When is a dynamical system periodic?

Like I mentioned in your other question already, don't worry about the language part too much if there is a way to express what you are trying to say in "natural" language (rather than trying to be overly formal). Here, I would say something like this:

The solution of this dynamical system is periodic in the sense that the $y(t)$ and $theta(t)$ components of the solution repeat after some time. This is analogous to the position of a marked point on a wobbly wheel of a car when letting $x(t)$ be the position along a straight road, $y(t)$ be the position perpendicular to the road, and $theta(t)$ the angle of the marked point against the vertical direction: $y(t)$ and $theta(t)$ are periodic if the car moves at a fixed speed, whereas $x(t)$ is not.

See how I explained the concept without trying to rely on a precisely definition of what "periodic in $y$" or "periodic in $x$" actually means?