Topological equivalence of gradient vectorfields
HTML articles powered by AMS MathViewer
- by Douglas S. Shafer
- Trans. Amer. Math. Soc. 258 (1980), 113-126
- DOI: https://doi.org/10.1090/S0002-9947-1980-0554322-6
- PDF | Request permission
Abstract:
This paper is a study of the behavior of the topological equivalence class of the planar gradient vectorfield $X = {\operatorname {grad} _g} V$, in a neighborhood of a degenerate singularity of V, as g varies over all Riemannian metrics. It is shown that under simple restrictions on V the topological equivalence class of X is determined by its first nonvanishing jet, and that only finitely many equivalence classes occur (for fixed V). In this case, when the degree of the first nonvanishing jet of V is less than five, necessary and sufficient conditions for change in equivalence class are given, both in terms of the coefficients of the homogeneous part of V and geometrically in terms of its level curves. A catalogue of possible phase portraits, up to topological equivalence, is included. Necessary conditions are given for change in higher degree.References
- A. A. Andronov et al., Qualitative theory of second order dynamic systems, John Wiley & Sons, New York, 1973.
- Freddy Dumortier, Singularities of vector fields on the plane, J. Differential Equations 23 (1977), no. 1, 53–106. MR 650816, DOI 10.1016/0022-0396(77)90136-X
- John Guckenheimer, Bifurcation and catastrophe, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 95–109. MR 0345139
- Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
- Nicolaas H. Kuiper (ed.), Manifolds—Amsterdam 1970, Lecture Notes in Mathematics, Vol. 197, Springer-Verlag, Berlin-New York, 1971. MR 0278311
- J. Palis and F. Takens, Topological equivalence of normally hyperbolic dynamical systems, Topology 16 (1977), no. 4, 335–345. MR 474409, DOI 10.1016/0040-9383(77)90040-4
- Floris Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 47–100. MR 339292
Bibliographic Information
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 258 (1980), 113-126
- MSC: Primary 58F14
- DOI: https://doi.org/10.1090/S0002-9947-1980-0554322-6
- MathSciNet review: 554322