Computational Bifurcations Occurring on Red Fixed Components in the λ-Parameter Plane for a Family of Optimal Fourth-Order Multiple-Root Finders under the Möbius Conjugacy Map

Mathematics

YoungHee Geum¹  YoungIk Kim¹ 

[1] Department of Mathematics, Dankook University, Cheonan 330-714, Korea;

关键词: parameter plane;    Möbius map;    bifurcation point;    circle;    cardioid;    fourth-order;   

DOI  :  10.3390/math8050763

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown