Structurally Stable Quadratic Systems without Limit Cycles

摘要

Let X denote the Banach space of planar polynomial systems of degree not greaterthan n and let Sigma(sub n) be the set of structurally stable systems in X. Let X(sub s) be the set of systems satisfying the following conditions on the Poincare sphere: (1) the number of singularities and periodic orbits is finite and each is hyperbolic; (2) there are no trajectories connecting saddle points (except possibly along the Poincare equator). It was proved by Peixoto that (1) X(sub s) is open and dense in X, (2) a necessary and sufficient condition for a system alpha to be in X(sub s) is that alpha is in Sigma(sub n). The problem of the topological classification of structurally stable quadratic systems without limit cycles was first mentioned and discussed by Tavares dos Santos. He found 25 different topological structures. Shi Songling showed on logical grounds that there are at most 65 different types of topological structures for structurally stable quadratic systems without limit cycles. In the paper the authors will show that the classes C(sub 1), C(sub 4), C(sub 9), and C(sub 14) in table 1 and the classes 1, 2 and 12 in table 2 of Shi Songling's paper are not empty by constructing examples in these classes. The authors results thus imply that there are at least 43 and at most 50 classes of structurally stable quadratic systems without limit cycles.