## 4 Bifurcation of Periodic Orbits  
In this section, we explore the existence and number of periodic orbits for a nonsmooth DVA model based on the Melnikov method. Let \(\bar{x}_{4}=\bar{x}_{3}-\bar{x}_{2},\bar{x}_{5}=\bar{x}_{2}-\bar{x}_{1}\), and system (2) is transformed into  
\[\begin{split}\ddot{\bar{x}}_{4}=&-\check{g}(\bar{x}_ {4})-(v+uv_{1}^{2})\bar{x}_{4}-\rho_{2}\check{g}(\bar{x}_{4})-(2\zeta_{4}+2 \zeta_{6})\dot{\bar{x}}_{4}-(2\zeta_{5}+2\zeta_{7})\check{h}(\dot{\bar{x}}_{4}) \\ &+v_{1}^{2}\bar{x}_{5}+2\zeta_{3}\dot{\bar{x}}_{5},\\ \dot{\bar{x}}_{5}=&\rho_{2}\check{g}(\bar{x}_{4})+ uv_{1}^{2}\bar{x}_{4}+2\zeta_{5}\check{h}(\dot{\bar{x}}_{4})+2\zeta_{4}\dot{ \bar{x}}_{4}-\Big{(}v_{1}^{2}+\rho_{1}v_{1}^{2}\Big{)}\bar{x}_{5}-(2\zeta_{2}+2 \zeta_{3})\dot{\bar{x}}_{5}\\ &+v_{2}^{2}\bar{x}_{1}+2\zeta_{1}\dot{\bar{x}}_{1}-\Big{(}v_{2}^ {2}w_{1}+2\zeta_{1}w_{2}\Big{)},\\ \ddot{\bar{x}}_{1}=&\rho_{1}v_{1}^{2}\bar{x}_{5}+2 \zeta_{2}\dot{\bar{x}}_{5}-v_{2}^{2}\bar{x}_{1}-2\zeta_{1}\dot{\bar{x}}_{1}+ \Big{(}v_{2}^{2}w_{1}+2\zeta_{1}w_{2}\Big{)},\end{split} \tag{6}\]  
where  
\[\check{g}(\bar{x}_{4})=\left\{\begin{array}{cc}\bar{x}_{4}-L,&\bar{x}_{4}> L\\ 0,&-L\leq\bar{x}_{4}\leq L,\\ \bar{x}_{4}+L,&\bar{x}_{4}<-L\end{array}\right.\]  
\[\check{h}(\dot{\bar{x}}_{4})=\left\{\begin{array}{cc}\bar{x}_{4},&\bar{x}_{4 }>L\\ 0,&-L\leq\bar{x}_{4}\leq L.\\ \dot{\bar{x}}_{4},&\bar{x}_{4}<-L\end{array}\right.\]  
System (6) is transformed into the following form:  
\[\ddot{\bar{x}}_{4}=-\widetilde{g}(\bar{x}_{4})-p_{1}\bar{x}_{4}-p_ {2}\tilde{g}(\bar{x}_{4})-p_{3}\dot{\bar{x}}_{4}-p_{4}\check{h}(\dot{\bar{x}}_ {4})+p_{5}\bar{x}_{5}+p_{6}\dot{\bar{x}}_{5},\] \[\ddot{\bar{x}}_{5}=e_{1}\check{g}(\bar{x}_{4})+e_{2}\bar{x}_{4}+e \check{g}\check{h}(\dot{\bar{x}}_{4})+e_{4}\dot{\bar{x}}_{4}-e_{5}\bar{x}_{5}-e _{6}\dot{\bar{x}}_{5}+e_{7}\bar{x}_{1}+e_{8}\dot{\bar{x}}_{1}-e_{9},\] \[\ddot{\bar{x}}_{1}=q_{1}\bar{x}_{5}+q_{2}\dot{\bar{x}}_{5}-q_{3} \dot{\bar{x}}_{1}-q_{4}\dot{\bar{x}}_{1}+q_{5}, \tag{7}\]  
where  
\[p_{1}=v+uv_{1}^{2},p_{2}=\rho_{2},p_{3}=2\zeta_{4}+2\zeta_{6},p_ {4}=2\zeta_{5}+2\zeta_{7},p_{5}=v_{1}^{2},p_{6}=2\zeta_{3},\] \[e_{1}=\rho_{2},e_{2}=uv_{1}^{2},e_{3}=2\zeta_{5},e_{4}=2\zeta_{4},e_{5}=v_{1}^{2}+\rho_{1}v_{1}^{2},e_{6}=2\zeta_{2}+2\zeta_{3},e_{7}=v_{2}^{2},\] \[e_{8}=2\zeta_{1},e_{9}=v_{2}^{2}w_{1}+2\zeta_{1}w_{2},q_{1}=\rho_ {1}v_{1}^{2},q_{2}=2\zeta_{2},q_{3}=v_{2}^{2},q_{4}=2\zeta_{1},q_{5}=v_{2}^{2} w_{1}+2\zeta_{1}w_{2}.\]  
The following scales transformation is introduced.  
\[p_{k}\to ep_{k},e_{i}\to\varepsilon^{2}e_{i},q_{j}\to \varepsilon^{2}q_{j},e_{6}\to\varepsilon e_{6},e_{8}\to ee_{8},q_{2}\to \varepsilon q_{2},q_{4}\to\varepsilon q_{4},k=1,\cdots,5,\] \[\big{\{}i\in N^{+}\mid i\leq 9,i\
eq 6,8\big{\}},\{j\in N^{+}\mid j \leq 5,j\
eq 2,4\big{\}}.\]  
And let \(\dot{\bar{x}}_{4}=z_{1},\dot{\bar{x}}_{5}=\varepsilon z_{2},\dot{\bar{x}}_{1}= \varepsilon z_{3},y_{1}=\bar{x}_{4},y_{2}=z_{1},y_{3}=\bar{x}_{5},y_{4}=z_{2},y_ {5}=\bar{x}_{1},y_{6}=z_{3}\). System (7) is transformed into  
\[\dot{y}_{1}=y_{2}\] \[\dot{y}_{2}=-\check{g}(y_{1})+\varepsilon\Big{(}-p_{1}y_{1}-p_{2} \check{g}(y_{1})-p_{3}y_{2}-p_{4}\hat{h}(y_{2})+p_{5}y_{3}+p_{6}y_{4}\Big{)}\] \[\dot{y}_{3}=\varepsilon y_{4}\] \[\dot{y}_{4}=\varepsilon(e_{1}\check{g}(y_{1})+e_{2}y_{1}+e_{3} \hat{h}(y_{2})+e_{4}y_{2}-e_{5}y_{3}-e_{6}y_{4}+e_{7}y_where  
\[\hat{g}(y_{1})=\left\{\begin{array}{ll}y_{1}-L,&y_{1}>L\\ 0,&-L\leq y_{1}\leq L\\ y_{1}+L,&y_{1}<-L\end{array}\right.,\hat{h}(y_{2})=\left\{\begin{array}{ll}y_{2},&y_{1}>L\\ 0,&-L\leq y_{1}\leq L.\\ y_{2},&y_{1}<-L\end{array}\right.\]