Performance of the Duffing oscillator to detect weak signals buried in heavy noise is analyzed quantitatively by LCEs. First in the case of noise, differential equations to compute LCE s are derived using RHR algorithm, so the quantitative criteria to identify system states are obtained. Then using LCEs, the threshold value of the forced periodic term is found accurately. Finally the system state and state change are analyzed using LCEs by keeping the threshold value and varying the noise intensity, and the minimum signal to noise ratio is determined. By contrast of phase trajectories and LCEs, it shows that phase trajectories disturbed by strong noise sometimes are ambiguous to our eyes, but through LCEs, the system state can be identified clearly and quantitatively especially in strong noise background. So the minimum signal to noise ratio can be obtained accurately.