I'm fairly new to dynamical systems and chaos. I'm dealing with a second order non-autonomous system of differential equations:
$$\dot{x} = y $$
$$\dot{y} = f(x,y,t)$$
I can further split $f$ as $$f(x,y,t) = g(x,t) + h(x,y,t)$$
I would like to find out if closed orbits and maybe a limit cycle exists for this system. Now, I understand that one way to check for closed orbits is to use the Poincaré-Bendixson theorem which is only applicable to 2-dimensional systems.
My question is, can this be applied to a two-dimensional non-autonomous system? Since a 2D non-autonomous system can be written as a 3D autonomous system with time as a state variable, would the PD theorem still be valid? If not, can you suggest another method that can be used to prove the existence or non-existence of closed orbits? Thank you!