Limit cycles of singular algebraic foliations on the real projective plane

Abstract

In this paper we consider cycles of singular algebraic foliations on the real projective plane. Our first result shows that there exists a limit cycle of singular algebraic foliations such that the limit cycle cannot be realized in any affine chart, that is, it fails to appear in any affine plane via the usual dehomogenization. On the other hand, under an appropriate convexity assumption, we verify that a cycle becomes fully visible, lying entirely in some affine chart. We then analyze the accumulation behavior of limit cycles of singular algebraic foliations on the real projective plane. Last we prove that algebraic foliations of degree at most two on the real projective plane have only finitely many limit cycles, and the maximum number of limit cycle is zero, one and at least six for algebraic foliations of degree zero, one and two on the real projective plane, respectively. This result extends Bamon's finiteness theorem for quadratic affine vector fields in [Publ. Math. IHÉS, (64)1986, pp. 111--142].