## 1 Introduction
To address the limitations outlined above, this study proposes a new iterative framework that integrates concepts from chaos theory and fractional calculus to solve practical nonlinear engineering problems. The approach combines chaotic perturbations with fractional-order memory to improve the exploration of complex solution spaces and ensure stable and reliable convergence. On the one hand, chaotic maps--such as the logistic map--are used to dynamically adjust step sizes or perturb search directions, helping prevent early convergence to suboptimal solutions or stagnation in local minima [13; 14]. On the other hand, the fractional calculus component introduces a memory effect that contributes to smoother and more reliable convergence, particularly when many iterations are required. Unlike traditional iterative strategies that rely on fixed step sizes and predominantly local behavior, the proposed chaos-fractional hybrid approach adapts its search dynamics based on the complexity of the problem. This integration enhances not only the robustness and accuracy of the solution but also the method's ability to handle highly nonlinear or chaotic systems. The main contributions of this work are summarized below:  
* A new fractional-order iterative technique based on the Caputo derivative is developed for solving nonlinear equations. A local convergence analysis shows that the method achieves an improved convergence order of \(2\gamma+1\), with \(\gamma\in(0,1]\). * The proposed schemes combine chaos theory and dynamical analysis to improve convergence stability and resilience and overcome the local sensitivity and high memory usage of traditional solvers. * The method incorporates efficient implementation strategies that reduce memory usage, making it suitable for large-scale or memory-constrained problems. * By integrating fractional memory and chaos-based perturbations, the methods achieve higher accuracy and faster error decay than conventional iterative techniques. * Extensive comparisons with established iterative techniques confirm the advantages of the proposed scheme in terms of memory efficiency, CPU time, convergence rate, and accuracy. * Benchmark problems from biomedical engineering are used to evaluate the method's performance and stability, with a focus on memory effects, error control, and the ability to capture long-term dynamics. The proposed chaos-enhanced fractional iterative method offers a promising approach to the numerical treatment of nonlinear engineering problems. It enables researchers and engineers to address more complex models with increased confidence in both convergence and solution accuracy, even when the underlying systems exhibit chaotic behavior or memory-dependent dynamics. Moreover, the framework broadens the applicability of iterative techniques to a wider class of problems, including nonlinear control system design, mechanical vibrations, structural stability analysis, and the modeling of viscoelastic materials. The rest of the paper is structured as follows:  
* Section 2 presents the mathematical formulation and computational procedure used to construct the proposed chaos-informed fractional iterative model. * Section 3 examines the role of chaotic dynamics in fractional settings and investigates the interaction between chaos and memory effects during iteration. It also includes a theoretical stability analysis that establishes convergence under specified conditions. * Section 4 evaluates the performance of the proposed method by comparing it with existing techniques in the context of selected biomedical engineering models. * Section 5 concludes the study and outlines directions for future research.