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

Abstract

Optimal fourth-order multiple-root finders with parameter λ were conjugated via the Möbius map applied to a simple polynomial function. The long-term dynamics of these conjugated maps in the λ -parameter plane was analyzed to discover some properties of periodic, bounded and chaotic orbits. The λ -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 λ -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.

Keywords

• parameter plane
• Möbius map
• bifurcation point
• circle
• cardioid
• fourth-order