# Dynamic Analysis of Modified Duffing System via Intermittent External Force and Its Application
## 3 Design and Dynamic Characteristic Analysis of MD System
### Dynamics Analysis of MD System with Respect to Parameter \(\tau\)  
Furthermore, if length of interval control time \(\tau\) is a variable and the others are fixed, i.e.,  
\[\begin{cases}\beta=0.2,\;k_{1}=0.6,\;k_{2}=0.1,\;A=2.1,\;\omega=1.1,\;\tau\in \left[0,\frac{4\pi}{3\omega}\right],\\ \left[T_{i},\;T_{i+1}\right]=\left[\frac{i\cdot 2\pi}{\omega},\;\frac{(i+1) \cdot 2\pi}{\omega}\right],\;t_{i}=\frac{i\cdot 2\pi+2\pi}{3\omega},\;\;(i=0,1,2,\cdots),\end{cases} \tag{14}\]  
then the bifurcation diagram with respect to parameter \(\tau\) and the corresponding Lyapunov exponent spectrum of MD System (10) are shown in Figure 7a,b. There are some parameter intervals in \(\left[0,\;\frac{4\pi}{3w}\right]\) that MD System (10) has a positive Lyapunov exponent, for example, it is chaotic in \(\tau\in\left[1.7,\;3.2\right]\).  
**Note 1:** If one considers replacing the original cosine function in Equation (9) with a constant, i.e., the intermittent external force is given by  
\[F=\begin{cases}A,&\text{if}\;\;t\in[t_{i},\;t_{i}+\tau]\subset[T_{i},\;T_{i+1} ],\\ 0,&\text{if}\;\;t\
otin[t_{i},\;t_{i}+\tau]\subset[T_{i},\;T_{i+1}],\end{cases} \tag{15}\]  
Figure 6: **(a)** Bifurcation diagram with respect to parameter \(\omega\) of Poincare cross-section of MD System (10) when \(x=y\) and \(t>500\). **(b)** Lyapunov exponent spectra with respect to parameter \(\omega\) of MD System (10) when other parameters are given in Equation (13).  
Figure 7: **(a)** Bifurcation diagram with respect to parameter \(\tau\) of Poincare cross-section of MD System (10) when \(x=y\) and \(t>500\). **(b)** Lyapunov exponent spectra with respect to parameter \(\tau\) when other parameters are given in Equation (14).  
where \([T_{i},\ T_{i+1}]\ \ (i=0,1,2,\cdots,T_{0}=0)\) is a periodic interval, \(\Delta T=T_{i+1}-T_{i}\ \ (i=0,1,2,\cdots)\), and \(\tau\) is the length of control time in an interval \([T_{i},\ T_{i+1}]\), then the MD system can be expressed as  
\[\left\{\begin{array}{ll}\left\{\begin{array}{ll}\dot{x}=-\beta x+y,\\ \dot{y}=k_{1}x-k_{2}x^{3}+A,&\text{if}\ \ t\in[t_{i},\ t_{i}+\tau]\subset[T_{i},\ T_{i+1}], \\ \left\{\begin{array}{ll}\dot{x}=-\beta x+y,\\ \dot{y}=k_{1}x-k_{2}x^{3},&\text{if}\ \ t\
otin[t_{i},\ t_{i}+\tau]\subset[T_{i},\ T_{i+1}]. \end{array}\right.\end{array}\right.\end{array}\right. \tag{16}\]  
Similarly, the initial values and parameters of MD System (16) are taken as  
\[\left\{\begin{array}{ll}(x_{0},\ y_{0},t_{0})=(0.1,\ 0.2,\ 0),\ A=2,\ \omega=1.1, \\ \beta=0.2,\ k_{1}=0.6,\ k_{2}=0.1,\ \tau=\frac{2\pi}{3\omega},\\ \left[T_{i},\ T_{i+1}\right]=\left[\frac{i\cdot 2\pi}{\omega},\ \frac{(i+1) \cdot 2\pi}{\omega}\right],\ t_{i}=\frac{i\cdot 2\pi+2\pi}{3\omega},\ \ (i=0,1,2,\cdots)\end{array}\right. \tag{17}\]  
where the length of a periodic time \(\Delta T=T_{i+1}-T_{i}=\frac{2\pi}{\omega}\), then the attractor is given in Figure 8a, and the corresponding time-domain waveform is shown in Figure 8b.  
In order to know about the dynamic characteristics of an MD system, one needs to fix all parameters except for magnitude \(A\) of external force, i.e.,  
\[\left\{\begin{array}{ll}\beta=0.2,\ k_{1}=0.6,\ k_{2}=0.1,\ \omega=1.1,\ \tau=\frac{2\pi}{3\omega},\ A\in[0,\ 10], \\ \left[T_{i},\ T_{i+1}\right]=\left[\frac{i\cdot 2\pi}{\omega},\ \frac{(i+1)\cdot 2 \pi}{\omega}\right],\ t_{i}=\frac{i\cdot 2\pi+2\pi}{3\omega},\ \ (i=0,1,2,\cdots). \end{array}\right. \tag{18}\]  
Then, the bifurcation diagram with respect to parameter \(A\) is given in Figure 9a, and the corresponding Lyapunov exponent spectrum with respect to parameter \(A\) is shown in Figure 9b. Obviously, there also exists a positive Lyapunov exponent in some intervals, such as \([1.5,2.3]\cup[8.6,10]\).  
Figure 8: (**a**) Chaotic attractor of MD System (16). (**b**) Corresponding time-domain waveform of MD System (16).