Generalisation of the Mandelbrot set to integral functions of quaternions

Jagannathan Gomatam; Isobel McFarlane

doi:10.3934/dcds.1999.5.107

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1999, Volume 5, Issue 1: 107-116. Doi: 10.3934/dcds.1999.5.107

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Generalisation of the Mandelbrot set to integral functions of quaternions

Jagannathan Gomatam^1, and
Isobel McFarlane¹

1.
Department of Mathematics, Glasgow Caledonian University, Glasgow

Received: May 31, 1997

Revised: March 31, 1998

Published: December 1998

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Abstract

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:

Julia and Mandelbrot sets,
quadratic map in the complex plane.

Mathematics Subject Classification: 15A33, 39A10, 39A11.

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