## 6 Simulation and Experiment of Weak Signal Detection Using Double-Coupled Duffing System
### Simulation Study of Multi-Component Noisy Signal Detection  
In this section, the multi-component signal is used for simulation and research. A multi-component noisy signal is constructed as shown in Equation (19):  
\[A_{test}(t)=0.01(\cos(10t))+0.5(\cos(50t))+n(t) \tag{19}\]  
Detect the 10 rad/s component in the signal and set the \(\omega\) parameter in the detection system to 10 according to the method in Section 4.4. Without introducing the VMD decomposition, directly excite the double-coupled Duffing system, and obtain the phase plane as shown in Figure 9.  
As shown in Figure 9, it can be observed that the double-coupled Duffing system is in a large-scale periodic state. At this time, the variance of the output value of the system is 2.5125. According to the judgment method in Section 4.3, its variance jump rate has exceeded the threshold, and it can be considered that the signal to be measured has been detected, but at the same time, it can also be observed that, due to the interference of other frequency components with high energy, the phase plane trajectory of the system has been deformed, which will affect the detection performance. Delete the periodic signal component of 10 rad/s in Equation (19) and excite the double-coupled Duffing system again to obtain the phase plan as shown in Figure 10.  
It can be seen from Figure 10 that the system is still in a large-scale periodic state after deleting the components of the frequency to be measured, which indicates that the system has lost its due detection ability under the interference of high-energy signals from other components. Figure 10: Phase plane diagram of Duffing system (The signal does not contain a frequency component of 10 rad/s). Figure 9: Phase plane diagram of Duffing system (The signal contain a frequency component of 10 rad/s). According to the method in Section 4.3, the signal is first decomposed by EMD, and nine IMF components are obtained after EMD decomposition. Set the number of IMF components of the VMD algorithm to 9 and perform VMD decomposition. The IMF components and their center frequencies are obtained as shown in Table 1. The time domain diagram and frequency domain diagram of each IMF are shown in Figure 11.  
Figure 11: VMD decomposition results. (**a**) Time domain diagram of VMD decomposition; (**b**) frequency domain diagram of VMD decomposition. By observing Table 1 and Figure 11, it can be seen that the center frequency of IMF7 coincides with the frequency of the high-energy component in the signal. The high-energy component can be clearly observed in IMF7, the frequency component to be measured can be observed in IMF9, and other components are useless noise components. According to the calculated center frequency of each component, the frequency to be measured is greater than the center frequency of IMF9 and less than the center frequency of IMF8. IMF8 and IMF9 are extracted and superimposed. After this operation, the high-energy frequency components in IMF7 are eliminated. Take the extracted features as the input of the double-coupled Duffing system and obtain the phase plane of the system output as shown in Figure 12.  
By observing Figure 12, the system is in a large-period state at this time, which proves that the system has detected the signal to be measured. Compared with the phase plane in Figure 9, the phase plane at this time is smoother, which greatly alleviates the problem of phase track ambiguity and deformation, which is beneficial for phase state discrimination. In order to fully prove the improvement of VMD decomposition on the detection performance of the double-coupled Duffing system, the frequency component of 10 rad/s in the signal to be measured is deleted. After the above steps are repeated, the double-coupled Duffing system is excited, and the phase plane is obtained as shown in Figure 13.  
It can be seen from Figure 13 that, after the frequency component to be measured is deleted, the system is in a chaotic state, which proves that the signal component to be measured is not detected at this time.