Convergence and Dynamics of Iterative Methods: Chaos and Fractals

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: closed (31 December 2021) | Viewed by 14580

Special Issue Editors

Department of Mathematics and Computation, University of La Rioja, Madre de Dios 53, 26006 Logroño, La Rioja, Spain
Interests: applied mathematics; mathematical problems; new trends in mathematical education; E-learning; dynamical behavior
Special Issues, Collections and Topics in MDPI journals
Escuela Superior de Ingeniería y Tecnología, Universidad Internacional de La Rioja, Avenida de la Paz 123, 26006 Logroño, La Rioja, Spain
Interests: applied mathematics; dynamics; iterative methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Solving nonlinear equations is a problem that appears frequently in different scientific disciplines. It is a well-known fact that the solutions of such equations can rarely be found in closed form, so we need to use iterative procedures to solve them. When talking about iterative procedures, the most well-known, studied, and used one is Newton’s method, due to its good properties related to the convergence and the easy way of computing it. However, sometimes this method cannot be used for solving equations since the computation of the inverse of the derivative can be too costly or even not possible, leading one to consider alternative approaches such as the secant method or other derivative-free methods. On the other hand, sometimes researchers need to use higher-speed and higher-order iterative methods that have been developed for such purposes.

In the study of these iterative procedures, the following fields are of interest:

  • Semilocal convergence, in which you should impose conditions on the function involved and the starting point or guess;
  • Local convergence, in which you must impose conditions on the solution and the function involved;
  • Global convergence, which is the most complicated one to prove;
  • Dynamical studies, where fractals appear.

The main target of this Special Issue is to show some research lines developed in this discipline, such as the analysis of the dynamical behavior of nonlinear equations that can allow researchers to find different new ways for finding solutions for these equations. We welcome manuscripts considering areas including (but not limited to) the following:

  • Study of real dynamics, by means of using Lyapunov exponents, Feigenbaum diagrams, or other tools such as the convergence plane;
  • Study of complex dynamics, by means of using parameter and dynamical planes;
  • Comparison of the real and complex dynamics of an iterative method;
  • Study of the dynamics of higher-order methods;
  • Development of different tools to help researchers in the study of dynamical behavior;
  • Comparison of the results obtained in different languages such as MATLAB, Python, C++, and Mathematica.

Prof. Dr. Ángel Alberto Magreñán
Prof. Dr. Iñigo Sarría
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • iterative methods
  • numerical analysis
  • semilocal convergence
  • dynamical behavior
  • fractal
  • fractal dimension

Published Papers (7 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

19 pages, 1403 KiB  
Article
The Dynamical Analysis of a Biparametric Family of Six-Order Ostrowski-Type Method under the Möbius Conjugacy Map
by Xiaofeng Wang and Xiaohe Chen
Fractal Fract. 2022, 6(3), 174; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6030174 - 21 Mar 2022
Cited by 3 | Viewed by 1415
Abstract
In this paper, a family of Ostrowski-type iterative schemes with a biparameter was analyzed. We present the dynamic view of the proposed method and study various conjugation properties. The stability of the strange fixed points for special parameter values is studied. The parameter [...] Read more.
In this paper, a family of Ostrowski-type iterative schemes with a biparameter was analyzed. We present the dynamic view of the proposed method and study various conjugation properties. The stability of the strange fixed points for special parameter values is studied. The parameter spaces related to the critical points and dynamic planes are used to visualize their dynamic properties. Eventually, we find the most stable member of the biparametric family of six-order Ostrowski-type methods. Some test equations are examined for supporting the theoretical results. Full article
(This article belongs to the Special Issue Convergence and Dynamics of Iterative Methods: Chaos and Fractals)
Show Figures

Figure 1

17 pages, 1481 KiB  
Article
Derivative-Free Kurchatov-Type Accelerating Iterative Method for Solving Nonlinear Systems: Dynamics and Applications
by Xiaofeng Wang and Xiaohe Chen
Fractal Fract. 2022, 6(2), 59; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6020059 - 24 Jan 2022
Cited by 6 | Viewed by 1746
Abstract
Two novel Kurchatov-type first-order divided difference operators were designed, which were used for constructing the variable parameter of three derivative-free iterative methods. The convergence orders of the new derivative-free methods are 3, (5+17)/24.56 and 5. [...] Read more.
Two novel Kurchatov-type first-order divided difference operators were designed, which were used for constructing the variable parameter of three derivative-free iterative methods. The convergence orders of the new derivative-free methods are 3, (5+17)/24.56 and 5. The new derivative-free iterative methods with memory were applied to solve nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) in numerical experiments. The dynamical behavior of our new methods with memory was studied by using dynamical plane. The dynamical planes showed that our methods had good stability. Full article
(This article belongs to the Special Issue Convergence and Dynamics of Iterative Methods: Chaos and Fractals)
Show Figures

