Chaos in dynamics of a family of transcendental meromorphic functions

Sajid, Mohammad; Kapoor, G. P.

doi:10.5899/2017/jnaa-00355

Cited by 6 publications

(6 citation statements)

References 23 publications

(49 reference statements)

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“…Bergweiler [11] provides criteria for the escaping set and the Julia set of an entire function to have positive Lebesgue measure. For transcendental meromorphic functions, the dynamics of a family shows in [58] and Herman rings with small periods as well omitted values determine in [16]. The dynamical properties of the extraneous fixed points characterize in [26] with respect to their stability, basins of attraction, cycles, etc.…”

Section: Different Approaches To Studying Dynamics Of One Variable Co...mentioning

confidence: 99%

On Some Advances in Dynamics of One Variable Complex Functions

Sajid¹

2022

Int. J. of Appl. Math.

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In this article, we present some recent advances in the dynamics of one variable complex functions which are especially associated to the Julia sets, the Fatou sets, the Sigel disks, the Herman rings, fixed points as well as the Mandelbrot sets. For the sake of interest, we consider mainly research works which have been done from 2015 to 2019. To achieve our purpose, some interesting and selected themes are considered to show some recent advances in the dynamics of one variable complex functions.

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Section: Different Approaches To Studying Dynamics Of One Variable Co...mentioning

confidence: 99%

On Some Advances in Dynamics of One Variable Complex Functions

Sajid¹

2022

Int. J. of Appl. Math.

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“…That is, the trajectories of points in a Julia set remain bounded during the iterations of f . Particularly, if f has an attractive point w, the Julia set can be defined as J( f ) = ∂A(w), where A(w) is the attractive domain of the attractive fixed point w [38,39]. The basic frame of ETA is mainly based on the following definition.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm

Zhao

Zhang

et al. 2021

Fractal Fract

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In this paper, a novel escape-time algorithm is proposed to calculate the connectivity’s degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm.

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“…Moreover, the importance of the real dynamics of functions can be observed in the dynamics of functions in the complex plane and often such type of investigations describe Fatou sets, Julia sets, and some other dynamical results in the complex plane [23][24][25][26][27] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Chaotic Behaviour and Bifurcation in Real Dynamics of Two-Parameter Family of Functions including Logarithmic Map

Sajid

2020

Abstract and Applied Analysis

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The focus of this research work is to obtain the chaotic behaviour and bifurcation in the real dynamics of a newly proposed family of functions fλ,ax=x+1−λxlnax;x>0, depending on two parameters in one dimension, where assume that λ is a continuous positive real parameter and a is a discrete positive real parameter. This proposed family of functions is different from the existing families of functions in previous works which exhibits chaotic behaviour. Further, the dynamical properties of this family are analyzed theoretically and numerically as well as graphically. The real fixed points of functions fλ,ax are theoretically simulated, and the real periodic points are numerically computed. The stability of these fixed points and periodic points is discussed. By varying parameter values, the plots of bifurcation diagrams for the real dynamics of fλ,ax are shown. The existence of chaos in the dynamics of fλ,ax is explored by looking period-doubling in the bifurcation diagram, and chaos is to be quantified by determining positive Lyapunov exponents.

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Chaos in dynamics of a family of transcendental meromorphic functions

Cited by 6 publications

References 23 publications

On Some Advances in Dynamics of One Variable Complex Functions

On Some Advances in Dynamics of One Variable Complex Functions

On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm

Chaotic Behaviour and Bifurcation in Real Dynamics of Two-Parameter Family of Functions including Logarithmic Map

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