Book Review
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MathSciNet review:
1566999
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Book Information:
Authors:
J. E. Marsden and
M. McCracken
Title:
The Hopf bifurcation and its applications
Additional book information:
Springer-Verlag, New York, 1976, xiii + 408 pp., $14.80.
J. C. Alexander and James A. Yorke, Global bifurcations of periodic orbits, Amer. J. Math. 100 (1978), no. 2, 263–292. MR 474406, DOI 10.2307/2373851
2. A. A. Andronov and A. A. Witt, Sur la théorie mathématiques des autooscillations, C. R. Acad. Sci. Paris 190 (1930), 256-258.
Salomon Bochner and Deane Montgomery, Groups of differentiable and real or complex analytic transformations, Ann. of Math. (2) 46 (1945), 685–694. MR 14102, DOI 10.2307/1969204
Nathaniel Chafee, The bifurcation of one or more closed orbits from an equilibrium point of an autonomous differential system, J. Differential Equations 4 (1968), 661–679. MR 229930, DOI 10.1016/0022-0396(68)90015-6
Michael G. Crandall and Paul H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180. MR 341212, DOI 10.1007/BF00282325
6. E. Conway, D. Hoff and J. Smoller, Large-time behavior of systems of nonlinear reaction-diffusion equations, S.I.A.M. J. Appl. Math. (to appear).
7. B. D. Hassard, Bifurcation of periodic solutions of the Hodgkin-Huxley model for the current-clamped squid giant axon, J. Theoret. Biology (to appear).
B. Hassard and Y. H. Wan, Bifurcation formulae derived from center manifold theory, J. Math. Anal. Appl. 63 (1978), no. 1, 297–312. MR 488152, DOI 10.1016/0022-247X(78)90120-8
Eberhard Hopf, Abzweigung einer periodischen Lösung von einer stationären eines Differentialsystems, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Nat. Kl. 95 (1943), no. 1, 3–22 (German). MR 39141
Gérard Iooss, Existence et stabilité de la solution périodique secondaire intervenant dans les problèmes d’evolution du type Navier-Stokes, Arch. Rational Mech. Anal. 47 (1972), 301–329 (French). MR 346350, DOI 10.1007/BF00281637
D. D. Joseph and D. H. Sattinger, Bifurcating time periodic solutions and their stability, Arch. Rational Mech. Anal. 45 (1972), 79–109. MR 387844, DOI 10.1007/BF00253039
12. V. I. Judovich, The birth of proper oscillations in a fluid, Prikl. Math. Meh. 35 (1971), 638-655.
13. A. Lindstedt, Differentialgleichungen der Störungstheorie. (8), Mém. Sci. St. Pétersbourgh 31 (1883).
14. E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci. 20 (1963), 130-141.
15. M. Poincaré, Les méthodes nouvelles de la mécanique céleste. Vols. I, II, Gauthier-Villars et Fils, Paris, 1892 and 1893; Dover Publications, Inc., New York, 1957. MR 19, 414.
David Ruelle and Floris Takens, On the nature of turbulence, Comm. Math. Phys. 20 (1971), 167–192. MR 284067
David H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Mathematics, Vol. 309, Springer-Verlag, Berlin-New York, 1973. MR 0463624
18. G. G. Stokes, On the theory of oscillatory waves, Cambridge Trans. 8 (1847), 441-473 (Papers vol. I, 197-229).
W. Velte, Über ein Stabilitätskriterium der Hydrodynamik, Arch. Rational Mech. Anal. 9 (1962), 9–20 (German). MR 155501, DOI 10.1007/BF00253330
John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582
- 1.
- J. Alexander and J. York, Global bifurcations of periodic orbits, Ann. of Math. (to appear). MR 0474406
- 2.
- A. A. Andronov and A. A. Witt, Sur la théorie mathématiques des autooscillations, C. R. Acad. Sci. Paris 190 (1930), 256-258.
- 3.
- S. Bochner and D. Montgomery, Groups of differentiable and real or complex analytic transformations, Ann. of Math. (2) 46 (1945), 685-694. MR 7, 241. MR 14102
- 4.
- N. Chafee, The bifurcation of one or more closed orbits from an equilibrium point of an autonomous differential system, J. Differential Equations 4 (1968), 661-679. MR 37 #5496. MR 229930
- 5.
- M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal. 52 (1974), 161-180. MR 49 #5962. MR 341212
- 6.
- E. Conway, D. Hoff and J. Smoller, Large-time behavior of systems of nonlinear reaction-diffusion equations, S.I.A.M. J. Appl. Math. (to appear).
- 7.
- B. D. Hassard, Bifurcation of periodic solutions of the Hodgkin-Huxley model for the current-clamped squid giant axon, J. Theoret. Biology (to appear).
- 8.
- B. D. Hassard and Y.-H. Wan, Bifurcation formulae derived from center manifold theory, J. Math. Anal. Appl. (to appear). MR 488152
- 9.
- E. Hopf, Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems, Ber. Verh. Sächs. Acad. Wiss. Leipzig. Math.-Nat. KL. 95 (1943), no. 1, 3-22. MR 12, 501. MR 39141
- 10.
- G. Iooss, Existence et stabilité de la solution périodique secondaire intervenant dans les problèmes d'evolution du type Navier-Sokes, Arch. Rational Mech. Anal. 47 (1972), 301-329. MR 49 #11075. MR 346350
- 11.
- D. D. Joseph and D. H. Sattinger, Bifurcating time periodic solutions and their stability, Arch. Rational Mech. Anal. 45 (1972), 79-109. MR 52 #8682. MR 387844
- 12.
- V. I. Judovich, The birth of proper oscillations in a fluid, Prikl. Math. Meh. 35 (1971), 638-655.
- 13.
- A. Lindstedt, Differentialgleichungen der Störungstheorie. (8), Mém. Sci. St. Pétersbourgh 31 (1883).
- 14.
- E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci. 20 (1963), 130-141.
- 15.
- M. Poincaré, Les méthodes nouvelles de la mécanique céleste. Vols. I, II, Gauthier-Villars et Fils, Paris, 1892 and 1893; Dover Publications, Inc., New York, 1957. MR 19, 414.
- 16.
- D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys. 20 (1971), 167-192; 23 (1971), 343-344. MR 44 #1297; 45 #1478. MR 284067
- 17.
- D. H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Math., vol. 309, Springer-Verlag, Berlin and New York, 1973. MR 463624
- 18.
- G. G. Stokes, On the theory of oscillatory waves, Cambridge Trans. 8 (1847), 441-473 (Papers vol. I, 197-229).
- 19.
- W. Velte, Über ein Stabilitätskriterium der Hydrodynamik, Arch. Rational Mech. Anal. 9 (1962), 9-20. MR 27 #5435. MR 155501
- 20.
- R. F. Williams, The structure of Lorenz attractors (preprint). MR 556583
Review Information:
Reviewer:
Nicholas D. Kazarinoff
Journal:
Bull. Amer. Math. Soc.
83 (1977), 998-1004
DOI:
https://doi.org/10.1090/S0002-9904-1977-14352-8
