Iooss
    • 1
  • J. E. Los
    • 1
  1. 1.Laboratoire de Mathématiques, U.A. 168Université de NiceNiceFrance

Personalised recommendations