Multiplicity of forced oscillations for the spherical pendulum (Q1270583)

Using the degree arguments on a manifold (a sphere), the existence of at least two geometrically different \(T\)-periodic solutions is proved for a \(T\)-periodically perturbed gravitational pendulum, provided that the perturbations are sufficiently small. The following particular case reflects well the spirit of the main theorem. Corollary. Given a \(T\)-periodic active force \(\varphi: \mathbb{R}\times TS\to \mathbb{R}^3\) on the sphere \(S\), there exists \(\lambda\varphi> 0\) such that, for any \(\lambda\in [0,\lambda\varphi]\), the perturbed gravitational pendulum equation \[ m\ddot x= -{m|\dot x|^2\over r^2} x+ h_g(x)- \mu\dot x+ \lambda\varphi(t, x,\dot x), \] where \(\mu\geq 0\), has at least two \(T\)-periodic solutions with different images.