Bifurcations in stochastic systems-models, analysis and simulation (Q1122229)

The present paper studies mechanical models in the presence of parametric noise described by second-order differential equations of the form \[ (1)\quad \ddot X_ t+2D\omega_ 1(1+\delta \dot X^ 2_ t/\omega^ 2_ 1)\dot X_ t+\omega^ 2_ 1(1+\gamma X^ 2_ t+\sigma \dot W_ t/\sqrt{\omega_ 1})X_ t=0. \] Examples of structural, aero- or fluid- dynamic problems leading to this type of equation are given. The stability of the linearization of (1) can be read off from the Lyapunov exponents, the stochastic analogues to the real parts of the eigenvalues. Lyapunov coordinates describing the radial and angular behavior are introduced and lead to an approximation of the biggest Lyapunov exponent \(\lambda\) using Fourier moments of the phase process. If the noise intensity \(\sigma\) is considered as a bifurcation parameter then there will be a certain critical value \(\sigma_{crit}\) such that \(\lambda (\sigma_{crit})=0\). There the equilibrium position loses its stability. The solution which becomes stable beyond this point is studied by simulations of the original nonlinear system written in Lyapunov coordinates using an Euler scheme. ``Silent'' and ``noise'' limit cycles are found, and the influence of different nonlinear terms on the shape and stability of this limit cycle are examined. Some indications are given how this can be generalized to multi-degree-of-freedom systems.