Largest Lyapunov exponent for second-order linear systems under combined harmonic and random parametric excitations

The principal resonance of a second-order linear stochastic oscillator to combined harmonic and random parametric excitations is investigated. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The effects of damping, detuning, bandwidth, and magnitudes of random excitation are analyzed. The method of path integration is used to obtain the steady state probability density function of the system, and then the largest Lyapunov exponent is calculated. The almost-sure stability or instability of the stochastic system depends on the sign of the largest Lyapunov exponent. The theoretical analyses are verified by numerical results.