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Additional file 1. SPIRIT 2013 Checklist: Recommended items to address in a clinical trial protocol and related documents*.

In the zero-order approximation, we use the perturbation method of parameter with small magnitude to prove that the harmonic frequency in the solution of the equation is close to that of the driving force when the chaotic system from Duffing–Holmes equation stays in the stable periodic state, which is the physical mechanism of the detection of the unknown frequency of weak harmonic signal using the chaotic theory. The result of the simulation experiment shows that the method proposed in this paper, by which one can determine the frequency of the stable system from the number of circulation change of the phase state directionally across a fixed phase state point (x, ) in fixed simulation time period, is successful. Analyzing the effects of the damping ratio on the chaotic detection result, one can see that for different frequency ranges it is necessary to carefully choose corresponding damping ratio α.