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Generalized Mandelbrot sets for meromorphic complex and quaternionic maps
International Journal of Bifurcation and Chaos (2002)
  • Walter Buchanan
  • Jagannathan Gomatam, Glasgow Caledonian University
  • Bonnie Steves, Glasgow Caledonian University
Abstract
The concepts of the Mandelbrot set and the definition of the stability regions of cycles for rational maps require careful investigation. The standard definition of the Mandelbrot set for the map f : z → z2+ c (the set of c values for which the iteration of the critical point at 0 remains bounded) is inappropriate for meromorphic maps such as the inverse square map. The notion of cycle sets, introduced by Brooks and Matelski [1978] for the quadratic map and applied to meromorphic maps by Yin [1994], facilitates a precise definition of the Mandelbrot parameter space for these maps. Close scrutiny of the cycle sets of these maps reveals generic fractal structures, echoing many of the features of the Mandelbrot set. Computer representations confirm these features and allow the dynamical comparison with the Mandelbrot set. In the parameter space, a purely algebraic result locates the stability regions of the cycles as the zeros of characteristic polynomials. These maps are generalized to quaternions. The powerful theoretical support that exists for complex maps is not generally available for quaternions. However, it is possible to construct and analyze cycle sets for a class of quaternionic rational maps (QRM). Three-dimensional sections of the cycle sets of QRM are nontrivial extensions of the cycle sets of complex maps, while sharing many of their features.
Keywords
  • quaternions,
  • meromorphic maps,
  • cycle sets
Disciplines
Publication Date
August 1, 2002
Publisher Statement
Originally published in: International Journal of Bifurcation and Chaos (2002), 12 (8), pp.1755-1777.
Citation Information
Walter Buchanan, Jagannathan Gomatam and Bonnie Steves. "Generalized Mandelbrot sets for meromorphic complex and quaternionic maps" International Journal of Bifurcation and Chaos (2002)
Available at: http://works.bepress.com/bonnie_steves/2/