PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 83, Number 1, May 1982 QUADRATIC MORSE-SMALE VECTOR FIELDS WHICH ARE NOT STRUCTURALLY STABLE CARMEN CHICONE1 AND DOUGLAS S. SHAFER Abstract. An example is given of a quadratic system in the plane which is Morse-Smale but not structurally stable. Also, it is proved that no such example exists for a quadratic system which is a gradient. 1. Introduction. A vector field von a manifold AÍ is called Morse-Smale if X generates a flow tp, such that the nonwandering set ß of tp, is the union of a finite number of hyperbolic critical points and a finite number of hyperbolic periodic orbits, and the stable and unstable manifolds of the critical points and closed orbits intersect transversally. Given a topology on the space of vector fields, X is called structurally stable if all sufficiently small perturbations of X have the same phase portrait as X, up to topological equivalence. If M is compact, a Morse-Smale vector field is structurally stable [8, 9], but a Morse-Smale vector field may fail to be structurally stable when M is not compact. Krych and Nitecki [7] have asked if there is a quadratic system on the plane R2 which is Morse-Smale but not structurally stable in the C-Whitney topology [5]. A quadratic system on R2 is an autonomous differential equation of form x = P(x,y), y=Q(x,y) where P and Q are polynomials of degree less than or equal to two. In §2 we show that such an example is provided by the quadratic system x — 2xy, y = 2xy — x2 + y2 + 1. In §3 we show that the quadratic gradient vector fields which are Morse-Smale are structurally stable in the C-Whitney topology. This strengthens the results in [3] which show that generically, in the coefficient topology, quadratic gradients are Morse-Smale. Finally, in §4 we show that the choice of topology can be crucial for stability by showing that the Morse-Smale quadratic gradient x--y, y = 3y2-x is not structurally stable in the coefficient topology. Received by the editors July 16, 1981. 1980 Mathematics Subject Classification. Primary 58F10, 34D30. Key words and phrases. Morse-Smale, structural stability, quadratic system. 'Research supported by a grant from the Research Council of the Graduate School, University of Missouri. ©1982 American Mathematical Society 0002-9939/81 /0000- 1044/Î03.00 125 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use