Lyapunov exponent of the random frequency oscillator: cumulant expansion approach

We consider a one-dimensional harmonic oscillator with a random frequency, focusing on both the standard and the generalized Lyapunov exponents, $\lambda$ and $\lambda^\star$ respectively. We discuss the numerical difficulties that arise in the numerical calculation of $\lambda^\star$ in the case of strong intermittency. When the frequency corresponds to a Ornstein-Uhlenbeck process, we compute analytically $\lambda^\star$ by using a cumulant expansion including up to the fourth order. Connections with the problem of finding an analytical estimate for the largest Lyapunov exponent of a many-body system with smooth interactions are discussed.