Non linear interactions between a Hopf bifurcation and a pitchfork-type stationary bifurcation can produce secondary bifurcations of periodic solutions, and tertiary bifurcations of periodic or aperiodic solutions lying on an invariant torus. A complete classification of the resulting bifurcation diagrams is presented, with emphasis on the cases which exhibit tertiary bifurcation. Calculations involving successive transformations to polar normal forms lead to existence theorems for the secondary and tertiary solutions and asymptotic formulae for the invariant torus.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ashkenazi M. and Othmer H.G. Spacial Patterns in coupled biochemical oscillators. J, Math. Biology 5, 305–350, (1978).Google Scholar
  2. [2]
    Bauer L., Keller H.B. and Reiss E.L. Multiple eigenvalues lead to secondary bifurcation. SIAM Review 17, 101–122 (1975).CrossRefGoogle Scholar
  3. [3]
    Baxter R., Eiscrike H. and Stokes A. A pictorial study of an invariant torus in phase space of four dimensions. Ordinary Differential Equations, NRL-MRC Conference. Academic Press (1972)Google Scholar
  4. [4]
    Bouc R., Defilippi M. and Iooss G. On a problem of forced nonlinear oscillations. Nonlinear Analysis 2, 211–224 (1978).CrossRefGoogle Scholar
  5. [5]
    Chow S.N. and Mallet-Paret J. Integral averaging and bifurcation. J. Differential Equations 26, 112–159 (1977).CrossRefGoogle Scholar
  6. [6]
    Cronin J. Bifurcation of periodic solutions. J. Math. Anal. and Appl. 68, 130–151 (1979).CrossRefGoogle Scholar
  7. [7]
    Golubitsky M. and Schaeffer D. Imperfect bifurcation in the presence of symmetry. Comm. Math. Phys. 67, 205–232 (1979).CrossRefGoogle Scholar
  8. [8]
    Guckenheimer J. On a codimension two bifurcation. University of California at Santa (ruz, preprint (1979).Google Scholar
  9. [9]
    Holmes P. Unfolding a degenerate nonlinear oscillator: a codimension two bifurcation. New York Academy of Sciences, to appear.Google Scholar
  10. [10]
    Iooss G. Bifurcation of Maps and Applications. North-Holland (1979).Google Scholar
  11. [11]
    Iooss G. and Joseph D.D. Elementary stability and Bifurcation Theory. To appear.Google Scholar
  12. [12]
    Iooss G. and Langford W.F. Conjectures on the routes to turbulence via bifurcations. New York Academy of Sciences, to appear.Google Scholar
  13. [13]
    Iooss G. and Langford W.F. On the interactions of two Hopf bifurcations. In preparation.Google Scholar
  14. [14]
    Keener J.P. Secondary bifurcation in nonlinear diffusion reaction equations. Studies in Appl. Math. 55, 187–211 (1976).Google Scholar
  15. [15]
    Keener J.P. Infinite period bifurcation and global bifurcation branches. University of Utah, preprint (1979).Google Scholar
  16. [16]
    Langford W.F. Periodic and steady-state mode interactions lead to tori. SIAM J. Appl. Math. 37, 22–48 (1979).CrossRefGoogle Scholar
  17. [17]
    Langford W.F., Arneodo A., Coullet P., Tresser C. and Coste J. A mechanism for a soft mode instability. Submitted to Phys. Lett.Google Scholar
  18. [18]
    Lin J. and Kahn P.B. Qualitative dynamics of three species predator-prey systems. J. Math. Biol. 5, 257–268 (1978).CrossRefGoogle Scholar
  19. [19]
    Marsden J.E. and Mc Cracken M. The Hopf Bifurcation and its Applications. Springer-Verlag, New-York (1976).CrossRefGoogle Scholar
  20. [20]
    Schaeffer D. and Golubitsky M. Boundary conditions and mode jumping in the buckling of a rectangular plate. Comm. Math. Phys. 69, 209–236 (1979).CrossRefGoogle Scholar
  21. [21]
    Shearer M. Coincident bifurcation of equilibrium and periodic solutions of evolution equations. Preprint (1979).Google Scholar

Copyright information

© Springer Basel AG 1980

Authors and Affiliations

  • W. F. Langford
    • 1
  • G. Iooss
    • 2
  1. 1.Department of MathenaticsMc Gill UniversityMontrealCanada
  2. 2.Institut de Mathématiques et Sciences PhysiquesUniversité de Nice Parc ValroseNice CedexFrance

Personalised recommendations