## 3 Stability of Equilibrium Points
### Stability  
From (4), the characteristic equation is shown as follows,  
\[\lambda^{6}+a_{1}\lambda^{5}+a_{2}\lambda^{4}+a_{3}\lambda^{3}+a_{4}\lambda^{2} +a_{5}\lambda+a_{6}=0, \tag{5}\]where  
\[a_{1}= 2\zeta_{1}+2\zeta_{2}+2\zeta_{3}+2\zeta_{4}+2\zeta_{6},\] \[a_{2}= v+\
u_{1}^{2}+uv_{1}^{2}+\
u_{2}^{2}+4\zeta_{1}\zeta_{3}+4\zeta_{1} \zeta_{4}+4\zeta_{2}\zeta_{4}+4\zeta_{1}\zeta_{6}+4\zeta_{2}\zeta_{6}+4\zeta_{3} \zeta_{6}+\
u_{1}^{2}\rho_{1},\] \[a_{3}= 2v\zeta_{1}+2\
u_{1}^{2}\zeta_{1}+2 uv_{1}^{2}\zeta_{1}+2v\zeta_{2 }+2 uv_{1}^{2}\zeta_{2}+2v\zeta_{3}+2v_{2}^{2}\zeta_{3}+2v_{2}^{2}\zeta_{4}+2v_{ 1}^{2}\zeta_{6}+2v_{2}^{2}\zeta_{6}\] \[+8\zeta_{1}\zeta_{3}\zeta_{6}+2\
u_{1}^{2}\zeta_{4}\rho_{1}+2v_{1} ^{2}\zeta_{6}\rho_{1},\] \[a_{4}= v\
u_{1}^{2}+v_{1}^{2}+\
u_{1}^{2}\
u_{2}^{2}+uv_{1}^{2}\
u_{2} ^{2}+uv_{1}\zeta_{3}+4v_{1}^{2}\zeta_{1}\zeta_{6}+4\
u_{2}^{2}\zeta_{3}\zeta_{6 }+vv_{1}^{2}\rho_{1}+uv_{1}^{4}\rho_{1},\] \[a_{5}= 2 vv_{1}^{2}\zeta_{1}+2 vv_{2}^{2}\zeta_{3}+2v_{1}^{2}\
u_{2}^{2} \zeta_{6},\] \[a_{6}= v\
u_{1}^{2}\
u_{2}^{2},\] \[\Delta_{1}= a_{1},\Delta_{3}=\begin{vmatrix}a_{1}&a_{3}&a_{5}\\ 1&a_{2}&a_{4}\\ 0&a_{1}&a_{3}\end{vmatrix},\Delta_{5}=\begin{vmatrix}a_{1}&a_{3}&a_{5}&0&0\\ 1&a_{2}&a_{4}&a_{6}&0\\ 0&a_{1}&a_{3}&a_{5}&0\\ 0&1&a_{2}&a_{4}&a_{6}\\ 0&0&a_{1}&a_{3}&a_{5}\end{vmatrix}.\]  
The parameters of a DVA play an important role in its design and application. According to the Lienard-Chipart criterion [24], the conditions that Equation (5) has negative real parts are \(a_{2}>0,a_{4}>0,a_{6}>0,a_{1}>0,\Delta_{3}>0,\Delta_{5}>0\). The asymptotically stable and unstable regions of \(E_{c}\) are discussed under  
\[P_{ar1}=(v,\rho_{1},\rho_{2},u,\zeta_{1},\zeta_{5},\zeta_{6},\zeta_{7})=(2,0. 2,0.01,1,0,0.0,1).\]  
According to the conditions \(a_{2}>0,a_{4}>0,a_{6}>0,a_{1}>0,\Delta_{3}>0,\Delta_{5}>0\), the asymptotically stable and unstable regions of \(E_{c}\) under \(P_{ar1}\) in the two-dimensional parameter space \((v_{1},v_{2})\) are shown in Figure 2.  
Figure 2 shows that the asymptotically stable and unstable regions of \(E_{c}\) have different distributions for different \(\zeta_{2},\zeta_{3},\zeta_{4}\) values, providing theoretical guidance for the parameter design of a DVA. The relationship between the system parameters \(\
u_{1}\) and \(\
u_{2}\) can be obtained, which is drawn with the stability conditions. As can be seen in Figure 2a, when the parameter \(\
u_{1}\) increases, the critical value of the parameter \(\
u_{2}\) also increases gradually. When the system is in the shadow part of the region, the equilibrium point is asymptotically stable.  
The vibration reduction effect of system (1) with and without the nonsmooth DVA in the asymptotically stable and unstable regions under \((\zeta_{2},\zeta_{3},\zeta_{4})=(0.1,0,0)\) when \(y(t)=0.5\sin(0.1t)\) is shown in Figure 3.  
Figure 3a shows that the vibration is reduced in the unstable region. Figure 3b shows the vibration of the mass \(m_{2}\) with nonsmooth DVA and without nonsmooth DVA in the asymptotically stable region.  
The asymptotically stable regions of \(E_{c}\) could be visualized in the 3D parameter space \((v_{1},v_{2},u)\) for different \(\xi_{2},\zeta_{4},\zeta_{6}\) values under the following condition, which is shown in Figure 4.  
\[P_{ar2}=(v,\rho_{1},\rho_{2},\zeta_{1},\zeta_{3},\zeta_{5},\zeta_{7})=(1,1,0.1, 2,0,0,0.1).\]  
Figure 3: The vibration reduction effect in the unstable and asymptotically stable regions: (**a**) \((v_{1},v_{2})=(1,2.5)\). (**b**) \((v_{1},v_{2})=(2,1)\).