## 3 Simulation Analysis and Method Comparison
### Comparison with Existing Chaotic Detection Methods  
The new method proposed in this paper is not consistent with the existing chaotic detection methods in principle and the implementation mode. To reflect the difference between these methods, the new method is compared with the methods used in the Refs. [8; 10; 11; 18] in some aspects, as shown in Table 1.  
By comparing the principles of these detection methods, it can be seen that method I is based on a single Duffing system to achieve amplitude detection of the weak signal with specific frequency. The measurable range \(S_{1}\) depends on the critical point chosen and the interval width to keep the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Mode** & **Measurable Range** & **Measurable Parameters** & **Time** \\ \hline Method I [8] & Duffing oscillator & \(S_{1}\) & \(A\) & \(T_{1}\) \\ Method II [10] & Duffing oscillator array & \(S_{2}\) & \(\omega\) & \(T_{2}\) \\ Method III [11] & Duffing oscillator array & \(S_{3}\) & \(\omega\) & \(T_{3}\) \\ Method IV [18] & Chua’s circuit & \(S_{4}\) & \(\omega\) & \(T_{4}\) \\ Proposed method & Lorenz system & \(S\) & \(\omega\) & \(T\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Detection methods comparison. Figure 5: (**a**) Bifurcation diagram of the amplitude of the sinusoidal signal \(A\); (**b**) the SNR curve with frequency variation. operation state; it can only be used for amplitude detection, and the time length for determining the system running state is related to the frequency and phase of the tested signal. The detection time \(T_{1}\) is not less than \(6/\omega\) to prevent misjudgement, in which \(\omega\) is the frequency value of the tested signal. Method II is based on the optimization algorithm, and method III is based on the intermittent chaos. The measurable range \(S_{2}\) and \(S_{3}\) is related to the number of oscillators used. The larger the range is, the more the number of oscillators needed, which will greatly increase the complexity of the system. Both methods can achieve frequency measurement and need to judge the running state of the system, which affects the detection efficiency. In method II, the detection time \(T_{2}\) is equal to \(m\times T_{1}\), and \(m\) is the iteration times of the optimization algorithm. In method III, the determination of intermittent chaos is more difficult, and the detection time \(T_{3}\) is approximately (5\(\sim\)6) \(T_{1}\). Method IV is based on two chaotic synchronization systems; its measurable range is affected by the divergent state of the Chua's circuit, and the amplitude of the tested signal cannot be greater than 0.147. Its detection time is limited by the adaptive controller and is usually set to \(T_{4}=(4-5)/\omega\). Correspondingly, the simulation results show that the measurable range of the new method \(S\) is much larger than that of other methods \(S\gg S_{2},S_{3},S_{4}\). When the detection time \(T=3/\omega\), the new method achieves a good noise reduction effect. Its detection efficiency is much higher than other methods \(T\approx T_{4}\gg T_{3}\gg T_{2}\approx m\times T_{1}\). To compare the detection efficiency and accuracy of the above methods, we used these methods to estimate the frequency of the same test signal (\(r(t)\) in Section 3.1), and the results obtained are shown in Figure 6.  
In Figure 6, there are four curves, which represent the estimated frequencies obtained by the different methods. Curve \(a\) obtained by method II is composed of a set of discrete points, which are the optimal particles after each iteration of the genetic algorithm. The final estimated value is 0.965 Hz, and the time required is more than 470 s. Curve \(b\) obtained by method III is stable when the time is more than 219 s and the detection value is 0.97 Hz. Curve \(c\) obtained by method IV changes around the actual value 1 Hz after 60 s, and fluctuation range is gradually decreasing. When the time reaches 150 s the estimated value is 0.98 Hz.