# Synchronization and Application of a Novel Hyperchaotic System Based on Adaptive Observers
## 4 Adaptive Observer Based Synchronization of the Proposed Hyperchaotic System
#### Numerical Simulation  
In this section of the paper, the effectiveness of adaptive observer-based synchronization for the proposed hyperchaotic system will be demonstrated. The dynamics of the hyperchaotic system are depicted in (27):  
\[ \begin{array}{l}\dot{x}_{1}=-12x_{1}+30x_{2}\\ \dot{x}_{2}=bx_{1}-x_{2}-x_{1}x_{3}\\ \dot{x}_{3}=ax_{2}+x_{1}x_{2}-x_{3}+x_{4}\\ \dot{x}_{4}=-x_{1}b+x_{1}x_{3}-2x_{4}\end{array}  \tag{27}\]  
It is established that the system exhibits hyperchaotic behavior for \(a=15\) and \(b=45\). Let us consider that the parameter \(b\) is unknown, and \(a\) has a perturbation of \(\Delta_{a}=0.03\sin(2t)\). The relevant matrices \(A\), \(B\) and \(C\) for the hyperchaotic system presented in system (27) are provided in (28)-(30):  
\[A= \begin{bmatrix}-12&30&0&0\\ 0&-1&0&0\\ 0&a&-1&1\\ 0&0&0&-2\end{bmatrix}  \tag{28}\]  
\[B= \begin{bmatrix}0&0\\ -1&0\\ 0&1\\ 1&0\end{bmatrix}  \tag{29}\]  
\[C_{1}= \begin{bmatrix}1&1&1&0\end{bmatrix}  \tag{30}\]  
Furthermore, we define \(f(x)= \begin{bmatrix}x_{1}x_{3}\\ x_{1}x_{2}\end{bmatrix} \), \(g(x)= \begin{bmatrix}-x_{1}\\ 0\end{bmatrix} \) and \(\eta(x,t)= \begin{bmatrix}0\\ \Delta_{a}x_{2}\end{bmatrix} \). It is evident that the pair \((C,A)\) is observable. Therefore, the gain matrix \(L\) is selected as \(L= \begin{bmatrix}-3.5107&0.1818&1.328&-0.8\end{bmatrix} ^{T}\) such that the eigenvalues of the matrix \(A-LC_{1}\) are \(-6\), \(-4\), \(-2+i\) and \(-2-i\). Moreover, the symmetric positive definite matrices provided in (31) and (32) satisfy (19):\[P= \begin{bmatrix}0.3475&-0.3392&0.3903&0.0379\\ -0.3392&120.7581&0.2553&-3.0015\\ 0.3903&0.2553&2.2728&0.2408\\ 0.0379&-3.0015&0.2408&4.6182\\ \end{bmatrix}  \tag{31}\]  
\[Q= \begin{bmatrix}6.7527&4.4022&5.7821&-3.9132\\ 4.4022&305.9790&-17.9037&-18.0623\\ 5.7821&-17.9037&7.5493&-5.6039\\ 0.0379&-18.0623&-5.6039&17.9912\\ \end{bmatrix}  \tag{32}\]  
As mentioned before, the adaptive observer-based response system is designed as (33):  
\[ \begin{bmatrix}\dot{\hat{x}}_{1}\\ \dot{\hat{x}}_{2}\\ \dot{\hat{x}}_{3}\\ \dot{\hat{x}}_{4}\end{bmatrix} = \begin{bmatrix}-12&30&0&0\\ 0&-1&0&0\\ 0&a&-1&1\\ 0&0&0&-2\\ \end{bmatrix}\begin{bmatrix}\hat{x}_{1}\\ \hat{x}_{2}\\ \hat{x}_{3}\\ \hat{x}_{4}\end{bmatrix} + \begin{bmatrix}0&0\\ -1&0\\ 0&1\\ 1&0\end{bmatrix}\begin{bmatrix}\hat{x}_{1}\hat{x}_{3}\\ \hat{x}_{1}\hat{x}_{2}\end{bmatrix} + \begin{bmatrix}0&0\\ -1&0\\ 0&1\\ 1&0\end{bmatrix}\begin{bmatrix}-\hat{x}_{1}\\ 0\\ 1&0\end{bmatrix}\begin{bmatrix}\hat{x}\\ 0\end{bmatrix}\begin{bmatrix}\hat{x}\\ 0\end{bmatrix}\begin{bmatrix}\hat{x}\\ 0\end{bmatrix} \\ +\frac{1}{2}\hat{k}\left[ \begin{array}{cc}0&0\\ -1&0\\ 0&1\\ 1&0\end{array} \right](y_{2}-C_{2}\hat{x})+L(y_{1}-C_{1}\hat{x})+r(t)B(y_{2}-C_ {2}\hat{x})\\ \\ \end{bmatrix} \tag{33}\]  
The values of \(r(t)\), \(\varepsilon_{y}=1\) and \(\varepsilon=0.01\) used here are related to the inequalities given in (26) and the laws of adaptation on \(\hat{b}\) and \(\overline{k}\) are (34) and (35):  
\[\dot{\hat{b}}=l_{b} \begin{bmatrix}-\hat{x}_{1}\\ 0\end{bmatrix} ^{T}(y_{2}-C_{2}\hat{x}) \tag{34}\]  
\[\dot{\overline{k}}=l_{k}\|y_{2}-C_{2}\hat{x}\|^{2} \tag{35}\]  
Here, \(l_{b}=1\) and \(l_{k}=1\). The values for the free parameters \(\varepsilon_{f}\) and \(\varepsilon_{g}\), as given in (25), are set to 1. As mentioned earlier, \(C_{2}= \begin{bmatrix}0.3771&-123.7596&-0.0145&7.6197\\ 0.3903&0.2553&2.2728&0.2408\end{bmatrix} \) based on the equation \(B^{T}P=C_{2}\). The initial conditions for the simulation are \(x_{1}(0)=1,x_{2}(0)=1,x_{3}(0)=1,x_{4}(0)=1\) we \(\hat{x}_{1}(0)=2,\hat{x}_{2}(0)=5,\hat{x}_{3}(0)=4,\hat{x}_{4}(0)=2\) and \(\hat{b}(0)=10\), \(\overline{k}(0)=10\). The synchronization errors are depicted in Figure 11.  
As can be seen in Figure 8, the synchronization error quickly goes to 0. Therefore, it can be said that the synchronization is successful. When other studies in the literature are examined, the synchronization speed and robustness have been very successful.