## 4 Numerical Experiments and Performance Evaluation
The model is typically expressed through nonlinear equations involving polynomials and logarithmic terms, where the glucose concentration is modeled as a function of insulin levels and time. These nonlinearities account for physiological effects such as saturation and thresholds in the feedback system:

\[\hbar(x)=(x-G_{0})^{3}-a\log(x+1), \tag{47}\]

where \(x\) represents the glucose concentration and \(G_{0}\) is the basal glucose level. This formulation plays a critical role in simulating and controlling glucose dynamics in diabetic patients and supports the development of therapeutic strategies. By fixing \(G_{0}=0.5\) and \(a=0.3\), the model becomes

\[\hbar(x)=(x-0.5)^{3}-0.3\log(x+1). \tag{48}\]

The Caputo-order fractional derivative of this function is given by

\[\big{[}D_{\gamma_{1}}^{\gamma}\big{]}\hbar(x)=\big{[}D_{\gamma_{1}}^{\gamma} \big{]}\hbar(x)=\frac{1}{\Gamma(\gamma+1)}\int\limits_{0}^{x}\bigg{[}\frac{3(t- 0.5)-\frac{0.3}{t+1}}{(x-t)^{\gamma}}\bigg{]}dt. \tag{49}\]

The exact root of the nonlinear model is \(-0.9999869889\), and the corresponding initial guess is \(x^{[0]}=-1.7i\). Tables 7-9 present the numerical results obtained using fractional schemes, both with and without chaos-optimized parameters, for solving Equation (48). Tables 9 and 10 present cross-validation results from both training and testing phases, illustrating the improved stability, consistency, memory efficiency, and accuracy of the proposed method. The dynamical behavior of the proposed scheme, in comparison with existing approaches, is presented in Table 8 and Figure 8. As shown in Table 8, the proposed methods exhibit improved performance in terms of reduced generation time (E-time) for the basin of attraction and higher percentage convergence. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Metric** & **Iterations** & **C-Time** & **M-Usage** & **P-Con** & **M-Err** \\ \hline EF\({}^{[1]}\) & 19 & 6.456 & 71.434 & 18.332 & \(1.13\times 10^{-7}\) \\ EF\({}^{[2]}\) & 20 & 6.113 & 84.332 & 15.545 & \(5.03\times 10^{-5}\) \\ FS\({}^{[1]}\) & 25 & 5.354 & 71.002 & 16.764 & \(3.54\times 10^{-9}\) \\ FS\({}^{[2]}\) & 18 & 4.355 & 65.332 & 21.655 & \(9.99\times 10^{-23}\) \\ FS\({}^{[3]}\) & 20 & 5.346 & 51.883 & 32.117 & \(7.06\times 10^{-27}\) \\ FS\({}^{[4]}\) & 19 & 4.887 & 59.764 & 25.002 & \(6.08\times 10^{-20}\) \\ FS\({}^{[5]}\) & 17 & 4.654 & 51.653 & 21.653 & \(0.76\times 10^{-23}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Numerical outcomes of fractional schemes without chaos-optimized parameter values for solving Equation (48). \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Metric** & **EF\({}^{[1]}\)** & **EF\({}^{[2]}\)** & **FS\({}^{[1]}\)** & **FS\({}^{[2]}\)** & **FS\({}^{[3]}\)** & **FS\({}^{[4]}\)** & **FS\({}^{[5]}\)** \\ \hline E-time & 6.5404 & 5.5465 & 2.4375 & 2.0055 & 1.9874 & 1.7655 & 0.8874 \\ M-usage & 86.432 & 76.435 & 55.353 & 43.654 & 33.546 & 32.009 & 24.887 \\ P-convergence & 75.645 & 57.2873 & 86.867 & 87.0769 & 84.055 & 75.069 & 86.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Dynamical performance analysis of fractional schemes FS\({}^{[1]}\)–FS\({}^{[5]}\) and EF\({}^{[1]}\)–EF\({}^{[2]}\) for solving Equation (48). Figure 8: Basins of attraction for fractional-order schemes FS\({}^{[1]}\)–FS\({}^{[5]}\) and EF\({}^{[1]}\)–EF\({}^{[2]}\) for solving Equation (48). Table 9 presents the error and consistency analysis, while Table 10 summarizes memory utilization, percentage convergence, and stability performance for the fractional schemes applied to the biomedical engineering model described in Section 4.2. The numerical results indicate that the proposed methods achieve lower error, improved memory efficiency, and higher convergence percentages and require fewer iterations to reach the desired accuracy, particularly when using chaos-optimized and fractional parameter values.