# zbMATH — the first resource for mathematics

Bifurcations and global stability of families of gradients. (English) Zbl 0706.58042
The purpose of the present paper is to prove that, among two-parameter families of gradients, the stable ones are dense. New techniques, especially concerning singular invariant foliations, are introduced to study the bifurcation diagrams and to prove stability.

##### MSC:
 37G99 Local and nonlocal bifurcation theory for dynamical systems 37C75 Stability theory for smooth dynamical systems 37C85 Dynamics induced by group actions other than $$\mathbb{Z}$$ and $$\mathbb{R}$$, and $$\mathbb{C}$$
Full Text:
##### References:
 [1] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko,Singularities of differentiable maps, vol. I, Birkhäuser, 1985. [2] T. Bröker,Differentiable germs and catastrophes, Cambridge University Press, 1975. · Zbl 0302.58006 [3] J. Guckenheimer, Bifurcations and catastrophes,Dynamical Systems, Acad. Press, 1973, 95–109. [4] B. A. Khesin, Local bifurcations of gradient vector fields,Functional Analysis and Appl.,20, 3 (1986), 250–252. · Zbl 0625.58011 · doi:10.1007/BF01078483 [5] N. H. Kuiper, C1-equivalence of functions near isolated critical points, in Symp. on Infinite Dim. Topology,Ann. of Math. Studies,69, Princeton University Press, 1972, 199–218. [6] R. Labarca, Stability of parametrized families of vector fields, in Dynamical Systems and Bifurcation Theory,Pitman Research Notes in Math. Series,160 (1987), 121–214. [7] J. Martinet, Singularities of smooth functions and maps,Lecture Notes Series,58, London Math. Society, 1982. · Zbl 0522.58006 [8] J. Martinet, Déploiements versels des applications différentiables et classification des applications stables, inLecture Notes in Math.,535, Springer-Verlag, 1975, 1–44. · doi:10.1007/BFb0080494 [9] J. Mather, Finitely determined maps germs,Publ. Math. I.H.E.S.,35 (1968), 127–156. · Zbl 0159.25001 [10] O. E. Lanford III, Bifurcation of periodic solutions into invariant tori: the work of Ruelle and Takens, in Nonlinear Problems in the Physical Sciences and Biology,Lecture Notes in Math.,322, Springer-Verlag, 1973, 159–192. · Zbl 0272.34039 · doi:10.1007/BFb0060566 [11] S. Newhouse, J. Palis andF. Takens, Stable families of diffeomorphisms,Publ. Math. I.H.E.S.,57 (1983), 5–71. [12] R. Palais, Local triviality of the restriction map for embeddings,Comm. Math. Helvet.,34 (1962), 305–312. · Zbl 0207.22501 · doi:10.1007/BF02565942 [13] R. Palais, Morse theory on Hilbert manifolds,Topology,2 (1963), 299–340. · Zbl 0122.10702 · doi:10.1016/0040-9383(63)90013-2 [14] J. Palis, On Morse-Smale dynamical systems,Topology,8 (1969), 385–405. · Zbl 0189.23902 · doi:10.1016/0040-9383(69)90024-X [15] J. Palis andF. Takens, Stability of parametrized families of gradient vector fields,Annals of Math.,118 (1983), 383–421. · Zbl 0533.58018 · doi:10.2307/2006976 [16] J. Palis andS. Smale, Structural stability theorems, inGlobal Analysis Proceedings Symp. Pure Math.,14, A.M.S., 1970, 223–231. [17] S. Sternberg, On the structure of local homeomorphisms of euclidean space II,Amer. Journal of Math.,80 (1958), 623–631. · Zbl 0083.31406 · doi:10.2307/2372774 [18] F. Takens, Moduli of stability for gradients, in Singularities and Dynamical Systems,Mathematics Studies,103, North-Holland, 1985, 69–80. [19] F. Takens, Singularities of gradient vector fields and moduli, in Singularities and Dynamical Systems,Mathematics Studies,103, North-Holland, 1985, 81–88. · Zbl 0614.58033 [20] F. Takens, Partially hyperbolic fixed points,Topology,10 (1971), 133–147. · Zbl 0214.22901 · doi:10.1016/0040-9383(71)90035-8 [21] R. Thom,Stabilité structurelle et morphogenèse, Benjamin, 1972. [22] G. Vegter,Bifurcations of gradient vector fields, Ph.D. Thesis, Groningen, 1983. · Zbl 0526.58034 [23] G. Vegter, Global stability of generic two-parameter families of gradients on three manifolds, in Dynamical Systems and Bifurcations,Lecture Notes in Math.,1125, Springer-Verlag, 1985, 107–129. · Zbl 0563.58018 · doi:10.1007/BFb0075638 [24] G. Vegter,The C p -preparation theorem, C p -unfoldings and applications, Report ZW-8013, Groningen, 1981.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.