## 2 Output Saturation Limits for Classical Stochastic Resonance Models  
### Potential Function Model for Bistable Systems  
The three necessary conditions for generating stochastic resonance are an input signal, noise, and a bi- or multi-stationary nonlinear system. Therefore, selecting a suitable nonlinear system is important for generating stochastic resonance phenomena. The bistable system potential function [9] is given by  
\[U(x)=-\frac{1}{2}ax^{2}+\frac{1}{4}bx^{4} \tag{1}\]where \(U\) denotes the potential energy of the bistable system, and \(x\) is the characteristic displacement of the bistable system, which is also used as an observation parameter of the system state. \(a\), \(b\) are the system parameters of the bistable system, and the potential well image is shown in Figure 1 when the system parameters \(a\), \(b\) are varied. As can be seen from Figure 1, the potential well height \(\Delta U\) of the bistable system is related to the parameters \(a\) and \(b\) with the specific relational expression [10]:  
\[\Delta U=\frac{a^{2}}{4b} \tag{2}\]  
Adjusting the values of variables a and b can alter the shape of the potential function, which is crucial in determining whether the particles can successfully make a jump [11]. When comparing the potential function images of group A (\(a=2\), \(b=0.5\)) and group B (\(a=2.4\), \(b=0.5\)), it is evident that as the system parameter increases, the depth of the potential well deepens and the width of the potential well widens, making it more difficult for particles to make the jump. Similarly, when comparing the potential function images of group A (\(a=2\), \(b=0.5\)) and group C (\(a=2\), \(b=2/3\)), it is observed that with an increase in the system parameters, the depth of the potential well becomes shallower, the width of the potential well becomes narrower, and it becomes less difficult for the particles to make the leap. In engineering, broad-spectrum background noise is usually modeled as Gaussian white noise \(\Gamma(t)\) with a mean value of 0 and an intensity of \(D\):  
\[\left\{ \begin{array}{l}\left\langle\Gamma(t)\right\rangle=0\\ \left\langle\Gamma(t)\cdot\Gamma(t+\tau)\right\rangle=2D\delta(\tau)\end{array} \right. \tag{3}\]  
where \(\delta(\tau)\) is the unit impulse function, \(t\) is time, \(\tau\) denotes the time between two stochastic processes, and all \(t\) and \(\tau\) appearing later express the same meaning. When excited only by a periodic signal, the periodic signal affects the potential function of the bistable system, which is expressed as [12]:  
\[U(x)=-\frac{1}{2}ax^{2}+\frac{1}{4}bx^{4}+x[A\cos(\omega_{0}t)] \tag{4}\]  
The potential curve of the bistable system is periodically tilted at the signal angular frequency \(\omega_{0}\), and the two potential wells periodically rise and fall at the signal angular  
Figure 1: Potential well image of a bistable system. frequency \(\omega_{0}\) with respect to the base height. At this point, there exists a dual potential well critical amplitude \(A_{C}\)[13]:  
\[A_{C}=\sqrt{\frac{4a^{3}}{27b}} \tag{5}\]  
When \(A>A_{C}\), the system cannot maintain the bistable characteristics; when \(A<A_{C}\), the system still maintains the bistable characteristics, and the signal particles make a periodic motion in only one of the potential wells at the signal frequency \(\omega_{0}\), depending on the initial conditions.