| Summary: | Optimal fourth-order multiple-root finders with parameter <inline-formula> <math display="inline"> <semantics> <mi>λ</mi> </semantics> </math> </inline-formula> were conjugated via the Möbius map applied to a simple polynomial function. The long-term dynamics of these conjugated maps in the <inline-formula> <math display="inline"> <semantics> <mi>λ</mi> </semantics> </math> </inline-formula>-parameter plane was analyzed to discover some properties of periodic, bounded and chaotic orbits. The <inline-formula> <math display="inline"> <semantics> <mi>λ</mi> </semantics> </math> </inline-formula>-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 <inline-formula> <math display="inline"> <semantics> <mi>λ</mi> </semantics> </math> </inline-formula>-dependent connected components. When a red fixed component in the parameter plane branches into a <i>q</i>-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.
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