Long-term orbit dynamics viewed through the yellow main component in the parameter space of a family of optimal fourth-order multiple-root finders

Young Hee Geum; Young Ik Kim

doi:10.3934/dcdsb.2020052

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2020, Volume 25, Issue 8: 3087-3109. Doi: 10.3934/dcdsb.2020052

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Long-term orbit dynamics viewed through the yellow main component in the parameter space of a family of optimal fourth-order multiple-root finders

Young Hee Geum and
Young Ik Kim^,

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

^* Corresponding author
^* Corresponding author

Received: April 2019

Revised: October 2019

Early access: February 2020

Published: August 2020

The first author (Y.H. Geum) is supported by research grant NRF-2018R1D1A1B07047715 from National Research Foundation of Korea

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Abstract

An analysis based on an elementary theory of plane curves is presented to locate bifurcation points from a main component in the parameter space of a family of optimal fourth-order multiple-root finders. We explore the basic dynamics of the iterative multiple-root finders under the Möbius conjugacy map on the Riemann sphere. A linear stability theory on local bifurcations is developed from the viewpoint of an arbitrarily small perturbation about the fixed point of the iterative map with a control parameter. Invariant conjugacy properties are established for the fixed point and its multiplier. The parameter spaces and dynamical planes are investigated to analyze the underlying dynamics behind the iterative map. Numerical experiments support the theory of locating bifurcation points of satellite and primitive components in the parameter space.

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Mathematics Subject Classification: Primary: 65H05, 65H99; Secondary: 41A25, 65B995.

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Figure 1. Bifurcations on the stability unit circle

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Figure 2. Stability circle $ \boldsymbol S $ for strange fixed point $ z = 1 $

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Figure 3. Stability surfaces of strange fixed points $ \xi = 1 $

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Figure 4. Stability surfaces of the strange fixed points $ \xi_j, 1 \le j \le 2 $ from the roots of $ T(\xi;\lambda) $

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Figure 6. Parameter space $ \mathcal{P} $, red and yellow main components $ \mathcal{H}_1 $

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Figure 5. Color chart defined in Table 1

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Figure 7. Bifurcation points $ {\lambda}_t $ of $ t $-periodic components in $ \mathcal{P} $

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Figure 8. Dynamical planes for various values of $ {\lambda} $-parameters

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Figure 9. Typical geometries for primitive and satellite components

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Table 1. Coloring scheme for a $ q $-periodic orbit with $ q \in \mathbb{N}\cup \{0 \} $

$ q $	$ C_q $
1	$ C_1 (\rm{fixed\; point}\; \infty) $ magenta		$ C_1(\rm{fixed \; point} \; 0) $ cyan		$ C_1(\rm{fixed \; point} \; 1) $ yellow		$ C_{1}(\rm{other\; strange\; fixed\; point}) $ red
	$ C_{2} $	$ C_{3} $	$ C_{4} $	$ C_{5} $	$ C_{6} $	$ C_{7} $	$ C_{8} $	$ C_{9} $
	orange	light green	brown	blue	green	dark yellow	antiquewhite	light pink
$ 2\le q\le 80 $	$ C_{10} $	$ C_{11} $	$ C_{12} $	$ C_{13} $	$ C_{14} $	$ C_{15} $	$ C_{16} $	$ C_{17} $
	khaki	melon	thistle	lavender	turquoise	plum	orchid	medium orchid
	$ C_{18} $	$ C_{19} $	$ C_{20} $	$ C_{21} $	$ C_{22} $	$ C_{23} $	$ C_{24} $	$ C_{25} $
	blue violet	dark orchid	purple	powder blue	sky blue	deep sky blue	dodger blue	royal blue
	$ C_{26} $	$ C_{27} $	$ C_{28} $	$ C_{29} $	$ C_{30} $	$ C_{31} $	$ C_{32} $	$ C_{33} $
	medium spring green	apple green	medium sea green	forest green	dark blue	olive drab	bisque	moccasin
	$ C_{34} $	$ C_{35} $	$ C_{36} $	$ C_{37} $	$ C_{38} $	$ C_{39} $	$ C_{40} $	$ C_{41} $
	light salmon	salmon	light coral	Indian red	dark red	peach puff	fire brick	sandy brown
	$ C_{42} $	$ C_{43} $	$ C_{44} $	$ C_{45} $	$ C_{46} $	$ C_{47} $	$ C_{48} $	$ C_{49} $
	wheat	tomato	orange red	chocolate	pink	pale violet red	deep pink	violet red
	$ C_{50} $	$ C_{51} $	$ C_{52} $	$ C_{53} $	$ C_{54} $	$ C_{55} $	$ C_{56} $	$ C_{57} $
	gainsboro	light gray	dark gray	gray	charteruse	electric indigo	electric lime	lime
	$ C_{58} $	$ C_{59} $	$ C_{60} $	$ C_{61} $	$ C_{62} $	$ C_{63} $	$ C_{64} $	$ C_{65} $
	silver	teal	pale turquoise	rosy brown	honeydew	lemon chiffon	misty rose	mintcream
	$ C_{66} $	$ C_{67} $	$ C_{68} $	$ C_{69} $	$ C_{70} $	$ C_{71} $	$ C_{72} $	$ C_{73} $
	gold	crimson	light crimson	lavenderblush	slateblue	light cyan	coral	light blue
	$ C_{74} $	$ C_{75} $	$ C_{76} $	$ C_{77} $	$ C_{78} $	$ C_{79} $	$ C_{80} $
	aquamarine	light yellow	peru	violet	papayawhip	dark orange	sea green
$ \stackrel{q = 0^\ast \mbox{ or}} {q> 80} $	black
∗: q = 0 implies a non-periodic but bounded orbit. These 84 colors are explicitly illustrated in Figure 5.

