## 5 The Design of Simple Parameter Update Laws
\begin{split} u_{1}&=(2\cos(x_{1}+1)x_{1}+2\sin(x_{ 1}+1)+1)\vartheta(x_{2}-x_{1})\\ &\quad+k(y_{1}-(2\sin(x_{1}+1)+1)x_{1})+\hat{a}(y_{1}-y_{2}),\\ u_{2}&=(-6\cos(2x_{2}-3)x_{2}-3\sin(2x_{2}-3)+1)( \hat{r}x_{1}-x_{2}-x_{1}x_{3})\\ &\quad+k(y_{2}-(3\sin(-2x_{2}+3)+1)x_{2})-(\hat{c}-\hat{a})y_{1} +y_{1}y_{3}-\hat{c}y_{2},\\ u_{3}&=(-3\cos(3x_{3}-1)x_{3}-\sin(3x_{3}-1)+1)(x_{ 1}x_{2}-\hat{a}x_{3})\\ &\quad+k(y_{3}-(-\sin(3x_{3}-1)+1)x_{3})-y_{1}y_{2}+\hat{b}y_{3}, \end{split}  \tag{40}\]  
\[ \begin{split}\dot{\phi}&=\gamma_{11}(\vartheta- \sigma)+\gamma_{12}(1-cos(y_{1}-(2 sin(x_{1}+1)+1)x_{1})),\\ \dot{\hat{r}}&=\gamma_{11}(\hat{r}-r)+\gamma_{12}(1- cos(y_{2}-(3 sin(-2x_{2}+3)+1)x_{2})),\\ \dot{\hat{\beta}}&=\gamma_{11}(\hat{\beta}-\beta)+ \gamma_{12}(1-cos(y_{3}-(-sin(3x_{3}-1)+1)x_{3})),\\ \dot{\hat{a}}&=\gamma_{21}(\hat{a}-a)+\gamma_{22}(1- cos(y_{1}-(2 sin(x_{1}+1)+1)x_{1})),\\ \dot{\hat{b}}&=\gamma_{21}(\hat{b}-b)+\gamma_{22}(1- cos(y_{2}-(3 sin(-2x_{2}+3)+1)x_{2})),\\ \dot{\hat{c}}&=\gamma_{21}(\hat{c}-c)+\gamma_{22}(1- cos(y_{3}-(-sin(3x_{3}-1)+1)x_{3})),\end{split}  \tag{41}\]  
where \(\vartheta,\hat{r},\hat{\beta},\hat{a},\hat{b}\) and \(\hat{c}\) are estimated values of the uncertain parameters in systems (38) and (39). The true values of the uncertain parameters of systems (38) and (39) are chosen as \(\sigma=10\), \(r=28\), \(\beta=8/3\) and \(a=35\), \(b=3\), \(c=28\). According to the previous analysis results in the section, if \(k,\gamma_{11},\gamma_{21},\gamma_{12},\gamma_{22}\) are chosen as \(k=\gamma_{11}=\gamma_{21}=-1\) and \(\gamma_{12}=\gamma_{22}=0.01\), GFPS between uncertain [PERSON] and Chen chaotic systems will be achieved under the control of the controller (40), and the uncertain parameters will be estimated using the parameter update laws (41) and (42). The initial conditions are given by \(x_{1}(0)=0.1\), \(x_{2}(0)=0.2\), \(x_{3}(0)=0.3\), \(y_{1}(0)=0.4\), \(y_{2}(0)=0.5\), \(y_{3}(0)=0.6\), \(\vartheta=9\), \(\hat{r}=25\), \(\hat{\beta}=3\), \(\hat{a}=33\), \(\hat{b}=2\) and \(\hat{c}=29\) to perform the numerical simulations. As shown in Figures 5 and 6, GFPS between systems (38) and (39) is achieved, and the uncertain parameters are also identified. If the scaling function matrix is taken as \(h(x)=diag\{0.02x_{1}^{2}+0.2x_{1}+1,-0.01x_{2}^{2}+0.1x_{2}-1,0.03x_{3}^{2}-0.1x _{3}+0.5\}\) and all other parameter conditions in systems (38) and (39) remain the same, GFPS between systems (38) and (39) still can be achieved and the uncertain parameters are also identified under the controller and parameter update laws given by Equations (3) and (36), respectively. The simulation results are shown in Figures 7 and 8, which illustrate the validity of the parameter update laws (36) proposed in the section. It is shown that the accuracy of the proposed scheme is better than the accuracy of the proposed scheme. Figure 5: GFPS between systems (38) and (39) is achieved with the scaling function matrix \(h(x)=diag\{2 sin(x_{1}+1)+1,3 sin(-2x_{2}+3)+1,-sin(3x_{3}-1)+1\}\). (**a**) (\(2 sin(x_{1}+1)+1\))x_1 vs. \(y_{1}\) (**b**) (\(3 sin(-2x_{2}+3)+1\))x_2 vs. \(y_{2}\) (**c**) (\(-sin(3x_{3}-1)+1\))x_3 vs. \(y_{3}\). Figure 6: The estimates of the uncertain parameters of systems (38) and (39) for GFPS with the scaling function matrix \(h(x)=diag\{2 sin(x_{1}+1)+1,3 sin(-2x_{2}+3)+1,-sin(3x_{3}-1)+1\}\). (**a**) uncertain parameters in system (38) (**b**) uncertain parameters in system (39). worth pointing out that the parameter update laws (36) are still valid even if the scaling function factors are other complex functions.