# American Institute of Mathematical Sciences

August  2020, 25(8): 3087-3109. doi: 10.3934/dcdsb.2020052

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

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

* Corresponding author

Received  April 2019 Revised  October 2019 Published  February 2020

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

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.

Citation: Young Hee Geum, Young Ik Kim. Long-term orbit dynamics viewed through the yellow main component in the parameter space of a family of optimal fourth-order multiple-root finders. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3087-3109. doi: 10.3934/dcdsb.2020052
##### References:

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##### References:
Bifurcations on the stability unit circle
Stability circle $\boldsymbol S$ for strange fixed point $z = 1$
Stability surfaces of strange fixed points $\xi = 1$
Stability surfaces of the strange fixed points $\xi_j, 1 \le j \le 2$ from the roots of $T(\xi;\lambda)$
Parameter space $\mathcal{P}$, red and yellow main components $\mathcal{H}_1$
Color chart defined in Table 1
Bifurcation points ${\lambda}_t$ of $t$-periodic components in $\mathcal{P}$
Dynamical planes for various values of ${\lambda}$-parameters
Typical geometries for primitive and satellite components
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.
 $q$ $C_q$ 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.
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 $3$ $\begin{pmatrix}-3.85567\\- 0.14285\end{pmatrix}^\ast$ $\begin{pmatrix}-3.85567\\0.14285 \end{pmatrix}$ $4$ $\begin{pmatrix}-3.91781 \\- 0.219178\end{pmatrix}$ $\begin{pmatrix}-3.91781\\0.219178 \end{pmatrix}$ $5$ $\begin{pmatrix}-3.98185 \\- 0.261606 \end{pmatrix}$ $\begin{pmatrix}-3.8306 \\- 0.0840949 \end{pmatrix}$ $\begin{pmatrix}-3.8306\\0.0840949 \end{pmatrix}$ $\begin{pmatrix}-3.98185\\0.261606 \end{pmatrix}$ $6$ $\begin{pmatrix}-4.04082 \\- 0.282784 \end{pmatrix}$ $\begin{pmatrix}-4.04082\\0.282784 \end{pmatrix}$ $7$ $\begin{pmatrix}-4.09231 \\- 0.290424 \end{pmatrix}$ $\begin{pmatrix}-3.88575 \\- 0.186408\end{pmatrix}$ $\begin{pmatrix}-3.82438 \\- 0.0597191 \end{pmatrix}$ $\begin{pmatrix}-3.82438\\0.0597191 \end{pmatrix}$ $\begin{pmatrix}-3.88575\\0.186408 \end{pmatrix}$ $\begin{pmatrix}-4.09231\\0.290424 \end{pmatrix}$ $8$ $\begin{pmatrix}-4.13604 \\- 0.289658\end{pmatrix}$ $\begin{pmatrix}-3.8381 \\- 0.105794 \end{pmatrix}$ $\begin{pmatrix}-3.8381\\0.105794 \end{pmatrix}$ $\begin{pmatrix}-4.13604\\0.289658 \end{pmatrix}$ $9$ $\begin{pmatrix}-4.17269 \\- 0.283871 \end{pmatrix}$ $\begin{pmatrix}-3.95018 \\- 0.24367 \end{pmatrix}$ $\begin{pmatrix}-3.8219 \\- 0.0463343 \end{pmatrix}$ $\begin{pmatrix}-3.8219\\0.0463343 \end{pmatrix}$ $\begin{pmatrix}-3.95018\\0.24367\end{pmatrix}$ $\begin{pmatrix}-4.17269\\0.283871 \end{pmatrix}$ $10$ $\begin{pmatrix}-4.20324 \\- 0.275251 \end{pmatrix}$ $\begin{pmatrix}-3.8754 \\- 0.173249\end{pmatrix}$ $\begin{pmatrix}-3.8754\\0.173249\end{pmatrix}$ $\begin{pmatrix}-4.20324\\0.275251\end{pmatrix}$ ${}^\ast$: $\begin{pmatrix}-3.85567\\- 0.14285\end{pmatrix} \equiv -3.85567- 0.14285 \; i, \; i = \sqrt{-1}$
 $\ell$ $q$ 0 1 2 3 4 5 6 7 8 9 $1$ -4.4 $2$ -3.81818 $3$ $\begin{pmatrix}-3.85567\\- 0.14285\end{pmatrix}^\ast$ $\begin{pmatrix}-3.85567\\0.14285 \end{pmatrix}$ $4$ $\begin{pmatrix}-3.91781 \\- 0.219178\end{pmatrix}$ $\begin{pmatrix}-3.91781\\0.219178 \end{pmatrix}$ $5$ $\begin{pmatrix}-3.98185 \\- 0.261606 \end{pmatrix}$ $\begin{pmatrix}-3.8306 \\- 0.0840949 \end{pmatrix}$ $\begin{pmatrix}-3.8306\\0.0840949 \end{pmatrix}$ $\begin{pmatrix}-3.98185\\0.261606 \end{pmatrix}$ $6$ $\begin{pmatrix}-4.04082 \\- 0.282784 \end{pmatrix}$ $\begin{pmatrix}-4.04082\\0.282784 \end{pmatrix}$ $7$ $\begin{pmatrix}-4.09231 \\- 0.290424 \end{pmatrix}$ $\begin{pmatrix}-3.88575 \\- 0.186408\end{pmatrix}$ $\begin{pmatrix}-3.82438 \\- 0.0597191 \end{pmatrix}$ $\begin{pmatrix}-3.82438\\0.0597191 \end{pmatrix}$ $\begin{pmatrix}-3.88575\\0.186408 \end{pmatrix}$ $\begin{pmatrix}-4.09231\\0.290424 \end{pmatrix}$ $8$ $\begin{pmatrix}-4.13604 \\- 0.289658\end{pmatrix}$ $\begin{pmatrix}-3.8381 \\- 0.105794 \end{pmatrix}$ $\begin{pmatrix}-3.8381\\0.105794 \end{pmatrix}$ $\begin{pmatrix}-4.13604\\0.289658 \end{pmatrix}$ $9$ $\begin{pmatrix}-4.17269 \\- 0.283871 \end{pmatrix}$ $\begin{pmatrix}-3.95018 \\- 0.24367 \end{pmatrix}$ $\begin{pmatrix}-3.8219 \\- 0.0463343 \end{pmatrix}$ $\begin{pmatrix}-3.8219\\0.0463343 \end{pmatrix}$ $\begin{pmatrix}-3.95018\\0.24367\end{pmatrix}$ $\begin{pmatrix}-4.17269\\0.283871 \end{pmatrix}$ $10$ $\begin{pmatrix}-4.20324 \\- 0.275251 \end{pmatrix}$ $\begin{pmatrix}-3.8754 \\- 0.173249\end{pmatrix}$ $\begin{pmatrix}-3.8754\\0.173249\end{pmatrix}$ $\begin{pmatrix}-4.20324\\0.275251\end{pmatrix}$ ${}^\ast$: $\begin{pmatrix}-3.85567\\- 0.14285\end{pmatrix} \equiv -3.85567- 0.14285 \; i, \; i = \sqrt{-1}$
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)
 $(q, k)$ 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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