Jordan curves and funnel sections

A continuous ordinary vector differential equation in Euclidean space has a funnel of solutions through each initial condition. Its cross-section at time t is a continuum. Many continua are known to be funnel sections: For instance the circle is a cross-section of a continuous ODE y′=f(t,y) where y is a variable in the plane, but it is not known whether every Jordan curve J is a planar funnel section. In this paper we give sufficient conditions that imply J is a planar funnel section – “pierceability.” We show that pierceability is not generic when we put a fairly interesting complete metric on the space of Jordan curves. We also give proofs of several statements in the first authorʼs paper on funnel sections that appeared in the JDE in 1975.