Consider 2d$2D$ dynamical systems X' = F(X)$X' = F(X)$ where X$X=(x,y)$ is a 2-vector, and there is an equilibrium point at the origin. Let L$L$ be the set of numbers x > 0$x > 0$ such that a limit cycle of the system meets the x$x$-axis at (x,0)$(x,0)$. Is L$L$ necessarily closed? More generally what sets L$L$ can arise in this way? [The reason I want to know is that I'm teaching dynamical systems.]
