American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data
  • DCDS Home
  • This Issue
  • Next Article
    Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion
October  2017, 37(10): 5455-5465. doi: 10.3934/dcds.2017237

Constructing attracting cycles for Halley and Schröder maps of polynomials

Jared T. Collins ,

Department of Mathematics and Computer Science, Freed-Hardeman University, Henderson, TN 38340, USA

* Corresponding author: Jared T. Collins

Received  September 2016 Revised  June 2017 Published  June 2017

We show that for any set of $n$ distinct points in the complex plane, there exists a polynomial $p$ of degree at most $n+1$ so that the corresponding Halley and Schröder map for $p$ has the given points as a super-attracting cycle. This improves the result in [1], which shows how to find such a polynomial of degree $3n$. Moreover we show that in general one cannot improve upon degree $n+1$.

Keywords: Complex dynamics, extraneous attractors, Halley's method, Schröder's method, polynomials.
Mathematics Subject Classification: Primary: 30D05; Secondary: 34M03, 37F10, 39B12.
Citation: Jared T. Collins. Constructing attracting cycles for Halley and Schröder maps of polynomials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5455-5465. doi: 10.3934/dcds.2017237
References:
[1]

S. Amat, S. Busquier and S. Plaza, A construction of attracting periodic orbits for some classical third-order iterative methods, J. Comput. Appl. Math., 189 (2006), 22-33.  doi: 10.1016/j.cam.2005.03.049.  Google Scholar

[2]

J. Campbell and J. Collins, Specifying attracting cycles for Newton maps of polynomials, J. Difference Equ. Appl., 19 (2013), 1361-1379.  doi: 10.1080/10236198.2012.751987.  Google Scholar

[3]

D. Hilbert, Über die vollen Invariantensysteme, Math. Ann., 42 (1893), 313-373.  doi: 10.1007/BF01444162.  Google Scholar

[4]

K. Kneisl, Julia sets for the super-Newton method, Cauchy's method and Halley's method, Chaos, 11 (2001), 359-370.  doi: 10.1063/1.1368137.  Google Scholar

[5]

S. Plaza and N. Romero, Attracting cycles for the relaxed Newton's method, J. Comput. Appl. Math., 235 (2011), 3238-3244.  doi: 10.1016/j.cam.2011.01.010.  Google Scholar

[6]

S. Plaza and V. Vergara, Existence of attracting periodic orbits for the Newton method, Sci. Ser. A Math. Sci., 7 (2001), 31-36.   Google Scholar

show all references

References:
[1]

S. Amat, S. Busquier and S. Plaza, A construction of attracting periodic orbits for some classical third-order iterative methods, J. Comput. Appl. Math., 189 (2006), 22-33.  doi: 10.1016/j.cam.2005.03.049.  Google Scholar

[2]

J. Campbell and J. Collins, Specifying attracting cycles for Newton maps of polynomials, J. Difference Equ. Appl., 19 (2013), 1361-1379.  doi: 10.1080/10236198.2012.751987.  Google Scholar

[3]

D. Hilbert, Über die vollen Invariantensysteme, Math. Ann., 42 (1893), 313-373.  doi: 10.1007/BF01444162.  Google Scholar

[4]

K. Kneisl, Julia sets for the super-Newton method, Cauchy's method and Halley's method, Chaos, 11 (2001), 359-370.  doi: 10.1063/1.1368137.  Google Scholar

[5]

S. Plaza and N. Romero, Attracting cycles for the relaxed Newton's method, J. Comput. Appl. Math., 235 (2011), 3238-3244.  doi: 10.1016/j.cam.2011.01.010.  Google Scholar

[6]

S. Plaza and V. Vergara, Existence of attracting periodic orbits for the Newton method, Sci. Ser. A Math. Sci., 7 (2001), 31-36.   Google Scholar

[1]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[2]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127
[3]

Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233
[4]

Hong Fu, Mingwu Liu, Bo Chen. Supplier's investment in manufacturer's quality improvement with equity holding. Journal of Industrial & Management Optimization, 2021, 17 (2) : 649-668. doi: 10.3934/jimo.2019127
[5]

Skyler Simmons. Stability of broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015
[6]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406
[7]

François Ledrappier. Three problems solved by Sébastien Gouëzel. Journal of Modern Dynamics, 2020, 16: 373-387. doi: 10.3934/jmd.2020015
[8]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020404
[9]

Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021016
[10]

Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283
[11]

Dmitry Dolgopyat. The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces. Journal of Modern Dynamics, 2020, 16: 351-371. doi: 10.3934/jmd.2020014
[12]

Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021002
[13]

Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013
[14]

Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020180
[15]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095
[16]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019
[17]

Qing-Hu Hou, Yarong Wei. Telescoping method, summation formulas, and inversion pairs. Electronic Research Archive, , () : -. doi: 10.3934/era.2021007
[18]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266
[19]

Chiun-Chuan Chen, Yuan Lou, Hirokazu Ninomiya, Peter Polacik, Xuefeng Wang. Preface: DCDS-A special issue to honor Wei-Ming Ni's 70th birthday. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : ⅰ-ⅱ. doi: 10.3934/dcds.2020171
[20]

Tingting Wu, Li Liu, Lanqiang Li, Shixin Zhu. Repeated-root constacyclic codes of length $ 6lp^s $. Advances in Mathematics of Communications, 2021, 15 (1) : 167-189. doi: 10.3934/amc.2020051

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (56)
  • HTML views (62)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences