General Mandelbrot Sets and Julia Sets Generated from Non-analytic Complex Iteration ⨍m(z)=z^n+c

2009 ◽  
Author(s):  
Dejun Yan ◽  
Junxing Zhang ◽  
Nan Jiang ◽  
Lidong Wang
Keyword(s):  
Julia Sets ◽  
Mandelbrot Sets

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 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.

MANDELBROT AND JULIA SETS OF ONE-PARAMETER RATIONAL FUNCTION FAMILIES ASSOCIATED WITH NEWTON'S METHOD

Fractals ◽  
2010 ◽  
Vol 18 (02) ◽  
pp. 255-263 ◽  
Author(s):  
XIANG-DONG LIU ◽  
ZHI-JIE LI ◽  
XUE-YE ANG ◽  
JIN-HAI ZHANG
Keyword(s):  
Rational Function ◽  
Newton’S Method ◽  
Newton's Method ◽  
Julia Sets ◽  
Rational Functions ◽  
Periodic Points ◽  
Mandelbrot Sets

In this paper, general Mandelbrot and Julia sets of one-parameter rational function families associated with Newton's method were discussed. The bounds of these general Mandelbrot sets and two formulas for calculating the number of different periods periodic points of these rational functions were given. The relations between general Mandelbrot sets and common Mandelbrot sets of zn + c (n ∈ Z, n ≥ 2), along with the relations between general Mandelbrot sets and their corresponding Julia sets were investigated. Consequently, the results were found in the study: there are similarities between the Mandelbrot and Julia sets of one-parameter rational function families associated with Newton's method and the Mandelbrot and Julia sets of zn + c (n ∈ Z, n ≥ 2).

CHAOS AND FRACTALS IN C–K MAP

2008 ◽  
Vol 19 (09) ◽  
pp. 1389-1409 ◽  
Author(s):  
XING-YUAN WANG ◽  
QING-YONG LIANG ◽  
JUAN MENG
Keyword(s):  
Power Spectrum ◽  
Fixed Points ◽  
Lyapunov Exponents ◽  
Correlation Dimension ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Parameter Plane ◽  
Boundary Equation ◽  
Mandelbrot Sets

The characteristic of the fixed points of the Carotid–Kundalini (C–K) map is investigated and the boundary equation of the first bifurcation of the C–K map in the parameter plane is given. Based on the studies of the phase graph, the power spectrum, the correlation dimension and the Lyapunov exponents, the paper reveals the general features of the C–K map transforming from regularity. Meanwhile, using the periodic scanning technology proposed by Welstead and Cromer, a series of Mandelbrot–Julia (M–J) sets of the complex C–K map are constructed. The symmetry of M–J set and the topological inflexibility of distributing of periodic region in the Mandelbrot set are investigated. By founding the whole portray of Julia sets based on Mandelbrot set qualitatively, we find out that Mandelbrot sets contain abundant information of structure of Julia sets.

scholarly journals CHECKERBOARD JULIA SETS FOR RATIONAL MAPS

2013 ◽  
Vol 23 (02) ◽  
pp. 1330004 ◽  
Author(s):  
PAUL BLANCHARD ◽  
FİGEN ÇİLİNGİR ◽  
DANIEL CUZZOCREO ◽  
ROBERT L. DEVANEY ◽  
DANIEL M. LOOK ◽  
...  
Keyword(s):  
Rotation Number ◽  
Julia Sets ◽  
Conjugacy Classes ◽  
Rational Maps ◽  
Root Of Unity ◽  
The Family ◽  
Mandelbrot Sets ◽  
The Given

In this paper, we consider the family of rational maps [Formula: see text] where n ≥ 2, d ≥ 1, and λ ∈ ℂ. We consider the case where λ lies in the main cardioid of one of the n - 1 principal Mandelbrot sets in these families. We show that the Julia sets of these maps are always homeomorphic. However, two such maps Fλ and Fμ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy μ = νj(d+1)λ or [Formula: see text] where j ∈ ℤ and ν is an (n - 1)th root of unity. We define a dynamical invariant, which we call the minimal rotation number. It determines which of these maps are conjugate on their Julia sets, and we obtain an exact count of the number of distinct conjugacy classes of maps drawn from these main cardioids.

scholarly journals New Fixed Point Results for Fractal Generation in Jungck Noor Orbit withs-Convexity

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Shin Min Kang ◽  
Waqas Nazeer ◽  
Muhmmad Tanveer ◽  
Abdul Aziz Shahid
Keyword(s):  
Nonlinear Dynamics ◽  
Fixed Point ◽  
Julia Sets ◽  
Mandelbrot Sets

We establish new fixed point results in the generation of fractals (Julia sets, Mandelbrot sets, and Tricorns and Multicorns for linear or nonlinear dynamics) by using Jungck Noor iteration withs-convexity.

Fractal Structures of General Mandelbrot Sets and Julia Sets Generated from Complex Non-Analytic Iteration Fm(z)=z¯m+c

2013 ◽  
Vol 347-350 ◽  
pp. 3019-3023
Author(s):  
De Jun Yan ◽  
Xiao Dan Wei ◽  
Hong Peng Zhang ◽  
Nan Jiang ◽  
Xiang Dong Liu
Keyword(s):  
Rotational Symmetry ◽  
Julia Sets ◽  
Boundary Curve ◽  
Mandelbrot Set ◽  
Escape Time ◽  
Fractal Structures ◽  
Mandelbrot Sets ◽  
The Stability ◽  
Definition Of ◽  
Analytic Iteration

In this paper we use the same idea as the complex analytic dynamics to study general Mandelbrot sets and Julia sets generated from the complex non-analytic iteration . The definition of the general critical point is given, which is of vital importance to the complex non-analytic dynamics. The general Mandelbrot set is proved to be bounded, axial symmetry by real axis, and have (m+1)-fold rotational symmetry. The stability condition of periodic orbits and the boundary curve of stability region of one-cycle are given. And the general Mandelbrot sets are constructed by the escape-time method and the periodic scanning algorithm, which present a better understanding of the structure of the Mandelbrot sets. The filled-in Julia sets Km,c have m-fold structures. Similar to the complex analytic dynamics, the general Mandelbrot sets are kinds of mathematical dictionary or atlas that map out the behavior of the filled-in Julia sets for different values of c.

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