Some formulas for Lyapunov exponents and rotation numbers in two dimensions and the stability of the harmonic oscillator and the inverted pendulum

Lyapunov exponents and rotation numbers of linear two-dimensional stochasti c differential equations are described by variants of Furstenberg-Khasminsk ii formulas exhibiting the interaction of drift and diffusion in terms of L ie brackets of their projections into projective space. In the case of one diffusion matrix of sheet type and general drift, the formulas simplify to expressions containing the moments of one-dimensional diffusions of potenti al type. Applications are given to the following systems perturbed by white noise: the harmonic oscillator and the inverted pendulum linearized in its unstable equilibrium position. Their Lyapunov exponents and rotation numbe rs are explicited in terms of hypergeometric functions, and are asymptotica lly expanded into series as functions of the noise parameter. A complete ac count of the stability diagrams of the systems is given. Lines of change of stability and of maximal stability are described in the planes spanned by the damping and noise respectively restoring force and noise parameters. Th e area in the planes where stabilization by noise for the inverted pendulum takes place is investigated.