## 3 Almost Anti-Periodic Oscillation of the Mechanical System
### Stability of the Almost Anti-Periodic Solutions
Assume that the solution of Equation (10) is stable, by Theorem 7, there exists \(K(s)\) such that  
\[= \bigg{\|}\sum_{j=-\infty}^{n+\tau-2}\bigg{[}\prod_{i=1}^{n+\tau-j- 1}M(n+\tau-i)\bigg{]}c(j)+c(n+\tau-1)\]0  
it is a contradiction with  
\[= \bigg{\|}\sum_{j=-\infty}^{n+\tau-2}\bigg{[}\prod_{i=1}^{n+\tau-j- 1}M(n+\tau-i)\bigg{]}c(j)+c(n+\tau-1)\]1  
for \(\lambda_{3}>1\). Hence, the solution of Equation (10) is unstable. The proof is completed. In what follows, we will provide two examples to support our obtained results of Theorem 8. **Example 1**. : _In Equation (4), let \(s=1\), \(x(s)=[q(1),q(2)]^{T}\)_  
\[B=C= \begin{bmatrix}2&0\\ 0&2\end{bmatrix} ,\;\;S(q(n))=2q(n),\;\;f(n)= \begin{bmatrix}\sin\frac{u\pi}{4} \\ \cos\frac{u\pi}{2}\end{bmatrix} \;\;q(1)= \begin{bmatrix}1\\ \frac{1}{2}\end{bmatrix} \;\;q(2)= \begin{bmatrix}-1\\ 0\end{bmatrix} .\]  
_In fact,_  
\[B^{-1}= \begin{bmatrix}\frac{1}{2}&0\\ 0&\frac{1}{2}\end{bmatrix} ,\;\;B^{-1}f(n)-B^{-1}S(q(n))= \begin{bmatrix}\frac {1}{2}\sin\frac{u\pi}{4}\\ \frac{1}{2}\cos\frac{u\pi}{2}\end{bmatrix} -q(n),\;\;R= \begin{bmatrix}0&I\\ 0&I\end{bmatrix} ,\;\;R^{k}=R.\]  
_Thus,_  
\[x(n)=[q(n),q(n+1)]^{T}=Rx(s)+R\sum\limits_{j=n-1-s}^{0}b(n-1-j),\]  
_and_  
\[q(n)= \begin{bmatrix}q_{1}(n)\\ q_{2}(n)\end{bmatrix} = \begin{bmatrix}-1+\sum\limits_{j=n-1-s}^{0}\frac{1}{2} \sin\frac{(n-1-j)\pi}{4}\\ \sum\limits_{j=n-1-s}^{0}\frac{1}{2}\cos\frac{(n-1-j)\pi}{2}\end{bmatrix} .\]  
**Example 2**. : _In Equation (6), let \(s=1\), \(q(n)=[q_{1}(n),q_{2}(n)]^{T}\), \(q(1)=[1,1]^{T}\), \(q(2)=[5,2]^{T}\), \(\omega=[0,1]^{T}\)_  
\[B(q(n))= C\big{(}q(n),q(n+1)-q(n)\big{)}\] \[= [2q(n),q(n)]+ \begin{bmatrix}1&0\\ 0&0\end{bmatrix} \ G(q(n))= \begin{bmatrix}-q_{2}(n)\cos\frac{n}{4\pi}\\ 1-q_{2}\sin\frac{n}{8\pi}\end{bmatrix} .\]  
_In fact,_  
\[H(n)=H^{k}(n)= \begin{bmatrix}0&I\\ 0&I\end{bmatrix} ,\ \ L\omega= \begin{bmatrix}0\\ \frac{1}{2}\end{bmatrix} ,\]  
\[B^{-1}(q(n))= \begin{bmatrix}1&\frac{-q_{1}(n)}{q_{2}(n)}\\ -\frac{1}{2}&\frac{\sin(q_{1}(n))-q_{1}(n)\sin\frac{n}{8\pi}}{q_{2}(n)\sin \frac{n}{8\pi}}\end{bmatrix} ,\ L\omega-G(q(n))= \begin{bmatrix}q_{2}(n)\cos\frac{n}{4\pi}\\ q_{2}\sin\frac{n}{8\pi}\end{bmatrix} ,\]  
\[B^{-1}(q(n))\big{(}L\omega-G(q(n))\big{)}= \begin{bmatrix}q_{2}(n)\cos\frac{n} {4\pi}-q_{1}(n)\sin\frac{n}{8\pi}\\ -\frac{q_{2}(n)}{2}\cos\frac{n}{4\pi}+\sin(q_{1}(n))-q_{1}(n)\sin\frac{n}{8 \pi}\end{bmatrix} .\]  
_Hence,_  
\[x(n)= \bigg{(}\prod_{j=s}^{n-1}H(j)\bigg{)}x_{0}+\sum_{j=s}^{n-2}\bigg{[} \prod_{l=1}^{n-(j+1)}H(n-i)\bigg{]}a(j)+a(n-1)\] \[= H(n-1)x_{0}+\sum_{j=s}^{n-2}H(n-i)a(j)+a(n-1)\]  
_and_  
\[q(n+1) =q(2)+\sum_{j=1}^{n-2}\Bigg{[} \begin{matrix}q_{2}(j)\cos\frac{j}{ 4\pi}-q_{1}(j)\sin\frac{j}{8\pi}\\ -\frac{q_{2}(j)}{2}\cos\frac{j}{4\pi}+\sin(q_{1}(j))-q_{1}(j)\sin\frac{j}{8 \pi}\end{matrix} \] \[+\Bigg{[} \begin{matrix}q_{2}(n-1)\cos\frac{n-1}{4\pi}-q_{1}(n-1) \sin\frac{n-1}{8\pi}\\ -\frac{q_{2}(n-1)}{2}\cos\frac{n-1}{4\pi}+\sin(q_{1}(n))-q_{1}(n-1)\sin\frac{ n-1}{8\pi}\end{matrix} \] \[=q(n)+\Bigg{[} \begin{matrix}q_{2}(n-1)\cos\frac{n-1}{4\pi}-q_{1}( n-1)\sin\frac{n-1}{8\pi}\\ -\frac{q_{2}(n-1)}{2}\cos\frac{n-1}{4\pi}+\sin(q_{1}(n))-q_{1}(n-1)\sin\frac{n- 1}{8\pi}\end{matrix} \Bigg{]}.\]  
_For the status of \(q_{1}(n)\) and \(q_{2}(n)\), see Figures 3 and 4._Figure 4: The status of the almost anti-periodic solution of the system (6). Figure 3: The status of the almost anti-periodic solution of the system (6).