Generalization of the Mandelbrot set: quaternionic quadratic maps

Jagannathan Gomatam, John Doyle, Bonnie Steves, Isobel McFarlane

Research output: Contribution to journalArticlepeer-review

38 Citations (Scopus)

Abstract

The iterated map Q → Q2 + C, where Q and C are complex 2 × 2 matrices representing quaternions, provides a natural generalisation of the Mandelbrot set to higher dimensions. Using the well-known expansion of the quaternion in terms of the generators of SU(2), the Pauli matrices, it is shown that the fixed point Q = Q2 + C is stable for C inside a cardioidal surface M3 in R4 and the boundary set ∂M3 sprouts domains of stability of multiple cycles. Stability calculations up to 3-cycle leading to explicit expressions for the associated Mandelbrot domain in R4 are presented here for the first time. These analyses lay down the theoretical frame work for characterizing the stability domain for general k-cycles.

Original languageEnglish
Pages (from-to)971-986
Number of pages16
JournalChaos, Solitons and Fractals
Volume5
Issue number6
DOIs
Publication statusPublished - Jun 1995

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • General Mathematics
  • General Physics and Astronomy
  • Applied Mathematics

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