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On Locating and Counting Satellite Components Born along the Stability Circle in the Parameter Space for a Family of Jarratt-Like Iterative Methods

Author

Listed:
  • Young Hee Geum

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

  • Young Ik Kim

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

Abstract

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.

Suggested Citation

  • Young Hee Geum & Young Ik Kim, 2019. "On Locating and Counting Satellite Components Born along the Stability Circle in the Parameter Space for a Family of Jarratt-Like Iterative Methods," Mathematics, MDPI, vol. 7(9), pages 1-16, September.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:839-:d:266168
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    References listed on IDEAS

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    1. Geum, Young Hee & Kim, Young Ik & Magreñán, Á. Alberto, 2016. "A biparametric extension of King’s fourth-order methods and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 282(C), pages 254-275.
    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. 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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