Search Results (6)

This investigation provides a comprehensive analytical framework for the topological morphology and global convergence dynamics governing a specific family of sixth-order iterative schemes designed for nonlinear equations with multiple roots. By invoking a Möbius conjugacy transformation upon the specialized polynomial class $f\left(z\right)={\left((z-p)(z-q)\right)}^m$, we project the iterative sequence onto the Riemann sphere $\widehat{\mathbb{C}}$, effectively recasting the algorithm as a discrete complex dynamic system. The core of this study lies in the bifurcation analysis of the associated parameter space. We meticulously chart the stability manifolds, tracing the evolution of critical orbits to distinguish between regions of predictable convergence and those characterized by chaotic instability. By examining the iterative methods generated by these rational endomorphisms, the research unveils the intricate fractal boundaries that delineate the basin of attraction, offering a profound insight into the structural robustness of higher-order methods. In the dynamical plane, the geometry of the basins of attraction is scrutinized to evaluate the robustness of the numerical flow and its sensitivity to the configuration of weight functions. By analyzing the fractal complexity of the boundaries within these basins, we provide a detailed characterization of the iterative morphology and its global reliability. The analytical findings are supported by high-resolution graphical representations and comparative numerical data, illustrating the superior performance and structural integrity of the proposed methods in solving nonlinear problems. Full article

Let $F_2\left(\mathbb{C}\right)$ denote the second symmetric product of the complex plane $\mathbb{C}$ endowed with the Hausdorff topology, i.e., $F_2\left(\mathbb{C}\right)=\{A\subset \mathbb{C}:|A|\le 2,A\ne \varnothing \}$. In this paper, we extended the concept of Möbius transformations to $F_2\left(\mathbb{C}\right)$. More precisely, given a Möbius transformation T of $\mathbb{C}$, we define the map $\tilde{T}\left(\{z,w\}\right)=\{T\left(z\right),T\left(w\right)\}$ within $F_2\left(\mathbb{C}\right)$. We describe some general properties of these maps, including the structure of their generators, characteristics related to transitivity, and the geometry of the conjugacy classes. Full article

In this paper, a family of Ostrowski-type iterative schemes with a biparameter was analyzed. We present the dynamic view of the proposed method and study various conjugation properties. The stability of the strange fixed points for special parameter values is studied. The parameter spaces related to the critical points and dynamic planes are used to visualize their dynamic properties. Eventually, we find the most stable member of the biparametric family of six-order Ostrowski-type methods. Some test equations are examined for supporting the theoretical results. Full article

Optimal fourth-order multiple-root finders with parameter $\lambda$ were conjugated via the Möbius map applied to a simple polynomial function. The long-term dynamics of these conjugated maps in the $\lambda$ -parameter plane was analyzed to discover some properties of periodic, bounded and chaotic orbits. The $\lambda$ -parameters for periodic orbits in the parameter plane are painted in different colors depending on their periods, and the bounded or chaotic ones are colored black to illustrate $\lambda$ -dependent connected components. When a red fixed component in the parameter plane branches into a q-periodic component, we encounter geometric bifurcation phenomena whose characteristics determine the desired boundary equation and bifurcation point. Computational results along with illustrated components support the bifurcation phenomena underlying this paper. Full article

Bifurcations have been studied with an extensive analysis of boundary curves of red, fixed components in the parametric space for a uniparametric family of simple-root finders under the Möbius conjugacy map applied to a quadratic polynomial. An elementary approach from the perspective of a plane curve theory properly describes the geometric figures resembling a circle or cardioid to characterize the underlying boundary curves that are parametrically expressed. Moreover, exact bifurcation points for satellite components on the boundaries have been found, according to the fact that the tangent line at a bifurcation point simultaneously touches the red fixed component and the satellite component. Computational experiments implemented with examples well reflect the significance of the theoretical backgrounds pursued in this paper. Full article

This paper is devoted to an analysis on locating and counting satellite components born along the stability circle in the parameter space for a family of Jarratt-like iterative methods. An elementary theory of plane geometric curves is pursued to locate bifurcation points of such satellite components. In addition, the theory of Farey sequence is adopted to count the number of the satellite components as well as to characterize relationships between the bifurcation points. A linear stability theory on local bifurcations is developed based upon a small perturbation about the fixed point of the iterative map with a control parameter. Some properties of fixed and critical points under the Möbius conjugacy map are investigated. Theories and examples on locating and counting bifurcation points of satellite components in the parameter space are presented to analyze the bifurcation behavior underlying the dynamics behind the iterative map. Full article