Maximal Lyapunov exponent of a single-degree-of-freedom linear vibroimpact system to a boundary random parametric excitation

The resonance response and maximal Lyapunov exponent of single-degree-of-freedom linear vibroimpact oscillator with a one-sided barrier to boundary random parametric excitation are investigated. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, or velocity jumps, thereby permitting the applications of asymptotic averaging over the period for slowly varying random process. The averaged equations are solved exactly and value of the maximal Lyapunov exponent is obtained in the case without random disorder. The FPK equations are solved exactly and the explicit asymptotic formulas for the maximal Lyapunov exponent and invariant measures are obtained for the case with random disorder. Theoretical analyses show that the maximal Lyapunov exponent will increase when the damping of the system, bandwidth of random excitation and restitution factor decrease. The maximal Lyapunov exponent will increase when the magnitudes of random excitation increase. The maximal Lyapunov exponent will reach the maximum values when the excitation frequency equals two times of the system frequency, therefore make the system become more unstable. The system will be almost sure stable (or unstable) if the maximal Lyapunov exponent is negative (or positive), therefore the stable bifurcation will be occur if the maximal Lyapunov exponent equal to zero and the stochastic stable bifurcation point can be obtained.