## 2 The PTSR in Underdamped Condition  
In prior research, we invented a PTSR system that could improve the output saturation problem, the potential function, as shown below [27]:  
\[\mathrm{U}(x)=\left\{\begin{array}{l}-\frac{3 mp^{2}q^{2}\left(p^{2}-q^{2} \right)-m\left(p^{6}-q^{6}\right)}{6\sqrt{2}\sqrt{3q^{2}-p^{2}-12q}}x+\frac{mq ^{4}\left(3p^{2}-q^{2}\right)}{12}-\frac{3 mp^{2}q^{4}\left(p^{2}-q^{2}\right)- mq\left(p^{6}-q^{6}\right)}{6\sqrt{2}\sqrt{3q^{2}-p^{2}-12q}},\;x\leq-q\\ \frac{m}{2}p^{2}q^{2}x^{2}-\frac{m}{4}\left(p^{2}+q^{2}\right)x^{4}+\frac{m} {6}\kappa^{6}\;,\;q<x<q\\ \frac{3 mp^{2}q^{2}\left(p^{2}-q^{2}\right)-m\left(p^{6}-q^{6}\right)}{6\sqrt{ 2}\sqrt{3q^{2}-p^{2}-12q}}x+\frac{mq^{4}\left(3p^{2}-q^{2}\right)}{12}-\frac{3 mp^{2}q^{4}\left(p^{2}-q^{2}\right)-mq\left(p^{6}-q^{6}\right)}{6\sqrt{2} \sqrt{3q^{2}-p^{2}-12q}}\;,\;x\geq q\end{array}\right.\right. \tag{1}\]  
where \(m\), \(p\), \(q\) are the system parameter for PTSR, and \(m>0\), \(p>0\), \(q>0\) and \(q>p\). Figure 1 displays its potential function.  
The SR in the system of Equation (1) in the underdamped state can be expressed as:  
\[\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}=\frac{\mathrm{d}\mathrm{U}(x)}{ \mathrm{d}x}-\gamma\frac{\mathrm{d}x}{\mathrm{d}t}+A\cos(2\pi f_{d}t)+\xi(t) \tag{2}\]  
where \(x\) is system output, \(A\) and \(f_{d}\) are the amplitude and frequency of sine signal, \(\gamma\) denotes the damping ratio, \(\mathrm{U}(x)\) is the \(\mathrm{U}(x)\) mentioned in Equation (1). \(\xi(t)\) represents additive white gaussian noise (AWGN) and it fits the following equation:  
\[\left\{\begin{array}{l}\left\langle\xi(t)\right\rangle=0\\ \left\langle\xi(t)\xi(t-\tau)\right\rangle=2D\delta(t)\end{array}\right. \tag{3}\]  
where \(\tau\) represents the time interval, \(D\) represents noise intensity.  
Figure 1: Potential function (\(m=1,p=0.5,q=1\)).  
First, turning Equation (2) into a system of first order differential equations for the convenience of solving:  
\[\left\{\begin{array}{c}\frac{\mathrm{d}x}{\mathrm{d}t}=-\frac{\mathrm{d}U(x)}{ \mathrm{d}x}-\gamma y+Acos(2\pi f_{d}t)+\xi(t)\end{array}\right. \tag{4}\]  
The fourth-order Runge-Kutta method commonly used in stochastic resonance theory is used to solve the above equation [28]:  
\[\left\{\begin{array}{c}k_{1}=y_{i}\\ l_{1}=S_{i}-\gamma k_{1}-\ \mathrm{U}^{\prime}(x_{i})\\ k_{2}=y_{i}+\frac{1}{2}hl_{1}\\ l_{2}=S_{i}-\gamma k_{2}-\ \mathrm{U}^{\prime}\Big{(}x_{i}+\frac{1}{2}hk_{1} \Big{)}\\ k_{3}=y_{i}+\frac{1}{2}hl_{2}\\ l_{3}=S_{i+1}-\gamma k_{3}-\ \mathrm{U}^{\prime}\Big{(}x_{i}+\frac{1}{2}hk_{2} \Big{)}\\ k_{4}=y_{i}+hl_{3}\\ l_{4}=S_{i+1}-\gamma k_{4}-\ \mathrm{U}^{\prime}(x_{i}+hk_{3})\\ x_{i+1}=x_{i}+\frac{1}{6}h(k_{1}+2k_{2}+2k_{3}+k_{4})\\ y_{i+1}=y_{i}+\frac{1}{6}h(l_{1}+2l_{2}+2l_{3}+l_{4})\end{array}\right. \tag{5}\]  
where \(S_{i}\) represents system input, \(x_{i}\) represents system output, \(h\) is step size, \(k_{i}\) and \(l_{i}\) are the slopes of \(x_{i}\) and \(y_{i}\), respectively. We note that \(i\) in the formula is the length of the data, which is the number of sampling points.  
The above calculation is equivalent to changing the overdamped state to the underdamped state on the basis of the PTSR method in the previous study.