A Characterization of theω-Limit Sets of Planar Continuous Dynamical Systems

If the family of curves ={Γi}iand the set of pointsSare given, we find necessary and sufficient conditions for the existence of aCkmapffrom the plane into itself, 0 k ∞ (ifk=0 we also assume thatfis locally Lipschitz), such that the curves from and the points fromSare respectively nondegenerate trajectories and singular points of the systemx′=f(x), and additionallyS ( i Γi) is theω-limit set of some trajectory of the system. In such cases, we also provide a detailed geometric description ofS ( i Γi) (after contraction of the connected components ofS).