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The Hopf bifurcation theorem in infinite dimensions. (English) Zbl 0385.34020

34C25Periodic solutions of ODE
34A99General theory of ODE
34G10Linear ODE in abstract spaces
Full Text: DOI
[1] Alexander, J.C. & J.A. Yorke, Global bifurcation of periodic orbits. Preprint, 1976. · Zbl 0321.58006
[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). · Zbl 0169.11301 · doi:10.1016/0022-0396(68)90015-6
[3] Crandall, M.G. & P.H. Rabinowitz, Bifurcation from simple eigenvalues. J. Functional Anal., 8, 321-340 (1971). · Zbl 0219.46015 · doi:10.1016/0022-1236(71)90015-2
[4] Crandall, M.G. & P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Rational Mech. Analysis, 52, 161-180 (1973). · Zbl 0275.47044 · doi:10.1007/BF00282325
[5] Crandall, M.G. & P.H. Rabinowitz, The Hopf Bifurcation Theorem. TSR 1604 (1976), Mathematics Research Center, University of Wisconsin, Madison. · Zbl 0385.34020
[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). · Zbl 0537.34044
[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.
[8] Friedman, A., Partial Differential Equations. Holt, Rinehart and Winston, Inc., New York, 1969. · Zbl 0224.35002
[9] Hartman, P., Ordinary Differential Equations. John Wiley, New York, 1964. · Zbl 0125.32102
[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).
[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). · Zbl 0258.35057 · doi:10.1007/BF00281637
[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).
[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).
[17] Ize, G., Bifurcation global de orbitas periodicas. Preprint, 1976.
[18] Joseph, D.D., Stability of Fluid Motions, Springer, Berlin-Heidelberg-New York, 1976. · Zbl 0345.76023
[19] Joseph, D.D., & D.A. Nield, Stability of bifurcating time periodic and steady solutions of arbitrary amplitude. Preprint, 1976. · Zbl 0344.34043
[20] Joseph, D.D., & D.H. Sattinger, Bifurcating time periodic solutions and their stability. Arch. Rational Mech. Anal., 45, 79-109 (1972). · Zbl 0239.76057 · doi:10.1007/BF00253039
[21] Marsden, J., The Hopf bifurcation for nonlinear semigroups. Bull. Amer. Math. Soc., 79, 537- 541 (1973). · Zbl 0262.76031 · doi:10.1090/S0002-9904-1973-13191-X
[22] Marsden, J., & M. McCracken, The Hopf Bifurcation and its Applications. Springer Applied Mathematical Sciences Lecture Notes Series, Vol. 19, 1976. · Zbl 0346.58007
[23] Poore, A.B., On the theory and application of the Hopf-Friedrichs bifurcation theory. Preprint, 1976. · Zbl 0358.34005
[24] Ruelle, D., & F. Takens, On the nature of turbulence. Comm. Math. Phys., 20, 167-192 (1971). · Zbl 0223.76041 · doi:10.1007/BF01646553
[25] Sattinger, D.H., Bifurcation of periodic solutions of the Navier-Stokes equations. Arch. Rational Mech. Analysis, 41, 66-80 (1971). · Zbl 0222.76022 · doi:10.1007/BF00250178
[26] Sattinger, D.H., Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics No. 309. Springer, New York, 1973. · Zbl 0248.35003
[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.