Phase diagram of the random frequency oscillator: The case of   Ornstein-Uhlenbeck noise
  We study the stability of a stochastic oscillator whose frequency is a random process with finite time memory represented by an   Ornstein-Uhlenbeck noise. This system undergoes a noise-induced bifurcation when the amplitude of the noise grows. The critical curve, that separates the absorbing phase from an extended non-equilibrium steady state, corresponds to the vanishing of the Lyapunov exponent that measures the asymptotic logarithmic growth rate of the energy.   We derive various expressions for this Lyapunov exponent by using different approximation schemes. This allows us to determine quantitatively the phase diagram of the random parametric oscillator. 
Kirone Mallick and Pierre-Emmanuel Peyneau