ODE on unit sphere $S^1$ Consider the equation
$$
\theta_t=\cos\theta+\mu
$$
where we think of $\theta\in S^1=\mathbb{R}/2\pi\mathbb{Z}$.

When I see it right, $\mathbb{R}/2\pi\mathbb{Z}$ means that we are identifying the real line with $S^1$? So this is an ODE which is original defined on $\mathbb{R}$ but we think of it as happening on the unit sphere?

Suppose you already noticed this but the important point is that the vector field, here $f(\theta)=\cos(\theta)+\mu$, is well-defined on the quotient space, i.e. that $f(\theta+2\pi k)=f(\theta)$ is independent of $k$. This generalizes to e.g. $\dot{x}=f(x)$ on $M={\Bbb R}^d /(2\pi {\Bbb Z})^d$ as long as $f(x+2\pi n)=f(x)$ where $n\in {\Bbb Z}^d$. Many other generalizations are possible.
