JULIA SET OF THE NEWTON TRANSFORMATION FOR SOLVING SOME COMPLEX EXPONENTIAL EQUATION

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 197-204 ◽  
Author(s):  
XINGYUAN WANG ◽  
XUEJING YU
Keyword(s):  
Exponential Function ◽  
Julia Set ◽  
Basins Of Attraction ◽  
Exponential Equation ◽  
Distribution Structure ◽  
Newton Transformation ◽  
Complex Exponential

We extend Kim's complex exponential function, come up with theory about the Julia set of Newton's transformation for general exponential equation, analyze the behavior of the roots of some complex exponential equation, and prove the symmetry, boundedness and embedding topology distribution structure of basins of attraction of the Julia set in theory.

JULIA SET OF THE NEWTON METHOD FOR SOLVING SOME COMPLEX EXPONENTIAL EQUATION

2009 ◽  
Vol 09 (02) ◽  
pp. 153-169 ◽  
Author(s):  
XINGYUAN WANG ◽  
WENJING SONG ◽  
LIXIAN ZOU
Keyword(s):  
Newton Method ◽  
Exponential Function ◽  
Julia Set ◽  
Julia Sets ◽  
Exponential Equation ◽  
Distribution Structure ◽  
Complex Exponential

We extend Kim's complex exponential function and come up with a theory about Julia sets of Newton method for general exponential equation. We analyze the behavior of the roots of some complex exponential equation, and prove the Julia Set's symmetry, boundedness and embedding topology distribution structure of attraction regions in theory.

Tying hairs for structurally stable exponentials

2000 ◽  
Vol 20 (6) ◽  
pp. 1603-1617 ◽  
Author(s):  
RANJIT BHATTACHARJEE ◽  
ROBERT L. DEVANEY
Keyword(s):  
Fixed Point ◽  
Julia Set ◽  
Basins Of Attraction ◽  
Crucial Importance ◽  
Exponential Functions ◽  
Complex Exponential ◽  
Structurally Stable

Our goal in this paper is to describe the structure of the Julia set of complex exponential functions that possess an attracting cycle. When the cycle is a fixed point, it is known that the Julia set is a ‘Cantor bouquet’, a union of uncountably many distinct curves or ‘hairs’. When the period of the cycle is greater than one, infinitely many of the hairs in the bouquet become pinched or attached together. In this paper, we develop an algorithm to determine which of these hairs are attached. Of crucial importance in this construction is the kneading invariant, a sequence that is derived from the topology of the basins of attraction of the attracting cycle.

scholarly journals An explosion point for the set of endpoints of the Julia set of λ exp (z)

1990 ◽  
Vol 10 (1) ◽  
pp. 177-183 ◽  
Author(s):  
John C. Mayer
Keyword(s):  
Exponential Function ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Julia Set ◽  
Periodic Points ◽  
Real Parameter ◽  
Topological Dimension ◽  
Complex Exponential ◽  
Totally Disconnected ◽  
Topological Sense

AbstractThe Julia set Jλ of the complex exponential function Eλ: z → λez for a real parameter λ(0 < λ < 1/e) is known to be a Cantor bouquet of rays extending from the set Aλ of endpoints of Jλ to ∞. Since Aλ contains all the repelling periodic points of Eλ, it follows that Jλ = Cl (Aλ). We show that Aλ is a totally disconnected subspace of the complex plane ℂ, but if the point at ∞ is added, then is a connected subspace of the Riemann sphere . As a corollary, Aλ has topological dimension 1. Thus, ∞ is an explosion point in the topological sense for Âλ. It is remarkable that a space with an explosion point occurs ‘naturally’ in this way.

THE ACCURACY OF COMPUTER ALGORITHMS IN DYNAMICAL SYSTEMS

1991 ◽  
Vol 01 (03) ◽  
pp. 625-639 ◽  
Author(s):  
MARILYN B. DURKIN
Keyword(s):  
Dynamical Systems ◽  
Exponential Function ◽  
Julia Set ◽  
Julia Sets ◽  
Simple Algorithm ◽  
Computer Algorithms ◽  
Simple Expansion ◽  
Mathematical Accuracy ◽  
Complex Exponential

We study the mathematical accuracy of computer algorithms used to produce pictures of Julia sets by analyzing two representatives cases of the complex exponential function. We first define the Julia set and give the simple algorithm used for the exponential function. We then define what it means for a picture to be "right" and consider the two totally different Julia sets of E0.3(z) = 0.3ez and E(z) = ez. We use a simple expansion argument together with the properties of the exponential function to show that each of these pictures is correct.

Periodic Point at Real Axis for a Generalized 3x+1 Function

2011 ◽  
Vol 55-57 ◽  
pp. 1670-1674 ◽  
Author(s):  
Shuai Liu ◽  
Zheng Xuan Wang
Keyword(s):  
Periodic Point ◽  
Real Axis ◽  
Exponential Function ◽  
Periodic Points ◽  
Divergence Points ◽  
Fractal Character ◽  
Convergence And Divergence ◽  
Complex Exponential

In order to study the fractal character of representative complex exponential function just as generalized 3x+1 function T(x). In this essay, we proved that T(x) has periodic points of every period in bound (n, n+1) when n>1 in real axis. Then, we found the distribution of 2-periods points of T(x) in real axis. We put forward the bottom bound of 2-periodic point’s number and proved it. Moreover, we found the number of T(x)’s 2-periodic points in different bounds to validate our conclusion. Then, we extended the conclusion to i-periods points and find similar conclusion. Finally, we proved there exist endless convergence and divergence points of T(x) in real axis.

scholarly journals A characterization of postcritically minimal Newton maps of complex exponential functions

2018 ◽  
Vol 39 (10) ◽  
pp. 2855-2880
Author(s):  
KHUDOYOR MAMAYUSUPOV
Keyword(s):  
Exponential Function ◽  
Julia Sets ◽  
Exponential Functions ◽  
Postcritically Finite ◽  
One To One ◽  
Complex Exponential

We obtain a unique, canonical one-to-one correspondence between the space of marked postcritically finite Newton maps of polynomials and the space of postcritically minimal Newton maps of entire maps that take the form $p(z)\exp (q(z))$ for $p(z)$, $q(z)$ polynomials and $\exp (z)$, the complex exponential function. This bijection preserves the dynamics and embedding of Julia sets and is induced by a surgery tool developed by Haïssinsky.

Misiurewicz Points for Complex Exponentials

1997 ◽  
Vol 07 (07) ◽  
pp. 1599-1615 ◽  
Author(s):  
Robert L. Devaney ◽  
Xavier Jarque
Keyword(s):  
Symbolic Dynamics ◽  
Topological Structure ◽  
Julia Set ◽  
Chaotic Regime ◽  
Singular Value ◽  
Entire Plane ◽  
Exponential Maps ◽  
Complex Exponential

In this paper we examine the structure of the chaotic regime or Julia set of certain complex exponential maps Eλ(z) = λez. In the case where λ is a Misiurewicz point (i.e. the singular value 0 is eventually periodic), it is known that the Julia set for the map is the entire plane. In this case the Julia set also possesses certain curves or "hairs" that are permuted by the map. We examine the dynamics on these hairs in detail. We describe a certain extended symbolic dynamics by which the topological structure of the hairs may be determined completely.

Sign in / Sign up

Export Citation Format

Share Document