scholarly journals A comparison of some fixed point iteration procedures by using the basins of attraction

2016 ◽  
Vol 32 (3) ◽  
pp. 277-284
Author(s):  
GHEORGHE ARDELEAN ◽  
◽  
OVIDIU COSMA ◽  
LASZLO BALOG ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Basins Of Attraction ◽  
Numerical Results ◽  
Complex Polynomials ◽  
Empirical Comparison ◽  
Fixed Point Iteration ◽  
Newton's Iteration ◽  
Iterative Processes ◽  
Points Approximation

Several iterative processes have been defined by researchers to approximate the fixed points of various classes operators. In this paper we present, by using the basins of attraction for the roots of some complex polynomials, an empirical comparison of some iteration procedures for fixed points approximation of Newton’s iteration operator. Some numerical results are presented. The Matlab m-files for generating the basins of attraction are presented, too.

A New Class of Three-Dimensional Maps with Hidden Chaotic Dynamics

2016 ◽  
Vol 26 (12) ◽  
pp. 1650206 ◽  
Author(s):  
Haibo Jiang ◽  
Yang Liu ◽  
Zhouchao Wei ◽  
Liping Zhang
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Chaotic Dynamics ◽  
Three Dimensional ◽  
Computational Procedure ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Computer Search ◽  
New Class ◽  
Special Concern

This paper studies a new class of three-dimensional maps in a Jerk-like structure with a special concern of their hidden chaotic dynamics. Our investigation focuses on the hidden chaotic attractors in three typical scenarios of fixed points, namely no fixed point, single fixed point, and two fixed points. A systematic computer search is performed to explore possible hidden chaotic attractors, and a number of examples of the proposed maps are used for demonstration. Numerical results show that the routes to hidden chaotic attractors are complex, and the basins of attraction for the hidden chaotic attractors could be tiny, so that using the standard computational procedure for localization is impossible.

scholarly journals Some fixed point iteration procedures

1991 ◽  
Vol 14 (1) ◽  
pp. 1-16 ◽  
Author(s):  
B. E. Rhoades
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Complete Coverage ◽  
Fixed Point Iteration

This paper provides a survey of iteration procedures that have been used to obtain fixed points for maps satisfying a variety of contractive conditions. The author does not claim to provide complete coverage of the literature, and admits to certain biases in the theorems that are cited herein. In spite of these shortcomings, however, this paper should be a useful reference for those persons wishing to become better acquainted with the area.

scholarly journals Geometrically Constructed Family of the Simple Fixed Point Iteration Method

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 694
Author(s):  
Vinay Kanwar ◽  
Puneet Sharma ◽  
Ioannis K. Argyros ◽  
Ramandeep Behl ◽  
Christopher Argyros ◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Nonlinear Equations ◽  
Iteration Method ◽  
Parameter Family ◽  
Arbitrary Parameter ◽  
Fixed Point Iteration ◽  
Iterative Schemes ◽  
Straight Line ◽  
Predictor Corrector

This study presents a new one-parameter family of the well-known fixed point iteration method for solving nonlinear equations numerically. The proposed family is derived by implementing approximation through a straight line. The presence of an arbitrary parameter in the proposed family improves convergence characteristic of the simple fixed point iteration as it has a wider domain of convergence. Furthermore, we propose many two-step predictor–corrector iterative schemes for finding fixed points, which inherit the advantages of the proposed fixed point iterative schemes. Finally, several examples are given to further illustrate their efficiency.

A new class of two-dimensional rational maps with self-excited and hidden attractors

2021 ◽  
Author(s):  
Li-Ping Zhang ◽  
Yang Liu ◽  
Zhou-Chao Wei ◽  
Hai-Bo Jiang ◽  
Qin-Sheng Bi
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Complex Dynamics ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Rational Maps ◽  
Two Dimensional ◽  
Hidden Attractors ◽  
New Class ◽  
Rational Term

Abstract This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stabilities of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov (Kaplan-Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.

scholarly journals Anderson Acceleration of the Arnoldi-Inout Method for Computing PageRank

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 636
Author(s):  
Xia Tang ◽  
Chun Wen ◽  
Xian-Ming Gu ◽  
Zhao-Li Shen
Keyword(s):  
Fixed Point ◽  
Convergence Analysis ◽  
Linear Combination ◽  
Computational Cost ◽  
Numerical Results ◽  
New Method ◽  
Fixed Point Iteration ◽  
Anderson Acceleration