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Table 2. Typical $ \ell/q $–bifurcation points $ {\lambda} $ for $ 1\le q \le 10 $ and $ 0 \le \ell \le 9 $

	$ \ell $
$ q $	0	1	2	3	4	5	6	7	8	9
$ 1 $	-4.4
$ 2 $		-3.81818
$ 3 $		$ \begin{pmatrix}-3.85567\\- 0.14285\end{pmatrix}^\ast $	$ \begin{pmatrix}-3.85567\\0.14285 \end{pmatrix} $
$ 4 $		$ \begin{pmatrix}-3.91781 \\- 0.219178\end{pmatrix} $		$ \begin{pmatrix}-3.91781\\0.219178 \end{pmatrix} $
$ 5 $		$ \begin{pmatrix}-3.98185 \\- 0.261606 \end{pmatrix} $	$ \begin{pmatrix}-3.8306 \\- 0.0840949 \end{pmatrix} $	$ \begin{pmatrix}-3.8306\\0.0840949 \end{pmatrix} $	$ \begin{pmatrix}-3.98185\\0.261606 \end{pmatrix} $
$ 6 $		$ \begin{pmatrix}-4.04082 \\- 0.282784 \end{pmatrix} $				$ \begin{pmatrix}-4.04082\\0.282784 \end{pmatrix} $
$ 7 $		$ \begin{pmatrix}-4.09231 \\- 0.290424 \end{pmatrix} $	$ \begin{pmatrix}-3.88575 \\- 0.186408\end{pmatrix} $	$ \begin{pmatrix}-3.82438 \\- 0.0597191 \end{pmatrix} $	$ \begin{pmatrix}-3.82438\\0.0597191 \end{pmatrix} $	$ \begin{pmatrix}-3.88575\\0.186408 \end{pmatrix} $	$ \begin{pmatrix}-4.09231\\0.290424 \end{pmatrix} $
$ 8 $		$ \begin{pmatrix}-4.13604 \\- 0.289658\end{pmatrix} $		$ \begin{pmatrix}-3.8381 \\- 0.105794 \end{pmatrix} $		$ \begin{pmatrix}-3.8381\\0.105794 \end{pmatrix} $		$ \begin{pmatrix}-4.13604\\0.289658 \end{pmatrix} $
$ 9 $		$ \begin{pmatrix}-4.17269 \\- 0.283871 \end{pmatrix} $	$ \begin{pmatrix}-3.95018 \\- 0.24367 \end{pmatrix} $		$ \begin{pmatrix}-3.8219 \\- 0.0463343 \end{pmatrix} $	$ \begin{pmatrix}-3.8219\\0.0463343 \end{pmatrix} $		$ \begin{pmatrix}-3.95018\\0.24367\end{pmatrix} $	$ \begin{pmatrix}-4.17269\\0.283871 \end{pmatrix} $
$ 10 $		$ \begin{pmatrix}-4.20324 \\- 0.275251 \end{pmatrix} $		$ \begin{pmatrix}-3.8754 \\- 0.173249\end{pmatrix} $				$ \begin{pmatrix}-3.8754\\0.173249\end{pmatrix} $		$ \begin{pmatrix}-4.20324\\0.275251\end{pmatrix} $
$ {}^\ast $: $ \begin{pmatrix}-3.85567\\- 0.14285\end{pmatrix} \equiv -3.85567- 0.14285 \; i, \; i = \sqrt{-1} $

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Table 3. Typical numerical values of $ \{\xi, {\lambda} \} $ for satellite and primitive components with $ q k \le 8 $

$ (q, k) $	Type	$ \xi $	$ \lambda $	Fig. No.
$ (2, 1) $	Primitive	$ -0.23512621166835 + 1.6020106612492319\; i $	$ -1.951962936703127 -1.7485319734836857\; i $	7(a)
$ (3, 1) $	Primitive	$ 0.454659800907803 + 0\; i $	$ -5.219261654371707 + 0\; i $	7(b)
$ (4, 1) $	Primitive	$ 0.341977833651053 + 0.4814672920477362\; i $	$ -3.564458921821090 + 1.646535885566244\; i $	7(c)
$ (5, 1) $	Primitive	$ 0.757190777481740 + 0.2122872817258981\; i $	$ -4.045956427432157 + 0.346070091333350\; i $	7(d)
$ (6, 1) $	Primitive	$ -1.10304555596122 + 1.0059215339291120\; i $	$ -0.568308288946734 - 0.953277135441219\; i $	7(e)
$ (7, 1) $	Primitive	$ 0.764464537114743 + 0.2097254339881594\; i $	$ -4.018513045186225 + 0.345332811593217\; i $	7(f)
$ (2, 2) $	Satellite	$ 0.851625771743808 + 0.2369760927165886\; i $	$ -3.752943559023020 + 0.074408072396025\; i $	7(g)
$ (3, 2) $	Satellite	$ 0.875562175989576 + 0.2566238834815281\; i $	$ -3.838763107389483 + 0.152045173864673\; i $	7(h)
$ (4, 2) $	Satellite	$ 1.201354999600545 - 0.2912383604676560\; i $	$ -3.921140825708978 + 0.257336038244326\; i $	7(i)

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