Maximal Lyapunov exponent and almost-sure stability for Stochastic Mathieu–Duffing Systems

The principal resonance of a Mathieu–Duffing oscillator to random excitation is investigated. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. The effects of damping, detuning, cubic term, and magnitudes of random excitation are analyzed. The explicit asymptotic formulas for the maximal Lyapunov exponent is obtained. The almost-sure stability or instability of the stochastic Mathieu–Duffing system depends on the sign of the maximal Lyapunov exponent. In the last part of the work, the numerical results are obtained.