Anderson(m0) extrapolation, an accelerator to a fixed-point iteration, stores m0+1 prior evaluations of the fixed-point iteration and computes a linear combination of those evaluations as a new iteration. The computational cost of the Anderson(m0) acceleration becomes expensive with the parameter m0 increasing, thus m0 is a common choice in most practice. In this paper, with the aim of improving the computations of PageRank problems, a new method was developed by applying Anderson(1) extrapolation at periodic intervals within the Arnoldi-Inout method. The new method is called the AIOA method. Convergence analysis of the AIOA method is discussed in detail. Numerical results on several PageRank problems are presented to illustrate the effectiveness of our proposed method.

scholarly journals Equivalence results for implicit Jungck–Kirk type iterations

2018 ◽  
Vol 22 (1) ◽  
pp. 75-89
Author(s):  
H. Akewe ◽  
A. A. Mogbademu
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Points ◽  
Fixed Point Iteration ◽  
Linear Spaces ◽  
Mann Iteration ◽  
Normed Linear Spaces ◽  
Weakly Compatible ◽  
The Common

We show that the implicit Jungck–Kirk-multistep, implicit Jungck–Kirk–Noor, implicit Jungck–Kirk–Ishikawa, and implicit Jungck–Kirk–Mann iteration schemes are equivalently used to approximate the common fixed points of a pair of weakly compatible generalized contractive-like operators defined on normed linear spaces. Our results contribute to the existing results on the equivalence of fixed point iteration schemes by extending them to pairs of maps. An example to show the applicability of the main results is included.

scholarly journals Three-Step Fixed Point Iteration for Generalized Multivalued Mapping in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Zhanfei Zuo
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Banach Spaces ◽  
Multivalued Mapping ◽  
Fixed Point Iteration ◽  
Iterative Processes ◽  
Multivalued Nonexpansive Mapping

The convergence of three-step fixed point iterative processes for generalized multivalued nonexpansive mapping was considered in this paper. Under some different conditions, the sequences of three-step fixed point iterates strongly or weakly converge to a fixed point of the generalized multivalued nonexpansive mapping. Our results extend and improve some recent results.

scholarly journals Generation of Julia and Mandelbrot Sets via Fixed Points

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 86 ◽  
Author(s):  
Mujahid Abbas ◽  
Hira Iqbal ◽  
Manuel De la Sen
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Iterative Process ◽  
Color Variation ◽  
Escape Time ◽  
Complex Polynomials ◽  
Mandelbrot Sets ◽  
C And N

The aim of this paper is to present an application of a fixed point iterative process in generation of fractals namely Julia and Mandelbrot sets for the complex polynomials of the form T ( x ) = x n + m x + r where m , r ∈ C and n ≥ 2 . Fractals represent the phenomena of expanding or unfolding symmetries which exhibit similar patterns displayed at every scale. We prove some escape time results for the generation of Julia and Mandelbrot sets using a Picard Ishikawa type iterative process. A visualization of the Julia and Mandelbrot sets for certain complex polynomials is presented and their graphical behaviour is examined. We also discuss the effects of parameters on the color variation and shape of fractals.

scholarly journals A solution to nonlinear Fredholm integral equations in the context of w-distances

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
P. Dhivya ◽  
M. Marudai ◽  
Vladimir Rakočević ◽  
Andreea Fulga
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Integral Equations ◽  
Best Proximity Point ◽  
Numerical Results ◽  
Fredholm Integral Equations ◽  
Fixed Point Result ◽  
Nonlinear Fredholm Integral Equations

AbstractIn this paper we propose a solution to the nonlinear Fredholm integral equations in the context of w-distance. For this purpose, we also provide a fixed point result in the same setting. In addition, we provide best proximity point results. We give examples and present numerical results to approximate fixed points.

A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3157-3172
Author(s):  
Mujahid Abbas ◽  
Bahru Leyew ◽  
Safeer Khan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Metric Space ◽  
Delay Differential Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Existence Of Periodic Solutions ◽  
Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

Sign in / Sign up

Export Citation Format

Share Document