Generalized Mandelbrot Sets and Julia Sets for Non-analytic Complex Maps

2010 ◽  
Author(s):  
Yan Dejun ◽  
Yang Rijing ◽  
Xin Huijie ◽  
Zheng Jiangchao
Keyword(s):  
Julia Sets ◽  
Mandelbrot Sets ◽  
Generalized Mandelbrot Sets

scholarly journals Smooth Decomposition of Generalized Fatou Set Explains Smooth Structure in Generalized Mandelbrot Set

1988 ◽  
Vol 43 (1) ◽  
pp. 14-16 ◽  
Author(s):  
J. Peinke ◽  
J. Parisi ◽  
B. Röhricht ◽  
O. E. Rössler ◽  
W. Metzler
Keyword(s):  
Parameter Space ◽  
Initial Conditions ◽  
Logistic Map ◽  
Mandelbrot Set ◽  
Fatou Set ◽  
Smooth Structure ◽  
Mandelbrot Sets ◽  
Generalized Mandelbrot Sets

Abstract Generalized Mandelbrot sets arise in perturbed (non-analytic) versions of the complex logistic map. Numerically, it contains smooth portions as shown previously. To exclude that this result is specific to particular initial conditions only, the structure of the analogue to the Fatou set is looked at in the region in question. The set of non-divergent points is being "eaten up" by a smooth invading boundary. Therefore, the same type of decomposition applies independent of position in parameter space, in the region in question.

The Construction for Generalized Mandelbrot Sets of the Frieze Group

2013 ◽  
Vol 756-759 ◽  
pp. 2562-2566
Author(s):  
Feng Ying Wang ◽  
Li Ming Du ◽  
Zi Yang Han
Keyword(s):  
Dynamical System ◽  
Parameter Space ◽  
Mandelbrot Sets ◽  
Group A ◽  
Novel Method ◽  
Definition Of ◽  
Generalized Mandelbrot Sets

By an analysis of symmetric features of equivalent mappings of the frieze group, a definition of their generalized Mandelbrot sets is given and a novel method for constructing generalized Mandelbrot sets of equivalent mappings of frieze group is presented via utilizing the Ljapunov exponent as the judgment standard. Based on generating parameter space of dynamical system, lots of patterns of generalized Mandelbrot sets are produced.

scholarly journals GEOMETRIC LIMITS OF MANDELBROT AND JULIA SETS UNDER DEGREE GROWTH

2012 ◽  
Vol 22 (12) ◽  
pp. 1250301 ◽  
Author(s):  
SUZANNE HRUSKA BOYD ◽  
MICHAEL J. SCHULZ
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Julia Sets ◽  
Rational Maps ◽  
Closed Unit Disk ◽  
Geometric Limit ◽  
The Family ◽  
Mandelbrot Sets

First, for the family Pn,c(z) = zn + c, we show that the geometric limit of the Mandelbrot sets Mn(P) as n → ∞ exists and is the closed unit disk, and that the geometric limit of the Julia sets J(Pn,c) as n tends to infinity is the unit circle, at least when |c| ≠ 1. Then, we establish similar results for some generalizations of this family; namely, the maps z ↦ zt + c for real t ≥ 2 and the rational maps z ↦ zn + c + a/zn.

scholarly journals Noise perturbed generalized Mandelbrot sets

2008 ◽  
Vol 347 (1) ◽  
pp. 179-187 ◽  
Author(s):  
Xingyuan Wang ◽  
Zhen Wang ◽  
Yahui Lang ◽  
Zhenfeng Zhang
Keyword(s):  
Mandelbrot Sets ◽  
Generalized Mandelbrot Sets

scholarly journals Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers

2019 ◽  
Vol 3 (1) ◽  
pp. 6 ◽  
Author(s):  
Vance Blankers ◽  
Tristan Rendfrey ◽  
Aaron Shukert ◽  
Patrick Shipman
Keyword(s):  
Dynamical Systems ◽  
Natural Number ◽  
Hyperbolic Plane ◽  
Julia Sets ◽  
Number System ◽  
Mandelbrot Set ◽  
Complex Numbers ◽  
Hyperbolic Numbers ◽  
Fractal Sets ◽  
Mandelbrot Sets

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form x + τ y for x , y ∈ R , and τ 2 = 1 but τ ≠ ± 1 , are the natural number system in which to encode geometric properties of the Minkowski space R 1 , 1 . We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.

Julia sets and Mandelbrot sets in Noor orbit

2014 ◽  
Vol 228 ◽  
pp. 615-631 ◽  
Author(s):  
Ashish ◽  
Mamta Rani ◽  
Renu Chugh
Keyword(s):  
Julia Sets ◽  
Mandelbrot Sets

scholarly journals Mandelbrot Sets and Julia Sets in Picard-Mann Orbit

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 64411-64421 ◽  
Author(s):  
Cui Zou ◽  
Abdul Aziz Shahid ◽  
Asifa Tassaddiq ◽  
Arshad Khan ◽  
Maqbool Ahmad
Keyword(s):  
Julia Sets ◽  
Mandelbrot Sets

scholarly journals Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions

2019 ◽  
Vol 3 (3) ◽  
pp. 42 ◽  
Author(s):  
L.K. Mork ◽  
Trenton Vogt ◽  
Keith Sullivan ◽  
Drew Rutherford ◽  
Darin J. Ulness
Keyword(s):  
Fixed Points ◽  
Parameter Space ◽  
Rotational Symmetry ◽  
Julia Sets ◽  
Phase Rotation ◽  
Mandelbrot Sets

Centered polygonal lacunary functions are a particular type of lacunary function that exhibit properties which set them apart from other lacunary functions. Primarily, centered polygonal lacunary functions have true rotational symmetry. This rotational symmetry is visually seen in the corresponding Julia and Mandelbrot sets. The features and characteristics of these related Julia and Mandelbrot sets are discussed and the parameter space, made with a phase rotation and offset shift, is intricately explored. Also studied in this work is the iterative dynamical map, its characteristics and its fixed points.

Two Partitioning Algorithms for Generating of M Sets of the Frieze Group

2013 ◽  
Vol 336-338 ◽  
pp. 2238-2241
Author(s):  
Feng Ying Wang ◽  
Li Ming Du ◽  
Zi Yang Han
Keyword(s):  
Dynamical System ◽  
Parameter Space ◽  
Mandelbrot Sets ◽  
Partitioning Algorithms ◽  
Generalized Mandelbrot Sets

Symmetric features of the frieze group equivalent mappings were analysed, and two partitioning algorithms are given for constructing generalized Mandelbrot sets of frieze group equivalent mappings in order to study the characteristics of generalized Msets. Based on generating parameter space of dynamical system, lots of patterns of generalized Mandelbrot sets are produced.

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