Skip to main content
Log in

The Hopf Bifurcation Theorem in infinite dimensions

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Alexander, J.C. & J.A. Yorke, Global bifurcation of periodic orbits. Preprint, 1976.

  2. Chafee, N., The bifurcation of one or more closed orbits from an equilibrium point of an autonomous differential equation. J. Differential Equations, 4, 661–679 (1968).

    Google Scholar 

  3. Crandall, M.G. & P.H. Rabinowitz, Bifurcation from simple eigenvalues. J. Functional Anal., 8, 321–340 (1971).

    Google Scholar 

  4. Crandall, M.G. & P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Rational Mech. Analysis, 52, 161–180 (1973).

    Google Scholar 

  5. Crandall, M.G. & P.H. Rabinowitz, The Hopf Bifurcation Theorem. TSR 1604 (1976), Mathematics Research Center, University of Wisconsin, Madison.

    Google Scholar 

  6. Crandall, M.G. & P.H. Rabinowitz, The principle of exchange of stability. Proceedings of the International Symposium on Dynamical Systems, Gainsville, Florida, 1976 (to appear).

  7. Fife, P.C., Branching phenomena in fluid dynamics and chemical reaction-diffusion theory. Proc. Sym. “Eigenvalues of Nonlinear Problems”, Edizioni Cremonese Rome, 23–83, 1974.

    Google Scholar 

  8. Friedman, A., Partial Differential Equations. Holt, Rinehart and Winston, Inc., New York, 1969.

    Google Scholar 

  9. Hartman, P., Ordinary Differential Equations. John Wiley, New York, 1964.

    Google Scholar 

  10. Henry, D., Geometric theory of semilinear parabolic equations, University of Kentucky Lecture Notes, 1974.

  11. Henry, D., Perturbation problems. Northwestern University Lecture Notes, 1974.

  12. Hopf, E., Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems, Ber. Math.-Phys. Kl. Sächs. Akad. Wiss. Leipzig, 94, 3–22 (1942).

    Google Scholar 

  13. Iooss, G., Existence et stabilité de la solution périodiques secondaire intervenant dans les problèmes d'evolution du type Navier-Stokes. Arch. Rational Mech. Analysis 47, 301–329 (1972).

    Google Scholar 

  14. Iooss, G., Bifurcation et Stabilite. Cours de 3 ème cycle, 1973–74, Orsay.

  15. Iudovich, V.I., The onset of auto-oscillations in a fluid. Prikl. Mat. Mek., 35, 638–655 (1971).

    Google Scholar 

  16. Iudovich, V.I., Investigation of auto-oscillations of a continuous medium occuring at loss of stability of a stationary mode. Prikl. Mat. Mek., 36, 450–459 (1972).

    Google Scholar 

  17. Ize, G., Bifurcation global de orbitas periodicas. Preprint, 1976.

  18. Joseph, D.D., Stability of Fluid Motions, Springer, Berlin-Heidelberg-New York, 1976.

    Google Scholar 

  19. Joseph, D.D., & D.A. Nield, Stability of bifurcating time periodic and steady solutions of arbitrary amplitude. Preprint, 1976.

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

    Google Scholar 

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

    Google Scholar 

  22. Marsden, J., & M. McCracken, The Hopf Bifurcation and its Applications. Springer Applied Mathematical Sciences Lecture Notes Series, Vol. 19, 1976.

  23. Poore, A.B., On the theory and application of the Hopf-Friedrichs bifurcation theory. Preprint, 1976.

  24. Ruelle, D., & F. Takens, On the nature of turbulence. Comm. Math. Phys., 20, 167–192 (1971).

    Google Scholar 

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

    Google Scholar 

  26. Sattinger, D.H., Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics No. 309. Springer, New York, 1973.

    Google Scholar 

  27. Schmidt, D.S., Hopf's bifurcation theorem and the center theorem of Liapunov. Preprint, 1976.

  28. Weinberger, H.F., The stability of solutions bifurcating from steady or periodic solutions. Proceedings of the International Symposium on Dynamical Systems, Gainsville Florida, 1976.

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by D.D. Joseph

This research was sponsored in part by the United States Army under Contract No. DAAG 29-75-C-0024, in part by the Office of Naval Research under Contract No. N 00014-76-C-0300, and in part by the National Science Foundation under grant No. MPS 73-8720.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Crandall, M.G., Rabinowitz, P.H. The Hopf Bifurcation Theorem in infinite dimensions. Arch. Rational Mech. Anal. 67, 53–72 (1977). https://doi.org/10.1007/BF00280827

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00280827

Keywords

Navigation