Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders
Bifurcations have been studied with an extensive analysis of boundary curves of red, fixed components in the parametric space for a uniparametric family of simple-root finders under the Möbius conjugacy map applied to a quadratic polynomial. An elementary approach from the perspective of a...
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| Format: | Article |
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MDPI AG
2020-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/1/51 |
| _version_ | 1811304600436736000 |
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| author | Min-Young Lee Young Ik Kim |
| author_facet | Min-Young Lee Young Ik Kim |
| author_sort | Min-Young Lee |
| collection | DOAJ |
| description | Bifurcations have been studied with an extensive analysis of boundary curves of red, fixed components in the parametric space for a uniparametric family of simple-root finders under the Möbius conjugacy map applied to a quadratic polynomial. An elementary approach from the perspective of a plane curve theory properly describes the geometric figures resembling a circle or cardioid to characterize the underlying boundary curves that are parametrically expressed. Moreover, exact bifurcation points for satellite components on the boundaries have been found, according to the fact that the tangent line at a bifurcation point simultaneously touches the red fixed component and the satellite component. Computational experiments implemented with examples well reflect the significance of the theoretical backgrounds pursued in this paper. |
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| format | Article |
| id | doaj.art-66691de51f8c4ac6baa1897e0984a663 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-13T08:10:19Z |
| publishDate | 2020-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-66691de51f8c4ac6baa1897e0984a6632022-12-22T02:55:00ZengMDPI AGMathematics2227-73902020-01-01815110.3390/math8010051math8010051Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root FindersMin-Young Lee0Young Ik Kim1Department of Applied Mathematics, Dankook University, Cheonan 330-714, KoreaDepartment of Applied Mathematics, Dankook University, Cheonan 330-714, KoreaBifurcations have been studied with an extensive analysis of boundary curves of red, fixed components in the parametric space for a uniparametric family of simple-root finders under the Möbius conjugacy map applied to a quadratic polynomial. An elementary approach from the perspective of a plane curve theory properly describes the geometric figures resembling a circle or cardioid to characterize the underlying boundary curves that are parametrically expressed. Moreover, exact bifurcation points for satellite components on the boundaries have been found, according to the fact that the tangent line at a bifurcation point simultaneously touches the red fixed component and the satellite component. Computational experiments implemented with examples well reflect the significance of the theoretical backgrounds pursued in this paper.https://www.mdpi.com/2227-7390/8/1/51parameter spacemöbius mapbifurcation pointjarratt’s methodcardioid-likecircle-like |
| spellingShingle | Min-Young Lee Young Ik Kim Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders Mathematics parameter space möbius map bifurcation point jarratt’s method cardioid-like circle-like |
| title | Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders |
| title_full | Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders |
| title_fullStr | Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders |
| title_full_unstemmed | Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders |
| title_short | Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders |
| title_sort | bifurcations along the boundary curves of red fixed components in the parameter space for uniparametric jarratt type simple root finders |
| topic | parameter space möbius map bifurcation point jarratt’s method cardioid-like circle-like |
| url | https://www.mdpi.com/2227-7390/8/1/51 |
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