American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Nonlinear stability and dynamical properties for a Kuramoto-Sivashinsky equation in space dimension two
  • DCDS Home
  • This Issue
  • Next Article
    Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation
January  1999, 5(1): 107-116. doi: 10.3934/dcds.1999.5.107

Generalisation of the Mandelbrot set to integral functions of quaternions

Jagannathan Gomatam 1, and  Isobel McFarlane 1,

1. 

Department of Mathematics, Glasgow Caledonian University, Glasgow, United Kingdom, United Kingdom

Received  June 1997 Revised  April 1998 Published  October 1998

The rich diversity of patterns and concepts intrinsic to the Julia and the Mandelbrot sets of the quadratic map in the complex plane invite a search for higher dimensional generalisations. Quaternions provide a natural framework for such an endeavour. The objective of this investigation is to provide explicit formulae for the domain of stability of multiple cycles of classes of quaternionic maps $F(Q)+C$ or $CF(Q)$ where $C$ is a quaternion and $F(Q)$ is an integral function of $Q$. We introduce the concept of quaternionic differentials and employ this in the linear stability analysis of multiple cycles.
Keywords: quadratic map in the complex plane., Julia and Mandelbrot sets.
Mathematics Subject Classification: 15A33, 39A10, 39A1.
Citation: Jagannathan Gomatam, Isobel McFarlane. Generalisation of the Mandelbrot set to integral functions of quaternions. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 107-116. doi: 10.3934/dcds.1999.5.107
[1]

Koh Katagata. Quartic Julia sets including any two copies of quadratic Julia sets. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2103-2112. doi: 10.3934/dcds.2016.36.2103
[2]

Tarik Aougab, Stella Chuyue Dong, Robert S. Strichartz. Laplacians on a family of quadratic Julia sets II. Communications on Pure & Applied Analysis, 2013, 12 (1) : 1-58. doi: 10.3934/cpaa.2013.12.1
[3]

François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262
[4]

Alexander Blokh, Lex Oversteegen, Vladlen Timorin. Non-degenerate locally connected models for plane continua and Julia sets. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5781-5795. doi: 10.3934/dcds.2017251
[5]

Héctor A. Tabares-Ospina, Mauricio Osorio. Methodology for the characterization of the electrical power demand curve, by means of fractal orbit diagrams on the complex plane of Mandelbrot set. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1895-1905. doi: 10.3934/dcdsb.2020008
[6]

Koh Katagata. Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5319-5337. doi: 10.3934/dcds.2019217
[7]

Yu-Hao Liang, Wan-Rou Wu, Jonq Juang. Fastest synchronized network and synchrony on the Julia set of complex-valued coupled map lattices. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 173-184. doi: 10.3934/dcdsb.2016.21.173
[8]

Luiz Henrique de Figueiredo, Diego Nehab, Jorge Stolfi, João Batista S. de Oliveira. Rigorous bounds for polynomial Julia sets. Journal of Computational Dynamics, 2016, 3 (2) : 113-137. doi: 10.3934/jcd.2016006
[9]

Robert L. Devaney, Daniel M. Look. Buried Sierpinski curve Julia sets. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 1035-1046. doi: 10.3934/dcds.2005.13.1035
[10]

Danilo Antonio Caprio. A class of adding machines and Julia sets. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5951-5970. doi: 10.3934/dcds.2016061
[11]

Nathaniel D. Emerson. Dynamics of polynomials with disconnected Julia sets. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 801-834. doi: 10.3934/dcds.2003.9.801
[12]

Igor E. Pritsker and Richard S. Varga. Weighted polynomial approximation in the complex plane. Electronic Research Announcements, 1997, 3: 38-44.

[13]

Tien-Cuong Dinh, Nessim Sibony. Rigidity of Julia sets for Hénon type maps. Journal of Modern Dynamics, 2014, 8 (3&4) : 499-548. doi: 10.3934/jmd.2014.8.499
[14]

Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293
[15]

Ali Messaoudi, Rafael Asmat Uceda. Stochastic adding machine and $2$-dimensional Julia sets. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5247-5269. doi: 10.3934/dcds.2014.34.5247
[16]

Ranjit Bhattacharjee, Robert L. Devaney, R.E. Lee Deville, Krešimir Josić, Monica Moreno-Rocha. Accessible points in the Julia sets of stable exponentials. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 299-318. doi: 10.3934/dcdsb.2001.1.299
[17]

Weiyuan Qiu, Fei Yang, Yongcheng Yin. Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3375-3416. doi: 10.3934/dcds.2016.36.3375
[18]

Rich Stankewitz, Hiroki Sumi. Random backward iteration algorithm for Julia sets of rational semigroups. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2165-2175. doi: 10.3934/dcds.2015.35.2165
[19]

Rich Stankewitz, Hiroki Sumi. Backward iteration algorithms for Julia sets of Möbius semigroups. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6475-6485. doi: 10.3934/dcds.2016079
[20]

Youming Wang, Fei Yang, Song Zhang, Liangwen Liao. Escape quartered theorem and the connectivity of the Julia sets of a family of rational maps. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5185-5206. doi: 10.3934/dcds.2019211

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (86)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy