Diffusion on d-sphere: Periodicity and inertial manifolds

On a $d$-sphere $S^d$, we consider the diffusion equation of the form $u^{\prime }\left(t\right)={(-{\Delta }_H{u}^{♭})}^{♯}+f\left(u\right)+g\left(t\right)$ for vector fields on $S^d$ and Hodge Laplacian ${\Delta }_H$. We prove the existence of a $T$-periodic solution to that equation under the action of a $T$-periodic external force $g$. Furthermore, we investigate the existence of an inertial manifold for the solutions nearby that periodic solution. The distribution of eigenvalues of Hodge Laplacian leads to the validity of the spectral gap condition yielding the existence of an inertial manifold.