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 language | English |
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Pages (from-to) | 971-986 |
Number of pages | 16 |
Journal | Chaos, Solitons and Fractals |
Volume | 5 |
Issue number | 6 |
DOIs | |
Publication status | Published - Jun 1995 |
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- General Mathematics
- General Physics and Astronomy
- Applied Mathematics