## 1 Introduction
### Transient Steady-State Response Analysis of Underdamped Stochastic Resonance System
#### 3.2.2 Influence of System Parameters and Input Signal Frequency on the System Response Speed  
There are two main factors affecting the system response speed, one is the system parameters, and the other is the frequency of external input signals. In order to further explore the influencing factors of system response speed, this paper takes the exponential decay term in Formula (36) for analysis:  
\[\begin{split} x_{LUBSR}(t)&=\ln\biggl{(}\frac{2v}{ ar^{2}}\biggr{)}e^{-\frac{\gamma}{2}tN-1}\sum\limits_{k=0}^{1}\frac{2\pi f_{k} \beta_{k}\gamma(\sin\theta_{k}+R_{1,2}\cos\theta_{k})}{4 DR_{1,2}{}^{2}(1+ \frac{\pi^{2}f_{k}^{2}}{R_{1,2}})\sqrt{(4b-\gamma^{2})}}\sin\biggl{(}\frac{ \sqrt{4b-\gamma^{2}}}{2}t\biggr{)}\\ &+\ln\biggl{(}\frac{2v}{ar^{2}}\biggr{)}e^{-\frac{\gamma}{2}tN-1 }\sum\limits_{k=0}^{1}\frac{r^{2}\beta_{k}(R_{1,2}\sin\theta_{k}-\cos\theta_{ k})}{2 DR_{1,2}{}^{2}(1+\frac{\pi^{2}f_{k}^{2}}{R_{1,2}})}\cos\biggl{(}\frac{ \sqrt{4b-\gamma^{2}}}{2}t\biggr{)}\end{split} \tag{39}\]  
Figure 3: Response curve graph of two systems.  
After combining and simplifying Formula (39), Formula (40) is obtained as follows:  
\[x_{UIUBSR}(t)=e^{-\frac{\gamma}{2}t}\sqrt{\left(C_{1}\right)^{2}+\left(C_{2} \right)^{2}}\sin\left(\frac{\sqrt{4b-\gamma^{2}}}{2}t+\phi\right) \tag{40}\]  
where \(\phi=\arctan\frac{r^{2}\left(R_{1,2}\sin\theta_{b}-\cos\theta_{b}\right)}{\pi f _{k}\gamma\left(\sin\theta_{k}+R_{1,2}\cos\theta_{b}\right)}\).  
By performing a Taylor series expansion on Formula (40), the Taylor series expansion of Formula (41) is obtained as follows:  
\[x_{\text{SUUBSR}}(t)=\sqrt{\left(C_{1}\right)^{2}+\left(C_{2}\right)^{2}}\sum _{L=0}^{\infty}\frac{\left(-1\right)^{3L+1}\gamma^{L}\left(\frac{\sqrt{4b- \gamma^{2}}}{2}t+\phi\right)^{2L+1}t^{L}}{2^{L}L!(2L+1)!} \tag{41}\]  
The influence diagram of system parameters and input signal frequency on the system response speed obtained from Formula (41) is shown in Figure 4. It can be seen from Figure 4a that the larger the damping coefficient value, the smaller the initial value of the transient response, and the faster the system response speed. When the damping coefficient value is too large, the system response speed becomes slow. In Figure 4c, as the parameter r increases, the initial value of the system transient response decreases, and the system response speed slows down. In Figure 4b,d, the parameters \(a\) and \(v\) do not affect the initial value of the transient response. As \(a\) increases, the system response speed becomes faster, and as \(v\) increases, the system response speed slows down. In Figure 4e, as the input signal frequency increases, the initial value of the transient response decreases, and the system response speed slows down.  
Furthermore, an approximate functional expression between time \(T_{d}\), \(\gamma\), and input signal frequency \(f_{k}\) can be obtained.  
\[T_{d}=\sqrt[3L+1]\sqrt[3L+1]\sum_{L=0}^{\infty}\frac{x_{si}2^{2L}L!(2L+1)!}{ \left(-1\right)^{3L+1}\gamma^{L}\sqrt{\left(4b-\gamma^{2}\right)^{2L+1}}\sqrt {\left(C_{1}\right)^{2}+\left(C_{2}\right)^{2}}} \tag{42}\]  
In Formula (42), \(L\) is a positive integer; in this case, \(T_{d}>0\), and it can be seen from the formula that the transient response time is related to the damping coefficient, particle transition rate, and signal frequency. The higher the input signal frequency and the greater the particle transition rate, the shorter the transient response time of the system. Meanwhile, the relationship between the damping coefficient and the transient response time is not completely monotonic. When the damping coefficient is within the range of \(\left(0,\sqrt{4b/3}\right)\), an increase in the damping coefficient leads to a shorter transient response time. However, when the damping coefficient is within the range of \(\left(\sqrt{4b/3},1\right)\), an increase in the damping coefficient gradually exhibits an overdamping characteristic, resulting in a longer transient time and a slower response speed.