In this paper, the determination of the conditions for the onset of chaos in a parametrically excited stochastic system with cubic nonlinearity was studied using a combination of Melnikov method and Lyapunov exponents. The unforced system possesses three fixed points. All the three points are hyperbolically fixed which confirms that the system is topologically and structurally stable. Furthermore, the numerical simulations were illustrated with Maple software which indicated that the method of Lyapunov exponent can narrow the range for the chaotic ethical threshold and detect the mutations in the system. The simulation results also confirm that the system is very sensitive to the sinusoidal signal in the high frequency cases.

Melnikov method, Lyapunov exponent, homoclinic orbits, QR-factorization method, threshold values, manifolds.