## 2 Theoretical Model  
### Coupled Multi-Stable System Model  
At present, there are many system models with different steady-state numbers for stochastic resonance, such as the monostable model, bistable model, tristable model, and periodic multi-stable model. Among these, the periodic multi-stable model exhibits unique advantages due to its multiple potential barriers and wells, enabling complex particle transitions across different energy states. This dynamic behavior can be mathematically described by the Langevin equation:  
\[\frac{dx}{dt}=-U^{\prime}(x)+s(t)+n(t), \tag{1}\]  
where \(U(x)\) represents the multi-stable potential function model, \(s(t)\) denotes the target signal to be detected, and \(n(t)\) represents Gaussian white noise with a zero mean and noise intensity \(D\). The multi-stable potential function \(U(x)\) can be mathematically expressed as  
\[U(x)=-a\cos(bx), \tag{2}\]  
where \(a\) and \(b\) are the system parameters. The potential function model is shown in Figure 1, which contains multiple potential wells and barriers.  
The coupled model enhances SR performance through strategic modification of the nonlinear system architecture. This innovative approach incorporates inter-system interactions between individual SR models, utilizing coupling coefficients as weighting parameters to optimize system performance. Specifically, the coupled SR model demonstrates superior capability in generating more reliable output signals and effectively extracting weak signal characteristics, particularly under conditions of intense noise interference. The mathematical representation of the coupled multi-stable system model is expressed as follows:  
\[\left\{\begin{array}{ll}\frac{dx}{dt}=-U^{\prime}(x)+\gamma(y-x)+s(t)\\ \frac{dy}{dt}=-V^{\prime}(x)+\gamma(y-x)\end{array}\right., \tag{3}\]  
where \(\gamma\) is the coupling coefficient; \(s(t)\) is the collected noisy input signal; \(x(t)\) is the final output signal of the system; and \(U(x)\) and \(V(y)\) are both periodic multi-stable models, which can be expressed as  
\[\left\{\begin{array}{ll}U(x)=-a\cos(bx)\\ V(x)=-c\cos(dx)\end{array}\right., \tag{4}\]  
where \(a\) and \(b\) represent the controlled system parameters, while \(c\) and \(d\) denote the control system parameters. Through the substitution of Equation (4) into Equation (3), the system output described by Equation (3) is numerically solved using the fourth-order Runge-Kutta method, as mathematically represented in Equation (5).  
Figure 1: Multi-stable potential function model.  
\[\left\{\begin{array}{l}k_{1}=h[-ab\sin(bx_{n})+\gamma(y_{n}-x_{n})+s_{n}]\\ j_{1}=h[-cd\sin(dy_{n})+\gamma(y_{n}-x_{n})]\\ k_{2}=h[-ab\sin(b(x_{n}+0.5k_{1}))+\gamma(y_{n}-(x_{n}+0.5k_{1}))+s_{n}]\\ j_{2}=h[-cd\sin(d(y_{n}+0.5j_{1}))+\gamma((y_{n}+0.5j_{1})-x_{n})]\\ k_{3}=h[-ab\sin(b(x_{n}+0.5k_{2}))+\gamma(y_{n}-(x_{n}+0.5k_{2}))+s_{n}]\\ j_{3}=h[-cd\sin(d(y_{n}+0.5j_{2}))+\gamma((y_{n}+0.5j_{2})-x_{n})]\\ k_{4}=h[-ab\sin(b(x_{n}+k_{3}))+\gamma(y_{n}-(x_{n}+k_{3}))+s_{n}]\\ j_{4}=h[-cd\sin(d(y_{n}+j_{3}))+\gamma((y_{n}+j_{3})-x_{n})]\\ x_{n+1}=x_{n}+\frac{1}{6}(k_{1}+2k_{2}+3k_{3}+k_{4})\\ y_{n+1}=y_{n}+\frac{1}{6}(j_{1}+2j_{2}+3j_{3}+j_{4})\end{array}\right., \tag{5}\]