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]

David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353
[2]

Christopher M. Kellett. Classical converse theorems in Lyapunov's second method. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2333-2360. doi: 10.3934/dcdsb.2015.20.2333
[3]

Matthias Eller. A remark on Littman's method of boundary controllability. Evolution Equations & Control Theory, 2013, 2 (4) : 621-630. doi: 10.3934/eect.2013.2.621
[4]

Darya V. Verveyko, Andrey Yu. Verisokin. Application of He's method to the modified Rayleigh equation. Conference Publications, 2011, 2011 (Special) : 1423-1431. doi: 10.3934/proc.2011.2011.1423
[5]

Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219
[6]

Lars Grüne, Peter E. Kloeden, Stefan Siegmund, Fabian R. Wirth. Lyapunov's second method for nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 375-403. doi: 10.3934/dcds.2007.18.375
[7]

Bernd Hofmann, Barbara Kaltenbacher, Elena Resmerita. Lavrentiev's regularization method in Hilbert spaces revisited. Inverse Problems & Imaging, 2016, 10 (3) : 741-764. doi: 10.3934/ipi.2016019
[8]

Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1007-1018. doi: 10.3934/dcds.2010.26.1007
[9]

R. Baier, M. Dellnitz, M. Hessel-von Molo, S. Sertl, I. G. Kevrekidis. The computation of convex invariant sets via Newton's method. Journal of Computational Dynamics, 2014, 1 (1) : 39-69. doi: 10.3934/jcd.2014.1.39
[10]

Jutta Bikowski, Jennifer L. Mueller. 2D EIT reconstructions using Calderon's method. Inverse Problems & Imaging, 2008, 2 (1) : 43-61. doi: 10.3934/ipi.2008.2.43
[11]

Liqun Qi, Zheng yan, Hongxia Yin. Semismooth reformulation and Newton's method for the security region problem of power systems. Journal of Industrial & Management Optimization, 2008, 4 (1) : 143-153. doi: 10.3934/jimo.2008.4.143
[12]

Paweł Góra, Abraham Boyarsky. Stochastic perturbations and Ulam's method for W-shaped maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1937-1944. doi: 10.3934/dcds.2013.33.1937
[13]

Joseph J Kohn. Nirenberg's contributions to complex analysis. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 537-545. doi: 10.3934/dcds.2011.30.537
[14]

Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569
[15]

Zhengyong Zhou, Bo Yu. A smoothing homotopy method based on Robinson's normal equation for mixed complementarity problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 977-989. doi: 10.3934/jimo.2011.7.977
[16]

Matthias Gerdts, Martin Kunkel. A nonsmooth Newton's method for discretized optimal control problems with state and control constraints. Journal of Industrial & Management Optimization, 2008, 4 (2) : 247-270. doi: 10.3934/jimo.2008.4.247
[17]

Kim Knudsen, Jennifer L. Mueller. The born approximation and Calderón's method for reconstruction of conductivities in 3-D. Conference Publications, 2011, 2011 (Special) : 844-853. doi: 10.3934/proc.2011.2011.844
[18]

José Mujica, Bernd Krauskopf, Hinke M. Osinga. A Lin's method approach for detecting all canard orbits arising from a folded node. Journal of Computational Dynamics, 2017, 4 (1&2) : 143-165. doi: 10.3934/jcd.2017005
[19]

Runchang Lin, Huiqing Zhu. A discontinuous Galerkin least-squares finite element method for solving Fisher's equation. Conference Publications, 2013, 2013 (special) : 489-497. doi: 10.3934/proc.2013.2013.489
[20]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (24)
  • HTML views (23)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences