Cubic vector fields symmetric with respect to a center

[摘要] In this paper we study cubic vector fields which are symmetric with respect to a center. Our perspective is from the viewpoint of invariant algebraic curves of such systems. We give here a new proof of the integrability of symmetric systems with respect to a center by the method of Darboux, which uses invariant algebraic curves. The first integrals of the systems are all elementary and we give here their complete list. We next study the global geometry of such systems. We give the bifurcation diagram of the phase portraits of the vector fields. We show that although most bifurcations correspond to bifurcations of the algebraic invariant curves, unlike what happens in the quadratic case, the changes in the invariant algebraic curves do not completely determine the bifurcation diagram. We prove that there appear other bifurcations of saddle connections, whose equations are transcendental. (C) 1995 Academic Press, Inc.

[发布日期] 1995-12-10