Figure 1

11 pages, 274 KiB  
Article
On Global Convergence of Third-Order Chebyshev-Type Method under General Continuity Conditions
by Fouad Othman Mallawi, Ramandeep Behl and Prashanth Maroju
Fractal Fract. 2022, 6(1), 46; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6010046 - 14 Jan 2022
Cited by 2 | Viewed by 1541
Abstract
There are very few papers that talk about the global convergence of iterative methods with the help of Banach spaces. The main purpose of this paper is to discuss the global convergence of third order iterative method. The convergence analysis of this method [...] Read more.
There are very few papers that talk about the global convergence of iterative methods with the help of Banach spaces. The main purpose of this paper is to discuss the global convergence of third order iterative method. The convergence analysis of this method is proposed under the assumptions that Fréchet derivative of first order satisfies continuity condition of the Hölder. Finally, we consider some integral equation and boundary value problem (BVP) in order to illustrate the suitability of theoretical results. Full article
(This article belongs to the Special Issue Convergence and Dynamics of Iterative Methods: Chaos and Fractals)
17 pages, 1524 KiB  
Article
Fractals as Julia Sets of Complex Sine Function via Fixed Point Iterations
by Swati Antal, Anita Tomar, Darshana J. Prajapati and Mohammad Sajid
Fractal Fract. 2021, 5(4), 272; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5040272 - 14 Dec 2021
Cited by 11 | Viewed by 3156
Abstract
We explore some new variants of the Julia set by developing the escape criteria for a function sin(zn)+az+c, where a,cC, n2, and z is a [...] Read more.
We explore some new variants of the Julia set by developing the escape criteria for a function sin(zn)+az+c, where a,cC, n2, and z is a complex variable, utilizing four distinct fixed point iterative methods. Furthermore, we examine the impact of parameters on the deviation of dynamics, color, and appearance of fractals. Some of these fractals represent the stunning art on glass, and Rangoli (made in different parts of India, especially during the festive season) which are useful in interior decoration. Some fractals are similar to beautiful objects found in our surroundings like flowers (to be specific Hibiscus and Catharanthus Roseus), and ants. Full article
(This article belongs to the Special Issue Convergence and Dynamics of Iterative Methods: Chaos and Fractals)
Show Figures

Figure 1

17 pages, 2877 KiB  
Article
A New Approach for Dynamic Stochastic Fractal Search with Fuzzy Logic for Parameter Adaptation
by Marylu L. Lagunes, Oscar Castillo, Fevrier Valdez, Jose Soria and Patricia Melin
Fractal Fract. 2021, 5(2), 33; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5020033 - 17 Apr 2021
Cited by 8 | Viewed by 2031
Abstract
Stochastic fractal search (SFS) is a novel method inspired by the process of stochastic growth in nature and the use of the fractal mathematical concept. Considering the chaotic stochastic diffusion property, an improved dynamic stochastic fractal search (DSFS) optimization algorithm is presented. The [...] Read more.
Stochastic fractal search (SFS) is a novel method inspired by the process of stochastic growth in nature and the use of the fractal mathematical concept. Considering the chaotic stochastic diffusion property, an improved dynamic stochastic fractal search (DSFS) optimization algorithm is presented. The DSFS algorithm was tested with benchmark functions, such as the multimodal, hybrid, and composite functions, to evaluate the performance of the algorithm with dynamic parameter adaptation with type-1 and type-2 fuzzy inference models. The main contribution of the article is the utilization of fuzzy logic in the adaptation of the diffusion parameter in a dynamic fashion. This parameter is in charge of creating new fractal particles, and the diversity and iteration are the input information used in the fuzzy system to control the values of diffusion. Full article
(This article belongs to the Special Issue Convergence and Dynamics of Iterative Methods: Chaos and Fractals)
Show Figures

Figure 1

15 pages, 1542 KiB  
Article
Local Convergence and Dynamical Analysis of a Third and Fourth Order Class of Equation Solvers
by Debasis Sharma, Ioannis K. Argyros, Sanjaya Kumar Parhi and Shanta Kumari Sunanda
Fractal Fract. 2021, 5(2), 27; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5020027 - 05 Apr 2021
Cited by 4 | Viewed by 1803
Abstract
In this article, we suggest the local analysis of a uni-parametric third and fourth order class of iterative algorithms for addressing nonlinear equations in Banach spaces. The proposed local convergence is established using an ω-continuity condition on the first Fréchet derivative. In [...] Read more.
In this article, we suggest the local analysis of a uni-parametric third and fourth order class of iterative algorithms for addressing nonlinear equations in Banach spaces. The proposed local convergence is established using an ω-continuity condition on the first Fréchet derivative. In this way, the utility of the discussed schemes is extended and the application of Taylor expansion in convergence analysis is removed. Furthermore, this study provides radii of convergence balls and the uniqueness of the solution along with the calculable error distances. The dynamical analysis of the discussed family is also presented. Finally, we provide numerical explanations that show the suggested analysis performs well in the situation where the earlier approach cannot be implemented. Full article
(This article belongs to the Special Issue Convergence and Dynamics of Iterative Methods: Chaos and Fractals)
Show Figures

Figure 1

10 pages, 562 KiB  
Article
A Characterization of the Dynamics of Schröder’s Method for Polynomials with Two Roots
by Víctor Galilea and José M. Gutiérrez
Fractal Fract. 2021, 5(1), 25; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract5010025 - 20 Mar 2021
Cited by 1 | Viewed by 1799
Abstract
The purpose of this work is to give a first approach to the dynamical behavior of Schröder’s method, a well-known iterative process for solving nonlinear equations. In this context, we consider equations defined in the complex plane. By using topological conjugations, we characterize [...] Read more.
The purpose of this work is to give a first approach to the dynamical behavior of Schröder’s method, a well-known iterative process for solving nonlinear equations. In this context, we consider equations defined in the complex plane. By using topological conjugations, we characterize the basins of attraction of Schröder’s method applied to polynomials with two roots and different multiplicities. Actually, we show that these basins are half-planes or circles, depending on the multiplicities of the roots. We conclude our study with a graphical gallery that allow us to compare the basins of attraction of Newton’s and Schröder’s method applied to some given polynomials. Full article
(This article belongs to the Special Issue Convergence and Dynamics of Iterative Methods: Chaos and Fractals)
Show Figures

Figure 1

Back to TopTop