## 3 Chaos-Based Parameter Estimation and Stability Analysis
### Dynamical Analysis for Parameter Tuning via Bifurcation and Chaotic Behavior  
In addition to conventional analyses based on spectral radius or contraction mappings, the study of _dynamical planes_ offers further insights into the stability properties of fractional-order iteration schemes. A dynamical plane is a graphical representation in which each point in the complex (or real) plane corresponds to an initial guess, and its color indicates the behavior of the iteration--specifically, whether it converges to a root (or attractor) or diverges [26; 27; 28]. In fractional-order schemes, the memory effect governed by the fractional order \(\gamma\), along with any introduced chaotic perturbations, significantly influences the structure of these planes. For stable schemes, the dynamical plane typically exhibits wide, smooth, and well-separated basins of attraction associated with different roots or solutions, reflecting robustness to initial guesses. The boundaries between these basins are often fractal, indicating sensitivity to initial conditions. Lowering the fractional order \(\gamma\) tends to smooth these fractal boundaries, thereby reducing sensitivity and enlarging regions of stable convergence. Properly tuned chaotic components can help prevent the iteration from becoming trapped in non-optimal attractors; however, excessive chaos may distort the basin structure, leading to fragmented or discontinuous convergence regions and signaling instability. Moreover, dynamical planes provide a more comprehensive view of global stability than local linearization techniques. As parameters such as \(\gamma\) and \(\alpha\) vary, bifurcation phenomena--abrupt transitions from connected, stable basins to highly fragmented and chaotic regions--can be visually identified. This visual framework enables researchers to distinguish between stable and unstable parameter regimes and assess the robustness of an iterative method in the presence of noise or imprecise initial guesses. In engineering contexts where initial conditions are uncertain or only approximately known, this global perspective is especially valuable. Consequently, dynamical plane analysis serves as a powerful complement to traditional mathematical stability proofs, particularly for fractional-order iterative methods incorporating chaotic dynamics. The following rational map \(G(x)\) is obtained by applying the fractional order to the polynomial \(p(x)=(x-\vartheta_{1})(x-\vartheta_{2})\), yielding  
\[G(x)=\frac{\alpha x^{8}+4x^{7}-2\alpha x^{4}+24x^{5}+4x^{3}+\alpha}{2x(\alpha x ^{4}-2\alpha x^{2}+8x^{3}+\alpha)(x^{2}+1)}, \tag{34}\]  
where \(\vartheta_{1},\vartheta_{2}\in\mathbb{R}\). By applying a Mobius transformation of the form \(U(x)=\frac{x-\vartheta}{x-\vartheta}\), the map \(G(x)\) becomes conjugate to the function \(O^{[*]}(x)\), given by  
\[O^{[*]}(x)=UoG\omega U^{-1}(x)=\frac{x^{3}(-x^{4}+2\alpha x^{2}-2x^{3}+2\alpha +2x+1)}{(2\alpha-1)x^{4}+2\alpha x^{2}-2x^{3}+2x+1}, \tag{35}\]  
for \(\gamma\approx 1\), and this representation is independent of the parameters \(a\) and \(b\)[29]. The rational maps corresponding to other fractional-order schemes are given as follows:  
\[O^{[*]}(x) = \frac{x^{3}(x^{8}+4x^{7}+4x^{6}-4x^{5}+1+(4\alpha-10)x^{4}-4x^{3} +(4\alpha+4)x^{3}+4x))}{1+x^{8}+4x^{7}+(4\alpha+4)x^{6}-4x^{5}+(4\alpha-10)x^{ 4}-4x^{3}+4x^{2}+4x)},\] (36) \[O^{[*]}(x) = \frac{\left( \begin{array}{c}(x^{8}+4.0x^{7}+(4.0-2.0\alpha)x^{6 }+(-4.0-4.0\alpha)x^{5}+\\ (-9.6-2.0\alpha)x^{4}-3.2x^{3}+(4.0+1.2\alpha)\\ x^{2}+(3.2+4.0\alpha)x+0.6+2.0\alpha)x^{4}\\ \end{array} \right)}{\left( \begin{array}{c}-1.+(-1.4+2.0 ax^{8}+(-4.8+4.0 \alpha)x^{7}+(-4.+2.8\alpha)x^{6}+\\ 4.8x^{5}+(10.4-2.0\alpha)x^{4}+(4.0-4.0\alpha)x^{3}\\ +(-4.0-2.0\alpha)x^{2}-4.x\\ \end{array} \right)},\] (37) \[O^{[*]}(x) = \frac{x^{3}\Bigg{(} \begin{array}{c}(x^{12}+6\alpha^{11}+12x^{10 }+2x^{9}-25x^{8}-28x^{7}+\\ 1+(-8\alpha+8)x^{6}+28x^{5}+7x^{4}-10x^{3}-4x^{2}+2x)\\ \end{array} \Bigg{)}}{\left(