Torsional instabilities in a two-degree-of-freedom system driven by a Hooke’s joint due to random input angular speed fluctuation are investigated. Linearised analytical models are used for calculating the largest Lyapunov exponent. Instability behaviour is then characterised by examining the sign of this exponent. Conditions for the onset of instability via sub-harmonic parametric resonances has been shown to coincide with those for the deterministic case. However, the onset of instability via sum as well as the difference type combination resonance is found to be different from that of the deterministic case. The instability conditions for the system under input angular speed fluctuation have been presented graphically in the excitation frequency-excitation amplitude-top Lyapunov exponent space. Predictions for the deterministic and the stochastic cases are compared. The effect of fluctuation probability density as well as that of inertia loads on the stability behaviour of the system has been examined.