## 3 Stability of Equilibrium Points
### Equilibrium Points  
According to the definition of equilibrium points of piecewise smooth systems [33], an equilibrium point is termed an admissible equilibrium point if \(\hat{x}^{*}=\mathbf{f}^{i}(\mathbf{x}^{*})\) and \(\mathbf{x}^{*}\in D_{i}\). If the system satisfies \(\hat{\mathbf{x}}^{*}=\mathbf{f}^{i}(\mathbf{x}^{*})\), \(\mathbf{x}^{*}\in D_{j},i\
eq j\) then it is called a virtual equilibrium point.  
Firstly, the equilibrium points of system (4) are  
\[E_{c}= \frac{\
u_{2}^{2}w_{1}+2w_{2}\zeta_{1}}{\
u_{2}^{2}}\Big{(} \partial_{1,1}^{6,1}(1)+\partial_{3,1}^{6,1}(1)+\partial_{5,1}^{6,1}(1)\Big{)},\] \[E_{r}= \partial_{1,1}^{6,1}\Bigg{(}\frac{\
u_{2}^{2}w_{1}+2w_{2}\zeta_{1 }}{\
u_{2}^{2}}-\frac{\rho_{1}\rho_{2}L}{\
u_{2}^{2}}+\frac{\rho_{1}L\big{(}w _{1}^{2}+\rho_{2}\big{)}}{\
u_{2}^{2}(v+1)}\Bigg{)}\] \[+\partial_{3,1}^{6,1}\Bigg{(}\frac{\
u_{2}^{2}w_{1}+2w_{2}\zeta_{ 1}}{\
u_{2}^{2}}-\frac{\big{(}\rho_{1}\
u_{1}^{2}+v_{2}^{2}\big{)}\rho_{2}L}{ \
u_{1}^{2}\
u_{2}^{2}}+\frac{\big{(}\rho_{1}\
u_{1}^{2}+v_{2}^{2}\big{)}\big{(} uv_{1}^{2}+\rho_{2}\big{)}L}{\
u_{1}^{2}v_{2}^{2}(v+1)}\Bigg{)}\] \[+\partial_{5,1}^{6,1}\Bigg{(}\frac{\
u_{2}^{2}w_{1}+2w_{2}\zeta_{ 1}}{\
u_{2}^{2}}-\frac{\big{(}\rho_{1}\
u_{1}^{2}+v_{2}^{2}\big{)}\rho_{2}L}{ \
u_{1}^{2}v_{2}^{2}}+\frac{L}{\
u_{1}^{2}v_{2}^{2}(v+1)}\] \[\cdot \Big{(}\rho_{1}w_{1}^{4}+w_{1}^{2}\
u_{2}^{2}+\rho_{1}\rho_{2}v_ {1}^{2}+v_{1}^{2}\
u_{2}^{2}+\rho_{2}v_{2}^{2}\Big{)}\Big{)},\] \[E_{l}= \partial_{1,1}^{6,1}\Bigg{(}\frac{\
u_{2}^{2}w_{1}+2w_{2}\zeta_{ 1}}{\
u_{2}^{2}}+\frac{\rho_{1}\rho_{2}L}{\
u_{2}^{2}}-\frac{\rho_{1}L\big{(} w_{1}^{2}+\rho_{2}\big{)}}{\
u_{2}^{2}(v+1)}\Bigg{)}\] \[+\partial_{3,1}^{6,1}\Bigg{(}\frac{\
u_{2}^{2}w_{1}+2w_{2}\zeta_{ 1}}{\
u_{2}^{2}}+\frac{\big{(}\rho_{1}\
u_{1}^{2}+v_{2}^{2}\big{)}\rho_{2}L}{ \
u_{1}^{2}v_{2}^{2}}-\frac{\big{(}\rho_{1}\
u_{1}^{2}+v_{2}^{2}\big{)}\big{(} uv_{1}^{2}+\rho_{2}\big{)}L}{\
u_{1}^{2}v_{2}^{2}(v+1)}\Bigg{)}.\] \[+\partial_{5,1}^{6,1}\Bigg{(}\frac{\
u_{2}^{2}w_{1}+2w_{2}\zeta_{ 1}}{\
u_{2}^{2}}+\frac{\big{(}\rho_{1}\
u_{1}^{2}+v_{2}^{2}\big{)}\rho_{2}L}{ \
u_{1}^{2}v_{2}^{2}}-\frac{L}{\
u_{1}^{2}v_{2}^{2}(v+1)}\] \[\cdot \Big{(}\rho_{1}w_{1}^{4}+w_{1}^{2}\
u_{2}^{2}+\rho_{1}\rho_{2}v_ {1}^{2}+v_{1}^{2}\
u_{2}^{2}+\rho_{2}v_{2}^{2}\Big{)}\Big{)}.\]  
Then, the existence of the admissible equilibrium point is discussed under different conditions of \(v\).  
Case 1: \(v>0\)  
* \(E_{c}\) is always an admissible equilibrium point;
* \(E_{r}\) is always a virtual equilibrium point, because \(\frac{L}{v+1}<L\);
* \(E_{l}\) is always a virtual equilibrium point, because \(-\frac{L}{v+1}>-L\).  
Case 2: \(v=0\)  
* \(E_{c}\) is always an admissible equilibrium point;
* \(E_{r}\) is always a boundary equilibrium point, because \(v=0\);
* \(E_{l}\) is always a boundary equilibrium point, because \(v=0\).  
Case 3: \(v<0\)  
* \(E_{c}\) is always an admissible equilibrium point;
* \(E_{r}\) is always an admissible equilibrium point, because \(\frac{L}{v+1}>L\);
* \(E_{l}\) is always a boundary equilibrium point, because \(-\frac{L}{v+1}<-L\).  
Next, the stability of an admissible equilibrium point \(E_{c}\) is discussed when \(v>0\).