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JULIA AND MANDELBROT SETS OF CHEBYSHEV FAMILIES
International Journal of Bifurcation and Chaos, 2001We present one-parameter families of rational functions of arbitrary degree d which are globally generalized polynomial-like of degree d and roughly speaking locally quadratic-like everywhere, where the parameter appears not only as a purely multiplicative factor but also in a more complicated nonlinear way.
Franz Peherstorfer, Christoph Stroh
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A note on inverted mandelbrot sets
The Visual Computer, 1990A simple technique is described for demonstrating artistically interesting behavior in chaotic systems defined by complex dynamics. In particular, the Mandelbrot set for the iterative process\(\zeta \to \zeta ^p + (1/u)^p \) is explored.
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Evolutionary Exploration of the Mandelbrot Set
2006 IEEE International Conference on Evolutionary Computation, 2006The Mandelbrot set is an infinitely complex fractal defined by a simple iterative algorithm operating on the complex numbers. Views of the Mandelbrot set are a common form of fractal art. Presented here is a collection of fitness functions that permit three-parameter evolutionary search of the Mandelbrot set to locate interesting views.
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The Mandelbrot Set in the Classroom
The Mathematics Teacher, 1991Inspired by Dewdney's article “A Tour of the Mandelbrot Set aboard the Mandelbus” in Scientific American (1989), we developed materials to add to our existing complex-number units, which introduce students to the wonderful Mandelbrot set. This article describes these materials and how they have been used in second-year-algebra and precalculus classes.
Marny Frantz, Sylvia Lazarnick
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2011
In this lecture, we shall discuss the geometric and topological features of the complex plane associated with dynamical systems whose evolution is governed by the iterative scheme \(z_n {\rm + 1}\,{\rm = }\,f{\rm (}zn{\rm ), }\,z_{0\,} {\rm = }\,p{\rm }\,{\rm where}\,{\rm }f{\rm (}z{\rm )}\) is a complex valued function and \(p\, \in \,C.\) Such ...
Ravi P. Agarwal +2 more
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In this lecture, we shall discuss the geometric and topological features of the complex plane associated with dynamical systems whose evolution is governed by the iterative scheme \(z_n {\rm + 1}\,{\rm = }\,f{\rm (}zn{\rm ), }\,z_{0\,} {\rm = }\,p{\rm }\,{\rm where}\,{\rm }f{\rm (}z{\rm )}\) is a complex valued function and \(p\, \in \,C.\) Such ...
Ravi P. Agarwal +2 more
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Julia Sets and the Mandelbrot Set
1986Quadratic Julia sets, and the Mandelbrot set, arise in a mathematical situation which is extremely simple, namely from sequences of complex numbers defined inductively by the relation $$z_n + = z_n^2 + c,$$ where c is a complex constant. I must say that, in 1980, whenever I told my friends that I was just starting with J.H.
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Mandelbrot set and Julia sets of fractional order
Nonlinear Dynamics, 2023Marius-F Danca +2 more
exaly