## 3 Influencing Factors of Multi-Stationary Stochastic Resonance System  
If the multi-stable SR system parameter \(b=4.8,\ c=0.04\) is given, the curve of its potential function and parameter \(a\) is plotted, as shown in Figure 2(a). Take \(a\) for 1.2, 2.6, and 4.2, respectively, and plot the potential function curve, as shown in Figure 2(b). It can be seen from the two figures that as the \(a\) increases, the depths of the three wells increase, the width of the well becomes wider, and the system is multi-stable; as \(a\) continues to increase, the two sides of the well depth will be much larger than the intermediate well depth until the intermediate well depth is reduced to zero, at which point the system switches to bi-stable. If the multi-stable SR system parameters are \(a=3,\ c=0.04\), the curve of the potential function and the parameter \(b\) is plotted as shown in Figure 2(c). It can be seen from the figure that as \(b\) increases continuously large, the system state gradually switches from multi-stable to mono-stable. Take \(b\) as 4, 5, and 7, respectively, and obtain the potential function curve, as shown in Figure 2(d). It can be seen from the figure that when \(b\) is small, the depth of the intermediate well is smaller than the depth of the left and right wells, and the system is multi-stable; but as \(b\) increases, the depth of the intermediate well remains unchanged, and the depth of the wells gradually decreases. Finally, when the depth of the two wells is zero, the system will switch to mono-stable. It can be seen from Equations (27) and (28) that by changing the system parameters \(a,b,c\), the three well depths of the multi-stable potential function can be changed, thereby changing the barrier height and further affecting whether the energy of the particles can be overcome by doing a periodic motion between potential wells. Since the multi-stable system has better detection capability than the mono-stable system and the bi-stable system, the system is selected in a multi-steady state as the system parameters are selected. In order to make the system in a multi-steady state, it is necessary to study the influence of system parameters \(a,b,c\) on the system to determine the range of values of each parameter. First, define the ratio of \(\Delta U_{M}\) and \(\Delta U_{L,R}\) to \(R\). Its expression is Equation (38)  
\[R=\frac{\Delta U_{M}}{\Delta U_{L,R}}, \tag{38}\]  
It can be seen from the above equation that when \(\Delta U_{M}=\Delta U_{L,R}\), the ratio \(R\) is 1, and the energy required for the periodic transition of the particles between each potential well is the same. When \(R>1\), which is \(\Delta U_{M}>\Delta U_{L,R}\), the energy required for the particles to transition from the two potential wells to the intermediate well is less than the energy required for the particles to transition from the intermediate well to the two wells. When \(R<1\), which is \(\Delta U_{M}<\Delta U_{L,R}\), the energy required for  
Figure 3: Potential function \(U(x)\) under different system parameters. the particles to transition from the intermediate potential well to the two potential wells is less than the energy required for the particles to transition from the two potential wells to the intermediate potential well. Assume that the input signal amplitude \(A_{i}\) and noise intensity \(D\) of the system are both 0, and the variation law of \(R\) with the system parameter \(a\) is shown in Figure 4.  
It can be seen from Figure 5 that when \(b\) and \(c\) take different values, as \(a\) increases continuously, the \(R\) value increases first and then decreases, and each \(R\) value corresponds to two different \(a\) values, When \(a=1\), its peak value is reached, that is, the ratio of \(U_{M}\) and \(U_{RL}\) is the largest, \(R=19.51\).