## 3 Chaos-Based Parameter Estimation and Stability Analysis
### Chaos-Based Parameter Optimization and Bifurcation Diagrams
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Figure** & **Scheme** & \(\mathbf{\sigma},\mathbf{\alpha}\) & \(\mathbf{S_{1}^{[*]}}\) & \(\mathbf{Z_{1}^{[*]}}\) & \(\mathbf{Z_{2}^{[*]}}\) & \(\mathbf{Z_{3-4,8}^{[*]}}\) & \(\mathbf{Z_{C}^{[*]}}\) \\ \hline Figure 2a & FS\({}^{[1]}\) & \(0.3,0.5\) & \([1,1.6]\) & \([1.6,2.5]\) & \([2.5,3.5]\) & \([3.5,4]\) & \(Nil\) \\ Figure 2b & FS\({}^{[2]}\) & \(0.3,1.0\) & \([1,2.5]\) & \([2.5,3.5]\) & \([3.5,3.6]\) & \(Nil\) & \(Nil\) \\ Figure 2c & FS\({}^{[3]}\) & \(0.3,0.3\) & \([1,2.4]\) & \([2.5,3.5]\) & \([3.5,3.6]\) & \([3.6,4.5]\) & \(Nil\) \\ Figure 2d & FS\({}^{[4]}\) & \(0.3,0.4\) & \([1,2.4]\) & \([2.6,5]\) & \(Nil\) & \(Nil\) & \(Nil\) \\ Figure 2e & FS\({}^{[5]}\) & \(0.3,0.5\) & \([1,2.5]\) & \([2.5,5]\) & \(Nil\) & \(Nil\) & \(Nil\) \\ Figure 2f & EF\({}^{[1]}\) & \(0.3,0.6\) & \([1,1.5]\) & \([2.5,3]\) & \(Nil\) & \(Nil\) & \(\left\{\begin{array}{l}[2,2.5]\\ [4,4.5]\end{array}\right.\) \\ Figure 2g & EF\({}^{[2]}\) & \(0.3,0.7\) & \([1,1.5]\) & \([1.5,2.7]\) & \(Nil\) & \(Nil\) & \([2.7,3.5]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The bifurcation of the iterative scheme containing the stable and unstable regions is analyzed. Figure 1: Flowchart of the fractional-order scheme for solving nonlinear equations, integrated with chaos and bifurcation analysis. The iterative sequence remains stable for small to moderate values of \(\alpha\), leading to smooth and predictable convergence toward fixed points or periodic orbits (see Figure 2a-e). As \(\alpha\) increases, the system experiences period-doubling bifurcations (Figure 2c-f), transitioning from simple periodic behavior (e.g., one-period cycles) to increasingly complex periodic patterns (e.g., two-period and four-period orbits). Further increases in \(\alpha\) lead to chaotic behavior in the iterative scheme (Figure 2f,g), characterized by a loss of periodicity and the emergence of irregular, scattered trajectories that exhibit high sensitivity to initial conditions. This chaotic regime reflects the system's strong nonlinearity and its tendency to diverge in certain parameter ranges. Deterministic chaos implies that small variations in \(\alpha\) or initial values can produce significantly different outcomes. However, the system also exhibits transitional zones where, through suitable adjustments of the parameters, the dynamics can return to stability. Within these zones, the system may resume convergence to stable fixed points or periodic orbits, indicating that chaotic behavior is both parameter-dependent and potentially reversible. Table 1 presents an analysis of the bifurcation behavior of the iterative scheme, highlighting both stable and unstable dynamical regions. The notation is as follows: \(\mathbb{S}_{1}^{[\alpha]}\)  
denotes stability; \(\mathrm{Z}_{1}^{[*]}\) marks the onset of bifurcation; \(\mathrm{Z}_{2}^{[*]}\) indicates period doubling; \(\mathrm{Z}_{3-4.8}^{[*]}\) corresponds to higher-order periodic transitions (e.g., four-period and eight-period cycles); and \(\mathrm{Z}_{\mathrm{C}}^{[*]}\) represents the chaotic regime, where the scheme becomes unstable for all values of \(\alpha\) and \(\sigma\). All bifurcation diagrams are generated for \(\gamma\approx 1\). These observations underscore the importance of carefully selecting and tuning damping parameters in iterative parallel methods, particularly in engineering applications where robustness and convergence stability are critical. The bifurcation analysis and visualizations presented in this study clearly illustrate how parameter variations influence the system's dynamical structure and convergence behavior. In practical applications, this approach enhances convergence stability by identifying parameter ranges that lead to chaotic dynamics and providing guidance for their adjustment.