## 1 Introduction  
Nonlinear engineering problems arise in a wide range of fields, including structural mechanics, fluid dynamics, control systems, energy systems, materials science, and biomedical engineering [1, 2, 3]. These problems are characterized by complex behaviors such as instability, bifurcations, multiple equilibria, and strong sensitivity to initial conditions. Accurately solving nonlinear systems is critical to ensuring the reliability, efficiency, and performance of engineering designs and processes. To address nonlinear equations of the form  
\[\hbar(x)=0, \tag{1}\]  
engineers and applied mathematicians often employ classical iterative methods, such as the Newton-Raphson method [4], fixed-point iterations [5], and their variants (see, e.g., [6, 7, 8]and references therein). These techniques are widely used due to their simplicity and well-established convergence properties under certain regularity assumptions. However, in practical applications, their performance may degrade significantly when the problem exhibits discontinuities, fractional-order dynamics [9], chaotic behavior [10], or memory effects. In such cases, traditional methods may converge slowly, produce inaccurate results, or fail to converge altogether. Chaos theory [11] shows that many engineering systems--especially those governed by nonlinear differential equations--are highly sensitive to initial conditions and can exhibit unpredictable yet deterministic behavior. Examples of chaotic dynamics include fluid turbulence, oscillations in electronic circuits, population dynamics in ecological models, and vibrations in mechanical systems. In such contexts, conventional solution methods based on local linearity or smooth variation may prove inadequate, necessitating the use of more advanced approaches. Moreover, many physical and engineering phenomena are not accurately described within the framework of integer-order models. Instead, they exhibit memory and hereditary properties that are better captured using fractional-order models. Fractional calculus [12] generalizes the concepts of derivatives and integrals to non-integer orders, enabling the modeling of systems whose current state depends not only on the immediate past but on the entire history of the system. This approach has proven effective in representing behaviors such as long-term memory in control systems, heat conduction in heterogeneous media, anomalous diffusion, and the mechanical response of viscoelastic materials. Although fractional-order methods have shown promise in modeling complex dynamics and memory effects, several limitations continue to hinder their broader application to nonlinear problems:  
* Computational Overhead: The nonlocal nature of fractional derivatives introduces considerable computational cost, particularly due to the need to account for historical data. * Parameter Selection Challenges: Determining appropriate parameter values is often difficult, as they are problem-specific and no established guidelines exist for their selection. This uncertainty can reduce the reliability of the resulting estimates. * Stability Issues in Nonlinear Contexts: In highly nonlinear settings, fractional-order schemes may suffer from instability or slow convergence unless properly tuned or combined with adaptive strategies. * Limited Integration with Chaos Theory: While there is growing recognition of chaotic and fractional-order behaviors in natural systems, most iterative solvers still rely on assumptions of smoothness, predictability, and integer-order dynamics. As a result, they may fail to capture essential chaotic phenomena. * Lack of Research on Hybrid Approaches: Beyond the limitations of existing solvers, there is limited research on developing iterative algorithms that jointly exploit fractional-order memory effects and chaos-informed dynamics. Such hybrid approaches could offer greater fidelity in modeling complex nonlinear systems.