Lyapunov exponents for two nonlinear systems driven by real noises

For a van der Pol-Duffing oscillator and a codimension-two bifurcation system excited parametrically by a small real noise, a model of enhanced generality is developed by assuming that the real noise is the first component of an output vector of a linear filter system, which conforms with the detailed balance condition. The strong mixing condition and the hypothesis (i.e. the infinitesimal generator of the noise has only one isolated simple zero eigenvalue) employed by Arnold and co-workers in 1986 are not considered. To tackle the complexity encountered in the present work, the asymptotic analysis approach and the eigenfunction expansion for the solution to the Fokker-Planck equations are applied to calculate the asymptotic expansions of the invariant measures and the maximal Lyapunov exponents for the relevant systems.