zbMATH — the first resource for mathematics

The index of stable critical points. (English) Zbl 1013.37016
Given \(k\in{\mathbb{Z}}\), this paper is devoted to the construction in dimension \(n\geq 3\) of a \(C^\infty\) vector field \(X:U\to{\mathbb{R}}^n\), \(U\) open in \({\mathbb{R}}^n\), with a unique zero in \(0\), with the property that \(0\) is a stable equilibrium (in positive and negative time) for the differential equation \(\dot x=X(x)\) and the local index of \(X\) at \(0\) is equal to \(k\).
Here, by local index at an isolated zero \(x_0\) of a \(C^1\) vector field \(X\) it is meant the topological degree of the \(S^{n-1}\)-valued map \(x\mapsto X(x)/\|X(x)\|\) defined on the boundary of a disk \(D\subset U\) centered at \(0\) and such that \(X^{-1}(0)\cap D=\{0\}\).

37C10 Dynamics induced by flows and semiflows
34D20 Stability of solutions to ordinary differential equations
55M25 Degree, winding number
57R25 Vector fields, frame fields in differential topology
57R70 Critical points and critical submanifolds in differential topology
Full Text: DOI
[1] Bobylev, N.A.; Krasnosel’skiı̆, M.A., Deformation of a system into an asymptotically stable system, Automat. remote control, 35, 1041-1044, (1974) · Zbl 0312.34032
[2] Brunella, M., Instability of equilibria in dimension three, Ann. inst. Fourier, 48, 1345-1357, (1998) · Zbl 0930.37025
[3] Cima, A.; Mañosas, F.; Villadelprat, J., A poincaré-hopf theorem for noncompact manifolds, Topology, 37, 261-277, (1998) · Zbl 0894.55002
[4] Dold, A., Lectures on algebraic topology, Grundlehren math. wiss., 200, (1972), Springer-Verlag Berlin · Zbl 0234.55001
[5] Erle, E., Stable equilibria and vector field index, Topology appl., 49, 231-235, (1993) · Zbl 0777.58032
[6] Krasnosel’skiı̆, M.A.; Zabreı̆ko, P.P., Geometrical methods of nonlinear analysis, (1984), Springer-Verlag Berlin
[7] Massey, W.S., Algebraic topology: an introduction, Graduate texts in math., 56, (1977), Springer-Verlag New York
[8] Milnor, J., Topology from the differentiable viewpoint, (1997), Princeton University Press Virginia
[9] Thews, K., Der abbildungsgrad von vektorfeldern zu stabilen ruhelagen, Arch. math., 52, 71-74, (1989) · Zbl 0633.34027
[10] Thews, K., On a topological obstruction to regular forms of stability, Nonlinear anal., 22, 347-351, (1994) · Zbl 0797.34060
[11] Wilson, F.W., On the minimal sets of non-singular vector fields, Ann. of math., 84, 529-536, (1966) · Zbl 0156.43803
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.