## 2 Tri-Stable Stochastic Resonance Theory
### Principle of Stochastic Resonance
#### 2.2.3 Multi-Stationary Stochastic Resonance Model  
When the external force \(f(x)=-U^{\prime}(x)\) of the particle in the nonlinear Langevin equation, \(\Gamma(t)=\xi(t)\) is taken into Equation (13)  
\[\dot{x}=-U^{\prime}(x)+s(t)+\xi(t), \tag{20}\]  
where \(U(x)\) is a class of nonlinear polystable potential functions, as shown in (20). Among them, \(a,b,c\) are the number of systems, and both are greater than 0. \[U(x)=\frac{a}{2}x^{2}-\frac{1+a}{4b}x^{4}+\frac{c}{6}x^{6}, \tag{21}\]Suppose the system input signal is \(s(t)\) and Equation (21) is taken into Equation (20). \[\dot{x}=-ax+\frac{1+a}{b}x^{3}-cx^{5}+s(t)+\xi(t), \tag{22}\]  
where \(s(t)\) is an input signal superimposed on multiple frequencies, representing an external force, as shown in Equation (23)  
\[s(t)=\sum_{i=1}^{n}A_{i}\sin\left(2\pi f_{i}t\right),i=1,2,\ldots,n, \tag{23}\]  
In the formula, \(A_{i}\) is the amplitude of the signal, \(f_{i}\) is the frequency of the signal, and \(n\) is the number of input signals. Suppose that the statistical characteristics of \(\xi(t)\) are shown in Equation (24), \(D\) is the noise intensity, and \(t^{\prime}\) is the time delay of \(t\). \[\left\langle\xi(t)\right\rangle\geq 0,\left\langle\xi(t)\xi(t^{\prime}) \right\rangle\geq 2D\delta(t-t^{\prime}), \tag{24}\]  
According to Formula (24) the Fourier expansion of \(\xi(t)\) can obtain its power spectrum. \[S(\omega)=\int e^{-i\omega\tau}2D\delta(\tau)d\tau=2D, \tag{25}\]  
From Equation (25), the power spectrum \(S(\omega)\) is independent of \(\omega\), that is, the spectrum is a white spectrum. Therefore, the lang of Equation (24), in which the time correlation function is \(\xi(t)\), is called white noise. At the same time, the nonlinear Langevin equation can be written as  
\[\dot{x}=-ax+\frac{1+a}{b}x^{3}-cx^{5}+s(t)+D\xi(t), \tag{26}\]  
The above formula is a kind of nonlinear Langevin equation with three steady-state points, which is essentially the periodic transition of the particle in the potential well of the nonlinear system under the action of all the forces. In the multi-stable SR system described in Equation (26), when the system is in equilibrium, there are three steady-state solutions (minimum points) and two unsteady solutions (maximum points), respectively, Equations (27) and (28):  
\[\left\{ \begin{array}{l}-x_{2}=-\sqrt{\frac{1}{2c}(\frac{1+a}{b}+\sqrt{( \frac{1+a}{b})^{2}-4 ac})}\\ x_{0}=0\\ x_{2}=\sqrt{\frac{1}{2c}(\frac{1+a}{b}+\sqrt{(\frac{1+a}{b})^{2}-4 ac})}\\ \end{array} \right. \tag{27}\]  
\[\left\{ \begin{array}{l}-x_{1}=-\sqrt{\frac{1}{2c}(\frac{1+a}{b}-\sqrt{( \frac{1+a}{b})^{2}-4 ac})}\\ x_{1}=\sqrt{\frac{1}{2c}(\frac{1+a}{b}-\sqrt{(\frac{1+a}{b})^{2}-4 ac})}\\ \end{array} \right. \tag{28}\]  
At this time, the corresponding intermediate well depth \(\Delta U_{M}\) and the potential well depth \(\Delta U_{L,R}\) are  
\[\Delta U_{M}=\frac{1}{24c^{2}}\left[\left(\left(\frac{1+a}{b}\right)^{2}-4 ac \right)^{\frac{3}{2}}-\frac{1+a}{b}\left(\left(\frac{1+a}{b}\right)^{2}-6 ac \right)\right], \tag{29}\]  
\[\Delta U_{L,R}=\frac{1}{12c^{2}}\left[\left(\frac{1+a}{b}\right)^{2}-4 ac \right]^{\frac{3}{2}}, \tag{30}\]  
When \(A_{i}=D=0,a=3,b=5,c=0.038\), the potential function curve of the multi-stable system is shown in Figure 2.  
When \(A_{i}=0;\ D\
eq 0\), that is, there is no periodic signal input and only the noise signal acts, the particles will randomly jump between the three potential wells of the multi-stable system without periodicity. When \(A_{i}\
eq 0;\ D=0\), when there is only periodic signal input and no noise signal is excited, the equilibrium state of the system will change, and the three potential wells will generate periodic tilt with the same frequency under the action of the periodic signal. When the particle energy is less than the energy required to cross the barrier, the particle can only periodically move in the same potential of the input signal in one of the potential wells of the multi-stable system.