## 4 Numerical Experiments and Performance Evaluation
Step 4: Visualize Result ;  Generate plot using: pcolor(x, y, angle(A)) ;
```  
**Algorithm 2**Two-step fractional-order scheme for solving nonlinear problems  
```
Input: Initial guess \(x^{[0]}\), tolerance \(\epsilon\), maximum iterations \(N_{\max}\), fractional order \(\gamma\) Output: Approximate solution \(x^{[j]}\) Set \(j\gets 0\), \(x^{[0]}\) as the initial guess, error \(\leftarrow\infty\) ; whileerror \(>\epsilon\) and \(j<N_{\max}\)do Step 1: Compute the Fractional Derivative ;  Evaluate the fractional derivative \(D^{[\gamma]}\) at \(x^{[j]}\), where \(\gamma\) is the fractional order ;  Compute the function value \(\hbar(x^{[j]})\) at the current point ; Step 2: Apply the Two-Step Fractional Iterative Method ;  Apply the iteration formulas: \[y^{[j]} =x^{[j]}-\left(\Gamma(\gamma+1)\frac{\hbar(x^{[j]})}{[D_{\tau n}^{ \gamma}]\,\hbar(x^{[j]})}\right)^{1/\gamma},\] \[x^{[j+1]} =y^{[j]}-\left(\Gamma(\gamma+1)\frac{\phi(\zeta)}{1-2\frac{\hbar( y^{[j]})}{\hbar(x^{[j]})}}\right)^{1/\gamma},\] where \(\alpha\in\mathbb{R}\) ; Step 3: Calculate the Error ;  Compute error as error \(=|x^{[j+1]}-x^{[j]}|\) ; Step 4: Check for Convergence ; iferror \(<\epsilon\)then  Convergence reached; break the loop ;  Increment iteration counter: \(j\gets j+1\) ; Return:\(x^{[j]}=x^{[j+1]}\) as approximate solution ;
```  
**Algorithm 3**Fractal generation via fractional scheme  
``` Input: Initial guess \(x^{[0]}\), tolerance \(\epsilon\), maximum iterations \(N_{\max}\), fractional order \(\gamma\) Output: Approximate solution \(x^{[j]}\) Set \(j\gets 0\), \(x^{[0]}\) as the initial guess, error \(\leftarrow\infty\) ; whileerror \(>\epsilon\) and \(j<N_{\max}\)do Step 1: Compute the Fractional Derivative ;  Evaluate the fractional derivative \(D^{[\gamma]}\) at \(x^{[j]}\), where \(\gamma\) is the fractional order ;  Compute the function value \(\hbar(x^{[j]})\) at the current point ; Step 2: Apply the Two-Step Fractional Iterative Method ;  Apply the iteration formulas: \[y^{[j]} =x^{[j]}-\left(\Gamma(\gamma+1)\frac{\hbar(x^{[j]})}{[D_{\tau n}^{ \gamma}]\,\hbar(x^{[j]})}\right)^{1/\gamma},\] \[\text{Percentage Error}=\frac{\left|x^{[j+1]}-x^{[j]}\right|}{\left|x^{[j]} \right|}\times 100.\]0

where \(x\) denotes the substrate concentration, and \(k_{1}\), \(k_{2}\), and \(k_{3}\) are rate constants characterizing the reaction. The logarithmic term accounts for more complex regulatory effects, such as enzyme cooperativity or allosteric modulation. This mathematical formulation plays a central role in modeling processes like drug metabolism, metabolic control, and signal transduction, where inhibitors and activators are critically involved. The nonlinear nature of the system underscores the importance of employing advanced computational methods for the accurate prediction and analysis of biochemical dynamics. For steady-state analysis, the system can be approximated by the following nonlinear function using the fixed parameter values \(k_{1}=0.5\), \(k_{2}=0.9\), \(k_{3}=0.3\), and \(K_{m}=1.7\):

\[\text{Percentage Error}=\frac{\left|x^{[j+1]}-x^{[j]}\right|}{\left|x^{[j]} \right|}\times 100.\]1

The corresponding Caputo fractional derivative of order \(\gamma\) is given by

\[\text{Percentage Error}=\frac{\left|x^{[j+1]}-x^{[j]}\right|}{\left|x^{[j]} \right|}\times 100.\]2

The exact root of this nonlinear model is 0.0, and the initial guess used for the iterative scheme is \(x^{[0]}=0.3\). The numerical results for the fractional-order schemes, both with and without chaos-based parameter tuning, are presented in Tables 2-4 for solving Equation (45). Tables 4 and 5 illustrate the cross-validation results obtained during training and testing, highlighting the proposed method's stability, consistency, memory efficiency, and accuracy.