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

摘要

Lyapunov exponents and rotation numbers of linear two-dimensional stochastic differential equations are described by variants of Furstenberg- Khasminskii formulas exhibiting the interaction of drift and diffusion in temrs of Lie 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 contain- Ing the moments of one-dimensional diffusions of potentilal type.