I'm trying to plot a system with a variable (or field) $\theta(x,t)\in[-\pi,\pi]$ which is an angle at every position $x$. The positional argument $x$ itself is a periodic coordinate $x\in[-\pi,\pi]$. I'm trying to show some topological properties of this field, and I felt that a natural way to do so would be to plot the curve $\theta$ versus $x$ for $\theta = \theta(x)$ at a fixed time $t$ on the surface of a torus ($S^1\times S^1$). This would clearly represent the periodic nature of $\theta$ and $x$. Is there a neat way to do this on Mathematica?
For example, my idea was to use the blue coordinate (refer image) as the position variable $x$ and the red coordinate at each $x$ to represent $\theta(x)$. In this way both $x$ and $\theta$ are periodic.
Edit: $t$ is time. I just put it there for completeness. I need to plot the variable at different time slices.