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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

Author

Listed:
  • Young Hee Geum

    (Department of Mathematics, Dankook University, Cheonan 330-714, Korea)

  • Young Ik Kim

    (Department of Mathematics, Dankook University, Cheonan 330-714, Korea)

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.

Suggested Citation

  • Young Hee Geum & Young Ik Kim, 2020. "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, MDPI, vol. 8(5), pages 1-17, May.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:5:p:763-:d:356439
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    References listed on IDEAS

    as
    1. Gao, Jianfang & Liang, Hui & Ma, Shufang, 2019. "Strong convergence of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations with constant delay," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 385-398.
    2. Behl, Ramandeep & Cordero, Alicia & Motsa, S.S. & Torregrosa, Juan R., 2015. "On developing fourth-order optimal families of methods for multiple roots and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 520-532.
    3. Argyros, Ioannis K. & Magreñán, Á. Alberto, 2015. "On the convergence of an optimal fourth-order family of methods and its dynamics," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 336-346.
    4. Young Ik Kim & Young Hee Geum, 2013. "A Two-Parameter Family of Fourth-Order Iterative Methods with Optimal Convergence for Multiple Zeros," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-7, February.
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