Hey askwhelp.
What is the definition of the periodic solution? Does it have a DE form? If so try and substitute it into that form and see that it satisfies those constraints.
Given the system of differential equations $x' = 2x + y^3$ and $y' = -y$ i found the flow $$\phi_t(x,y) = ((x_0 + 1/5y_0^3)e^{2t} - 1/5 y_0^3e^{-3t}, y_0 e^{-t})$$. I am wondering are there any periodic solutions? how do i check?