On Hopf bifurcation in the problem of motion of a heavy particle on a rotating sphere: the viscous friction case

Abstract

Abstract We investigate the Hopf bifurcation of a mass on a rotating sphere under the influence of gravity and viscous friction. After determining the equilibria, we study their stability and calculate the first Lyapunov coefficient to determine the post-critical behavior. It is found that the bifurcating periodic branches are initially stable. For several inclination angles of the sphere’s rotation axis, the periodic solutions are calculated numerically, which shows that for large inclination angles turning points occur, at which the periodic solutions become unstable. We also investigate the limiting case of small friction coefficients, when the mass moves close to the equator of the rotating sphere.