On the almost-sure stability condition for a co-dimension two-bifurcation system under the parametric excitation of a real noise

In this paper, the almost-sure stability condition for a co-dimension two-bifurcation system on a three-dimensional central manifold, which is parametrically excited by a real noise, is investigated. A model of enhanced generality is developed by assuming the real noise as the first component of an output of a linear filter system—a zero-mean stationary Gaussian diffusion vectoral process, which conforms to the detailed balance condition. The strong mixing condition, which is the essential theoretic basis for the stochastic averaging method, is removed in the present study. To solve the complicated problem encountered in this work, the asymptotic analysis approach and the eigenfunction expansion of the solutions to the relevant Fokker–Planck equations are employed in the construction of the asymptotic expansions of the invariant measures and the maximal Lyapunov exponents for the relevant system.