## 6 Simulation and Experiment of Weak Signal Detection Using Double-Coupled Duffing System  
### Simulation Study of Single-Component Noisy Signal Detection  
In order to verify the detection ability of the method proposed in this paper, a single-component noisy signal is used for testing. The composition of the signal is shown in Equation (17):  
\[A_{s}(t)=Acos(\omega t)+n(t) \tag{17}\]  
where \(A\cos(\omega t)\) is the periodic signal with amplitude \(A\) and frequency \(\omega\), and \(n(t)\) is the white noise with a variance of 0.01. Set the frequency of the periodic signal to 1 rad/s and use the method in Section 3.3 to detect the single-component signal in Equation (17). Since the signal has only one significant component, and there is no phase difference between the component and the built-in driving force, and the frequency is the same, there is no need to carry out frequency normalization, phase shift, VMD decomposition and other operations. When the amplitude of the periodic signal in the single-component noisy signal is 0.006, the signal to be measured is used to excite the traditional Holmes-Duffing system and the double-coupled Duffing system established in this paper, and the phase plane diagram of the system output is shown in Figure 6.  
According to Figure 6, by comparing the phase plan, it is found that the double-coupled Duffing system is in the large-period state, and the variance of the output value of the system is 2.3415. Although there is irregular movement in some periods, by calculating the variance jump rate, according to the criteria proposed in this paper, it can be considered that the signal to be tested is detected at this time. However, the [PERSON] system is still in chaos under the same excitation, the output value variance is only 2.2198, and its variance jump rate does not exceed the threshold, so it is considered that the signal to be measured is not detected at this time. The above analysis proves that the double-coupled Duffing system established in this paper has better anti-noise performance. In order to calculate the detection capability of the proposed method, it is necessary to calculate the lowest signal-to-noise ratio that can be detected by the method. The equation for calculating the signal-to-noise ratio is shown in Equation (18). \[SNR=10 lg\frac{A^{2}}{2\delta^{2}} \tag{18}\]  
where \(SNR\) is the signal-to-noise ratio of the signal, and \(\delta\) is the standard deviation of white noise. Calculate the simulated signal of Equation (17), constantly modify the amplitude of the periodic signal, calculate 100 times for each amplitude strength, count the output value variance of the double-coupled Duffing system, and draw the box-type diagram as  
Figure 6: Phase plane diagram of Duffing system. (**a**) Holmes–Duffing system; (**b**) double-coupled Duffing system. shown in Figure 7. It can be seen from Figure 7 that, with the increase in the amplitude of the periodic signal, the variance of the output value is on the rise as a whole, and the distribution is more concentrated. According to the analysis in the second chapter, it is considered that the signal to be measured is detected when the variance of the output value is greater than 2.3289. According to this standard, combined with statistical analysis, the double-coupled Duffing system established in this paper has the ability to detect signals with a signal-to-noise ratio of \(-\)23.74 dB. Starting from 0, gradually increase the amplitude of the periodic signal in steps of 0.0001 and repeat the calculation 100 times for each amplitude of the signal. To ensure the rationality of the simulation experiment, 100 random number seeds were generated before calculation, and white noise in 100 repeated calculations under all periodic signal amplitudes was generated using these 100 random number seeds.