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Nonlinear Dynamics

, Volume 82, Issue 1–2, pp 157–171 | Cite as

A study of dynamics of the tricomplex polynomial \(\eta ^p+c\)

  • Pierre-Olivier Parisé
  • Dominic Rochon
Original Paper

Abstract

In this article, we give the exact interval of the cross section of the so-called Mandelbric set generated by the polynomial \(z^3+c\) where \(z\) and \(c\) are complex numbers. Following that result, we show that the Mandelbric defined on the hyperbolic numbers \(\mathbb {D}\) is a square with its center at the origin. Moreover, we define the Multibrot sets generated by a polynomial of the form \(Q_{p,c}(\eta )=\eta ^p+c\) (\(p \in \mathbb {N}\) and \(p \ge 2\)) for tricomplex numbers. More precisely, we prove that the tricomplex Mandelbric has four principal slices instead of eight principal 3D slices that arise for the case of the tricomplex Mandelbrot set. Finally, we prove that one of these four slices is an octahedron.

Keywords

Tricomplex dynamics Generalized Mandelbrot sets Multicomplex numbers  Hyperbolic numbers 3D fractals 

Notes

Acknowledgments

DR is grateful to the Natural Sciences and Engineering Research Council of Canada (NSERC) for financial support. POP would also like to thank the NSERC for the award of a Summer undergraduate Research grant.

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Département de mathématiques et d’informatiqueUniversité du Québec à Trois-RivièresTrois-RivièresCanada

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