Book Review
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[1] V. I. Arnold, Lectures on bifurcations in versal families, Russian Math. Surveys 27 (1972), 54-123.
George D. Birkhoff, Dynamical systems, American Mathematical Society Colloquium Publications, Vol. IX, American Mathematical Society, Providence, R.I., 1966. With an addendum by Jurgen Moser. MR 0209095
Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 251, Springer-Verlag, New York-Berlin, 1982. MR 660633
C. Elphick, E. Tirapegui, M. E. Brachet, P. Coullet, and G. Iooss, A simple global characterization for normal forms of singular vector fields, Phys. D 29 (1987), no. 1-2, 95–127. MR 923885, DOI 10.1016/0167-2789(87)90049-2
Michael Field, Symmetry breaking for compact Lie groups, Mem. Amer. Math. Soc. 120 (1996), no. 574, viii+170. MR 1317939, DOI 10.1090/memo/0574
M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math. 32 (1979), no. 1, 21–98. MR 508917, DOI 10.1002/cpa.3160320103
M. Golubitsky and D. Schaeffer, Imperfect bifurcation in the presence of symmetry, Comm. Math. Phys. 67 (1979), no. 3, 205–232. MR 539727
John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
Jerrold Marsden and Jürgen Scheurle, The construction and smoothness of invariant manifolds by the deformation method, SIAM J. Math. Anal. 18 (1987), no. 5, 1261–1274. MR 902331, DOI 10.1137/0518092
[10] J. Menck, Analysis of non-hyperbolic equilibria in dynamical systems by use of symmetries and computer algebra, Ph.D. Thesis, Univ. of Hamburg, 1992 (in German).
Kenneth J. Palmer, Linearization near an integral manifold, J. Math. Anal. Appl. 51 (1975), 243–255. MR 374564, DOI 10.1016/0022-247X(75)90156-0
M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101–120. MR 142859, DOI 10.1016/0040-9383(65)90018-2
[13] H. Poincaré, Sur les courbes définies par les équations differentielles, C. R. Acad. Sci. Paris Sér. I Math. 90 (1880), 673-675.
S. Smale, Structurally stable systems are not dense, Amer. J. Math. 88 (1966), 491–496. MR 196725, DOI 10.2307/2373203
Ivan Kupka, Contribution à la théorie des champs génériques, Contributions to Differential Equations 2 (1963), 457–484 (French). MR 165536
A. N. Šošitaĭšvili, The bifurcation of the topological type of the singular points of vector fields that depend on parameters, Trudy Sem. Petrovsk. Vyp. 1 (1975), 279–309 (Russian). MR 0478239
Floris Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 47–100. MR 339292
René Thom, Stabilité structurelle et morphogénèse, Mathematical Physics Monograph Series, W. A. Benjamin, Inc., Reading, Mass., 1972 (French). Essai d’une théorie générale des modèles. MR 0488155
- [1]
- V. I. Arnold, Lectures on bifurcations in versal families, Russian Math. Surveys 27 (1972), 54-123.
- [2]
- G. D. Birkhoff, Dynamical systems, Amer. Math. Soc. Colloq. Publ., vol. 9, Amer. Math. Soc., Providence, RI, 1927; rev. ed., 1966. MR 0209095 (35:1)
- [3]
- S. N. Chow and J. K. Hale, Methods of bifurcation theory, Springer-Verlag, New York, 1982. MR 660633 (84e:58019)
- [4]
- C. Elphick, E. Tirapegui, M. Brachet, P. Coullet, and G. Iooss, A simple global characterization for normal forms of singular vector fields, Phys. D 29 (1987), 95-127. MR 923885 (90d:58111a)
- [5]
- M. Field, Symmetry breaking for compact Lie groups, preprint, 1993. MR 1317939 (96j:58121)
- [6]
- M. Golubitsky and D. Schaeffer, A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math. 32 (1979), 21-98. MR 508917 (80j:58061)
- [7]
- -, Imperfect bifurcation in the presence of symmetry, Comm. Math. Phys. 67 (1979), 205-232. MR 539727 (80j:58017)
- [8]
- J. Guckenheimer and P. J. Holmes, Nonlinear oscillations, dynamical systems and bifurcation of vector fields, Springer-Verlag, New York, 1983. MR 709768 (85f:58002)
- [9]
- J. E. Marsden and J. Scheurle, The construction and smoothness of invariant manifolds by the deformation method, SIAM J. Math. Anal. 18 (1987), 1261-1274. MR 902331 (88m:58096)
- [10]
- J. Menck, Analysis of non-hyperbolic equilibria in dynamical systems by use of symmetries and computer algebra, Ph.D. Thesis, Univ. of Hamburg, 1992 (in German).
- [11]
- K. J. Palmer, Linearization near an integral manifold, J. Math. Anal. Appl. 51 (1975), 243-255. MR 0374564 (51:10764)
- [12]
- M. M. Peixoto, Structural stability on two-manifolds, Topology 1 (1962), 101-120. MR 0142859 (26:426)
- [13]
- H. Poincaré, Sur les courbes définies par les équations differentielles, C. R. Acad. Sci. Paris Sér. I Math. 90 (1880), 673-675.
- [14]
- S. Smale, Structurally stable systems are not dense, Amer. J. Math. 86 (1966), 491-496. MR 0196725 (33:4911)
- [15]
- -, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1963), 97-116. MR 0165537 (29:2818b)
- [16]
- A. N. Sositaisvili, Bifurcations of a topological type of a vector field near a singular point, Trudy Sem. Petrovsk. 1 (1975), 279-308 (in Russian). MR 0478239 (57:17724)
- [17]
- F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ Math. 43 (1974), 47-100. MR 0339292 (49:4052)
- [18]
- R. Thom, Stabilité structurelle et morphogénèse, W. A. Benjamin, Reading, MA, 1972. MR 0488155 (58:7722a)
Review Information:
Reviewer:
J"urgen Scheurle
Journal:
Bull. Amer. Math. Soc.
31 (1994), 142-146
DOI:
https://doi.org/10.1090/S0273-0979-1994-00501-4
