# Synchronization and Application of a Novel Hyperchaotic System Based on Adaptive Observers
## 4 Adaptive Observer Based Synchronization of the Proposed Hyperchaotic System  
Hyperchaotic systems, which form a separate category within nonlinear systems, are typically characterized by a set of nonlinear differential equations. It is a prevalent practice to decompose a hyperchaotic dynamical system into two components: the linear component and the nonlinear component. In instances where the hyperchaotic dynamical system involves unknown parameters and is influenced by unidentified disturbances, it is often expressed in the form of (10) [36]:  
\[\dot{x}=Ax+Bf(x)+Bg(x)\theta+B\eta(x,t) \tag{10}\]  
Here, \(x\ \in\ R^{n}\) represents the state vector; \(\theta\ \in\ R^{p}\) is the vector of unknown parameters; \(\eta\ \in\ R^{r}\) is the external disturbance vector; and \(A\ \in\ R^{nxn}\) and \(B\ \in\ R^{nxr}\) are known matrices. Additionally, \(f(\cdot)\ \in\ R^{r}\) is a nonlinear function vector, and \(g(\cdot)\in\ R^{rxp}\) is a matrix of functions. The matching condition is satisfied by the external disturbance vector, the parameter uncertainty vector and the nonlinear function vector. It is noteworthy that many hyperchaotic systems can be reformulated into the same structure as (10). For instance, the Lorenz hyperchaotic system can be expressed in the form of (11):  
\[ \begin{bmatrix}\dot{x}_{1}\\ \dot{x}_{2}\\ \dot{x}_{3}\\ \dot{x}_{4}\end{bmatrix} = \begin{bmatrix}-a&a&0&1\\ 0&-1&0&0\\ 0&0&0&0\\ 0&0&0&r\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\\ 0&-1&0\end{bmatrix} + \begin{bmatrix}0&0&0\\ -1&0&0\\ 0&1&0\\ 0&-1&0\end{bmatrix}\begin{bmatrix}x_{1}x_{3}\\ x_{1}x_{2}\\ 0&1&0\\ 0&-1&0\end{bmatrix}\begin{bmatrix}-x_{1}&0\\ 0&-x_{3}\\ 0&0\end{bmatrix}\begin{bmatrix}c\\ b\end{bmatrix}  \tag{11}\]  
Here, \(a\) is known but \(c\) and \(b\) are unknown. Figure 10: (**a**) Multisim \(x_{1}\ x_{2}\) phase diagram of the hyperchaotic system; (**b**) Multisim \(x_{1}\ x_{4}\) phase diagram of the hyperchaotic system. We tackle the synchronization challenge of (10) employing the driver-response configuration. In this setup, if the system represented by (10) is designated as the drive system, a corresponding response system needs to be formulated to synchronize with the drive system using a designated driving signal. To achieve this, certain essential assumptions must be fulfilled. Firstly, the Lipschitz condition is articulated. It is required that the nonlinear function vector \(f(x)\) and the matrix \(g(x)\) adhere to the Lipschitz condition:  
\[ \begin{split}||f(x)\ -f(\hat{x})||&\leq k_{f}||x-\hat{x}||\\ ||g(x)\ -g(\hat{x})||&\leq k_{g}||x-\hat{x}||\end{split}  \tag{12}\]  
Here, the Lipschitz constants \(k_{f}\) and \(k_{g}\) are considered unknown. Secondly, the uncertain parameter \(\theta\) and the external disturbance \(\eta\) are subject to norm bounds defined by two unknown positive constants, denoted as \(\delta_{\theta}\) and \(\delta_{\eta}\), respectively. Thirdly, in pursuit of the objective of hyperchaotic synchronization, it is typically necessary to design artificial output information. The initial step involves determining a matrix \(C_{1}\in\mathfrak{R}^{1,m}\) such that the matrix \((C_{1},A)\) is observable, and there exists a matrix \(L\) such that the Lyapunov equation \(\left(A-LC_{1}\right)^{T}P+P(A-LC_{1})=-Q\) has a positive definite matrix solution \(P\) for a given positive definite matrix \(Q\). The second step is to specify \(B^{T}P=C_{2}\). If there exists a matrix \(H\in\mathfrak{R}^{rxp}\) such that \(C_{2}=HC_{1}\), then the output matrix of the drive system must be \(C=C_{1}\); otherwise, it will be \(C=\left(C_{1}^{T}\ C_{2}^{T}\right)^{T}\). In our study, we assume the absence of such an \(H\) matrix. Therefore, we set our system output as \(y=\left(y_{1}^{T}\ y_{2}^{T}\right)^{T}=\left(C_{1}^{T}\ C_{2}^{T}\right)^{T}x\). To synchronize the system presented in (8), an adaptive observer-based response system is designed.