Example of a dynamical system which has an $\omega$-limit which is a cylinder of closed orbits

Question

I have been studying dynamical systems and have recently come accross the following theorem:

Suppose $n=3$. Let $L$ be a compact limit set which contains no equilibrium. Then:

• $L$ is either a closed orbit or a cylinder of closed orbits,

• $L$ is either a closed orbit if the system is cooperative and L is an w-limit set,

• $L$ is a closed orbit if all closed orbits are hyperbolic.

I have been trying to find a concrete example of a system which has a limit set which is a cylinder of closed orbits. Can anyone point me in the right direction? I have been playing around with cylindrical coordinates but I still can't seem to get anywhere.

Answer 1

Hint: If you can come up with a planar system (i.e. $n=2$) that has a circle as $\omega$-limit set, then you can trivially extend that planar system to a three-dimensional system by adding a third component with trivial dynamics, i.e. $\dot{z} = 0$. Then, for every value of $z$, the system has a closed orbit (circle), which yields a cylinder of closed orbits.

Answer 2

Hint: Consider a $2$-torus with periodic orbits, and foliate its neighborhood by $2$-tori. Now you can consider orbits that spiral towards the central torus with inclination tending more and more to the inclination of the periodic orbits, but always traveling along the height of the tori. The limit set of all such orbits will be the central torus. I believe that you can use this idea to write down a system explicitly.