## 4 Establishment of Double-Coupled Duffing Weak Signal Detection System
### Determination of Chaotic Threshold of Double-Coupled Duffing System  
According to the above weak signal detection principle of the double-coupled Duffing system, the amplitude of the built-in power needs to be adjusted to the chaos threshold first. There are many factors that affect the chaos threshold of the double-coupled Duffing system, such as the solution method, the calculation step, the initial value of the system, etc. In this paper, the maximum Lyapunov exponent is used to determine the chaos threshold of the double-coupled Duffing system. The coupling term of the double-coupled Duffing system will make the two Duffing systems related, which will cause a weak difference in response and have a certain impact on the chaos threshold, but this impact is extremely weak and can be ignored. Therefore, the chaos threshold of a single Duffing system is used as the chaos threshold of the double-coupled Duffing system in this paper. For a single Duffing system, there are two Lyapunov exponents, the larger of which is the maximum Lyapunov exponent. The maximum Lyapunov exponent will change from a positive value to a negative value if and only if the system moves from a chaotic state to a  
Figure 3: Variation of the variance output value with the amplitude of the built-in force. large-scale periodic state with the amplitude change in the built-in power. According to this feature, by increasing the amplitude of the built-in power, when the maximum Lyapunov exponent changes from a positive value to a negative value, the amplitude of the built-in power corresponding to the last positive Lyapunov exponent is the chaos threshold. Under the system parameters in this paper, the frequency of the built-in force is set as 1 rad/s. According to the above calculation method, when the amplitude of the built-in force is 0.826, the maximum Lyapunov index is positive, and the system is in a chaotic state. When the amplitude of the built-in force is 0.827, the maximum Lyapunov index is negative, and the system is in a large-scale periodic state. Therefore, the chaotic threshold of the system is 0.826. The phase trajectory diagram when the built-in force is 0.826 and 0.827 is shown in Figure 4.  
### Determination of Chaotic Threshold of Double-Coupled Duffing System  
First, under any initial value, adjust the built-in force of the system to make the system in a large-scale periodic state and record the track range of periodic motion at this time. Then, traverse the initial value points within this range and calculate whether it can make the variance of the system's output value increase steadily with the growth of the built-in force. The calculation method is shown in Equations (5) and (6):  
\[Y=\sum_{n=0}^{m-1}sp(D(F_{0}+nd)-D(F_{0}+(n-1)d))/m \tag{5}\]  
\[sp(x)= \begin{cases}1&x>0\\ 0&x\leq 0\end{cases}  \tag{6}\]  
where \(Y\) is the fluctuation index, \(sp(x)\) is a judgment function, \(m\) is the number of calculations, \(D(F)\) is the variance of the output value of the double-coupled Duffing system when the amplitude of built-in force is \(F\), \(F_{0}\) is the chaos threshold and can be calculated by the method shown in Section 4.2, and \(d\) is the calculation step of the amplitude of the built-in force. The \(Y\) value essentially reflects the fluctuation of the variance of the output value of the double-coupled Duffing system when the built-in driving force increases from the chaos threshold to \(F_{0}+md\). If it is 0, it means that the variance of the output value of the system increases steadily with the increase in the built-in driving force. The larger the \(Y\) value, the greater the fluctuation in the variance of the output value. After traversing the initial value points within the calculation range, we can obtain regions that can make the variance of the output value increase steadily. As shown in Figure 5, The red area is stable, select points close to the large periodic orbit as the initial value of the system in these regions.