# Synchronization and Application of a Novel Hyperchaotic System Based on Adaptive Observers
## 2 Materials and Methods
### Dynamic Analysis of the Proposed Hyperchaotic System
\begin{table}
\begin{tabular}{c c c} \hline \hline \(E_{i}\) & **Equilibrium Points** & **Eigenvalues** \\ \hline \(E_{1}\) & \([0,0,0,0]\) & \((-2.0,-1.0,-43.65,\ 30.65)\) \\ \(E_{2}\) & \([5.55,2.22,44.6,-1.11]\) & \((-0.0252\pm j10.98,-2.03,-13.91)\) \\ \(E_{3}\) & \([-20.05,-8.02,44.6,4.01]\) & \((3.68\pm j16.71,-1.96,-21.40)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Eigenvalues of the system and equilibrium points. Figure 4: Eigenvalues (\(\lambda\)) of the hyperchaotic system. and parameter \(a\) was assumed to vary between 14 and 16. The evolution of Lyapunov exponents according to the variation in \(a\) parameter is shown in Figure 5.  
As can be seen from Figure 5, the system has two positive Lyapunov exponent values of \(a\) parameter between 14 and 15.3. Therefore, taking the \(a\) parameter value of the system as 15 and the \(b\) parameter value as 45, the change in the Lyapunov exponents of the system with respect to time is plotted in Figure 6. The initial conditions were set as \(x_{1}(0)=1\), \(x_{2}(0)=1\), \(x_{3}(0)=1\), \(x_{4}(0)=1\). The Lyapunov exponents are determined as \(L_{1}=1.3264\), \(L_{2}=0.0227\), \(L_{3}=-2.8806\) and \(L_{4}=-21.5094\). Two positive Lyapunov exponents are found when the calculated Lyapunov exponents of the recently suggested hyperchaotic system are examined. As a result, the behavior of the system is hyperchaotic. Hyperchaotic systems are inevitably drawn to fractals. The degree of system predictability can be inferred from the fractal dimension of the attractor. The data would have been generated by a deterministic system if the fractal dimension is minimal. The  
Figure 5: Lyapunov exponents according to the variation in \(a\) parameter. Figure 6: Diagram of Lyapunov exponents of the system. data would have been produced by a random system if the fractal dimension is large. Because this structure is not like other geometrical formations, the researchers use fractal dimensions to explain it [32]. The proposed four-dimensional system exhibits a Lyapunov fractal dimension [33], which is computed using (8):  
\[D_{KY}=j+\frac{\sum_{i=1}^{l}L_{i}}{\left|L_{j+1}\right|}=3+\frac{1.3264+0.0227-2.8806}{\left|-21.5094\right|}=2.92 \tag{8}\]  
This shows that the proposed system's Lyapunov dimension is fractional. The new system contains non-periodic orbits and close paths that diverge due to its fractal character. As a result, this nonlinear system is truly chaotic. In engineering applications, chaotic phase behavior can pose challenges in terms of control and predictability. Systems exhibiting chaotic dynamics may be more difficult to control, as small changes in input or conditions can lead to unpredictable outcomes. Engineers need to account for and understand chaotic behavior in order to design robust and reliable systems. The phase space diagrams of the proposed hyperchaotic system in various dimensions are shown in Figure 7. For the phase diagram analysis, \(a=15\), \(b=45\) and the initial values of the system are taken as \(x_{1}(0)=1,\ x_{2}(0)=1,\ x_{3}(0)=1,\ x_{4}(0)=1\). The system's phase diagrams reveal that it is a hyperchaotic system with dynamic oscillations and a variety of strange attractors. The phase diagrams for different \(a\) and \(b\) parameter values are shown in Figure 8. For the phase diagram analysis, \(a=5\) and \(b=6\) values were taken and initial values taken as \(x_{1}(0)=1,\ x_{2}(0)=1,\ x_{3}(0)=1,\ x_{4}(0)=1\). As can be seen from Figure 8, the chaotic properties of the system change for different parameter values.