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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

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 and Continuous Dynamical Systems, 1999, 5 (1) : 107-116. doi: 10.3934/dcds.1999.5.107
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