In this paper, the author proved the existence of a global compact attractor of the following system
And then the author deduced the existence of a periodic solution:
I feel like he somehow used a variant of the Poincaré-Bendixson Theorem for systems with diffusion despite the infinite dimensional aspect of such systems. I know that the the Poincaré-Bendixson Theorem fails even for dimension 3 ODE systems. Maybe here the diffusion does not affect the behavior. Do such Poincaré-Bendixson type theorems exist?