## 3 Chaos-Based Parameter Estimation and Stability Analysis
### Chaos-Based Parameter Optimization and Bifurcation Diagrams  
Stability and bifurcation analysis in iterative schemes has emerged as a significant area of research in the development of numerical methods for solving nonlinear engineering models. These phenomena are often investigated through bifurcation diagrams, Lyapunovexponent plots [22], and detailed visualizations of basins of attraction, which illustrate how small variations in initial guesses or iteration parameters can substantially influence convergence outcomes. Fractal basin boundaries and the high sensitivity to initial conditions show how chaotic dynamics in iterative methods can be highly complex and difficult to predict. When these concepts are applied to fractional-order iterative methods for solving nonlinear equations, the presence of memory effects associated with fractional derivatives introduces additional challenges. In such schemes, each iteration depends not only on the current state but also on the full history of the process. This memory dependence alters the stability characteristics and the conditions under which chaotic behavior may emerge. Key factors influencing the dynamics of fractional iterative schemes include the fractional order parameter \(\alpha\), the choice of initial values, the memory kernel, the structure of the nonlinearity, and the specific definition of the fractional derivative employed (e.g., Caputo and Riemann-Liouville [23]). Compared with classical integer-order methods, fractional-order schemes solve the same underlying problem by using derivatives that incorporate memory effects. These additional features can significantly influence the system's behavior: they may shift the point at which bifurcations occur, delay or hasten the onset of chaotic dynamics, and lead to more complex and detailed patterns in the system's evolution (see Table 1). To investigate the emergence of chaos in fractional-order iterative models, we adopt a general computational framework (see the flowchart in Figure 1). This approach involves selecting a base iterative method, introducing fractional-order modifications, systematically varying key parameters, computing the resulting Lyapunov spectra and bifurcation diagrams, and analyzing the associated stability regimes. To study bifurcation and chaos in the context of fractional-order schemes, we consider the nonlinear function  
\[\hat{h}(x)=x^{3}+\sigma x+1. \tag{33}\]  
This function serves as a representative case for exploring the dynamical behavior of iterative processes, particularly under the influence of the damping parameter \(\alpha\), as discussed in [24; 25]. When the parameter \(\alpha\) is varied within the fractional iterative method, the system exhibits significant changes in behavior. As \(\alpha\) crosses certain critical values, the system transitions through different dynamical regimes--namely, stability, bifurcation, and chaos. These transitions are summarized in Table 1. Figure 2a-g present the corresponding bifurcation diagrams, where the y-axis indicates the steady-state or long-term values of the system variable and the x-axis represents values of the parameters \(\sigma\) and \(\alpha\). These diagrams illustrate how the nature of the solution--whether a fixed point, a periodic orbit, or a chaotic trajectory--evolves as the parameters change.