Is there any differential equations which their co-domains are $n$-spheres, in particular the $2$-sphere $\mathbb{S}^2$, that represents certain physical phenomena or has applications in non-math field(s)? Google-ing "DE on sphere" gives me the differential equations with the "sphere" on the domain instead.
ODEs with spherical co-domains (like this although I don't get its applications, and/or this) are also welcomed but I would prefer partial ones, as simple as possible, at this moment. Thanks a lot.
Try searching "liquid crystals", "harmonic map", "wave map", "heat map", "Schrödinger map". Each of these keywords points to some manifold-valued differential equation.