## 4 Numerical Experiments and Performance Evaluation
Tables 18-19 present the results of cross-validation during training and testing, highlighting the improved stability, consistency, memory efficiency, and accuracy of the proposed method compared with existing approaches. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Metric** & **Iterations** & **C-Time** & **M-Usage** & **P-Con** & **M-Err** \\ \hline EF\({}^{[1]}\) & 18 & 4.932 & 41.657 & 27.029 & \(2.52\times 10^{-19}\) \\ EF\({}^{[2]}\) & 17 & 4.605 & 44.543 & 21.798 & \(2.44\times 10^{-16}\) \\ FS\({}^{[1]}\) & 15 & 4.187 & 35.345 & 37.769 & \(0.03\times 10^{-21}\) \\ FS\({}^{[2]}\) & 13 & 3.653 & 29.124 & 35.115 & \(9.48\times 10^{-31}\) \\ FS\({}^{[3]}\) & 11 & 3.882 & 27.675 & 45.007 & \(5.78\times 10^{-39}\) \\ FS\({}^{[4]}\) & 7 & 3.325 & 35.214 & 48.998 & \(8.87\times 10^{-35}\) \\ FS\({}^{[5]}\) & 7 & 2.001 & 31.764 & 58.768 & \(0.61\times 10^{-49}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 17: Numerical outcomes of fractional schemes without using the chaos-optimized parameter for solving (55). methods for nonlinear models: low convergence rates and difficulties in selecting suitable parameter values. By integrating fractional-order iterative methods with concepts from chaos theory and dynamical systems analysis, we proposed hybrid approaches that enhance local convergence (up to order \(2\gamma+1\)) while reducing residual errors, CPU time, and memory usage. The effectiveness of the proposed methods is demonstrated through extensive numerical experiments, particularly on nonlinear biological systems such as glucose-insulin regulation, tumor-immune interactions, neuronal dynamics, and cardiac models. The dynamical plane analysis (Figures 3 and 4) and bifurcation diagrams (Table 1 and Figure 2) provide useful guidance for identifying parameter regions that promote stability and efficient convergence, even under chaotic conditions, especially for fractional values of \(\gamma\). The developed schemes (FS\({}^{[1]}\)-FS\({}^{[5]}\)) consistently outperform existing fractional-order methods (EF\({}^{[1]}\)-EF\({}^{[2]}\)) in terms of residual error reduction, faster convergence, higher percentage of convergence, lower memory usage, reduced computational time, and improved robustness across a range of fractional parameters (see Tables 3-21 and Figures 6 and 7). The results indicate that as \(\gamma\) approaches 1, convergence improves significantly, and the use of chaotic analysis facilitates optimal parameter tuning. Future work will explore symbolic-numeric integration and adaptive learning strategies for dynamic parameter adjustment. Additionally, improving scalability through GPU acceleration and extending the framework to stochastic fractional models [34] may enable applications in biomedical control, neuroprosthetics, and intelligent diagnostic systems. Conceptualization, M.S. and B.C. ; methodology, M.S. ; software, M.S. ; validation, M.S. ; formal analysis, B.C. ; investigation, M.S. ; resources, B.C. ; writing--original draft preparation, M.S. and B.C. ; writing--review and editing, B.C. ; visualization, M.S. and B.C. ; supervision, B.C. ; project administration, B.C. ; funding acquisition, B.C. All authors have read and agreed to the published version of the manuscript. [PERSON]'s work is supported by the European Regional Development and Cohesion Funds (ERDF) 2021-2027 under Project AI4 AM-EFRE1052. He is a member of _Gruppo Nazionale per il Calcolo Scientific_ (GNCS) of _Istituto Nazionale di Alta Matematica_ (INdAM), and this work was partially supported by INdAM-GNCS under the _Progetti di Ricerca 2024_ program. Data are contained within the article. The authors declare that there are no conflicts of interest regarding the publication of this article. ## References

* [PERSON] and [PERSON] (2013) [PERSON]; [PERSON] (Eds.) _Computational Structural Mechanics & Fluid Dynamics: Advances and Trends_; American Society of Mechanical Engineers (ASME): New York, NY, USA, 2013. * [PERSON] et al.