Generalisation of the Mandelbrot set to integral functions of quaternions

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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

Abstract
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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 - A, 1999, 5 (1) : 107-116. doi: 10.3934/dcds.1999.5.107

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