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Fractal Fract
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Fractional Dynamical Systems: Applications and Theoretical Results
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Fractional Dynamical Systems: Applications and Theoretical Results

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

Deadline for manuscript submissions: closed (20 March 2022) | Viewed by 30163

Special Issue Editors

Prof. Dr. Jehad Alzabut

E-Mail Website
Guest Editor
1. Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
2. Group of Mathematics, Faculty of Engineering, OSTIM Technical University, Ankara 06374, Turkey
Interests: mathematical models describing biological; medical and ecological phenomena; qualitative properties (oscillation, stability, periodicity, controllability, existence and uniqueness and chaos); differential equations; difference equations; delay differential equations; delay difference equations; impulsive differential equations; impulsive difference equations; dynamic equations on time scales; partial differential equations; partial difference equations; differential and difference equations of fractional order
Prof. Dr. Shahram Rezapour

E-Mail Website
Guest Editor
1. Department of Medical Research, China Medical University, Taichung City, Taiwan
2. Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz 94171-71946, Iran
Interests: approximation theory; fixed point theory; fractional differential equations and inclusions; fractional finite difference; modeling theory
Prof. Dr. George M. Selvam

E-Mail Website
Guest Editor
Department of Mathematics, Sacred Heart College, Tirupattur-635 601, India
Interests: discrete dynamical systems; fractional calculus; mathematical biology

Special Issue Information

Dear Colleagues,

The fractional dynamic is a field of study in mathematics and physics that investigates the behavior of objects and systems by using differentiations of fractional orders. Due to its widespread applications in science and technology, research within the fractional dynamical systems has led to new developments that have attracted the attention of a considerable audience of professionals such as mathematicians, physicists, applied researchers and practitioners. Unlike integer-order models, fractional-order models have the potential to capture nonlocal relations in time and space with power law memory kernels. This means that they provide more realistic and adequate descriptions for many real-world phenomena. In spite of the tremendous amount of published results focused on fractional differential equations and dynamical systems, we believe that many challenging open problems remain. Indeed, the theory and application of these systems are still very active areas of research.

The main objective of this Special Issue is to fill a void in the literature by making relevant information available for an important area of research. The Special Issue on “Fractional Dynamical Systems: Applications and Theoretical Results” provides an international forum for researchers to contribute with original research focusing on the latest achievements in the theory and application of fractional dynamical systems.

Prof. Dr. Jehad Alzabut
Prof. Dr. Shahram Rezapour
Prof. Dr. George M. Selvam
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

  • Fractional differential/difference equations
  • Fractional stability and control
  • Fractional Oscillation and boundedness
  • Fractional chaos and bifurcation
  • Fractional iterative methods and numerical computations
  • Fractional modelling and simulation
  • Fractional inequalities
  • Fractional stochastic analysis

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Published Papers (14 papers)

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Research

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18 pages, 387 KiB  
Article
Multi-AUV Dynamic Maneuver Countermeasure Algorithm Based on Interval Information Game and Fractional-Order DE
by Lu Liu, Jian Wang, Lichuan Zhang and Shuo Zhang
Fractal Fract. 2022, 6(5), 235; https://doi.org/10.3390/fractalfract6050235 - 25 Apr 2022
Cited by 39 | Viewed by 2627
Abstract
The instability of the underwater environment and underwater communication brings great challenges to the coordination and cooperation of the multi-Autonomous Underwater Vehicle (AUV). In this paper, a multi-AUV dynamic maneuver countermeasure algorithm is proposed based on the interval information game theory and fractional-order [...] Read more.
The instability of the underwater environment and underwater communication brings great challenges to the coordination and cooperation of the multi-Autonomous Underwater Vehicle (AUV). In this paper, a multi-AUV dynamic maneuver countermeasure algorithm is proposed based on the interval information game theory and fractional-order Differential Evolution (DE), in order to highlight the features of the underwater countermeasure. Firstly, an advantage function comprising the situation and energy efficiency advantages is proposed on account of the multi-AUV maneuver strategies. Then, the payoff matrix with interval information is established and the payment interval ranking is achieved based on relative entropy. Subsequently, the maneuver countermeasure model is presented along with the Nash equilibrium condition satisfying the interval information game. The fractional-order DE algorithm is applied for solving the established problem to determine the optimal strategy. Finally, the superiority of the proposed multi-AUV maneuver countermeasure algorithm is verified through an example. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
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16 pages, 667 KiB  
Article
On the Existence and Stability of a Neutral Stochastic Fractional Differential System
by Manzoor Ahmad, Akbar Zada, Mehran Ghaderi, Reny George and Shahram Rezapour
Fractal Fract. 2022, 6(4), 203; https://doi.org/10.3390/fractalfract6040203 - 6 Apr 2022
Cited by 53 | Viewed by 2284
Abstract
The main purpose of this paper is to investigate the existence and Ulam-Hyers stability (U-Hs) of solutions of a nonlinear neutral stochastic fractional differential system. We prove the existence and uniqueness of solutions to the proposed system by using fixed point theorems and [...] Read more.
The main purpose of this paper is to investigate the existence and Ulam-Hyers stability (U-Hs) of solutions of a nonlinear neutral stochastic fractional differential system. We prove the existence and uniqueness of solutions to the proposed system by using fixed point theorems and the Banach contraction principle. Also, by using fundamental schemes of fractional calculus, we study the (U-Hs) to the solutions of our suggested system. Besides, we study an example, best describing our main result. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
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12 pages, 1417 KiB  
Article
Fractal Analysis of Local Activity and Chaotic Motion in Nonlinear Nonplanar Vibrations for Cantilever Beams
by Mingming Zhang, Pan Kong, Anping Hou and Yuru Xu
Fractal Fract. 2022, 6(4), 181; https://doi.org/10.3390/fractalfract6040181 - 24 Mar 2022
Viewed by 1894
Abstract
Many problems in practical engineering can be simplified as the cantilever beam model, which is generally studied by theoretical analysis, experiment, and numerical simulation. This paper discusses the local activity of the nonlinear nonplanar motion of a cantilever beam at the equilibrium point. [...] Read more.
Many problems in practical engineering can be simplified as the cantilever beam model, which is generally studied by theoretical analysis, experiment, and numerical simulation. This paper discusses the local activity of the nonlinear nonplanar motion of a cantilever beam at the equilibrium point. Firstly, the equilibrium point of the model and the Jacobian matrix have been calculated. The stability of the characteristic root corresponding to the characteristic polynomial has been analyzed. Secondly, the corresponding complexity function of the model at the equilibrium point has been given. Then, the local activity region of the model at the equilibrium point can be obtained by using the theory of the local activity. Based on the actual engineering research background, the damping coefficient is generally taken as 0 < c < 1. The cantilever beam model is the local activity at the equilibrium point only if the parameters of the model satisfy a certain condition. In the numerical simulation, it is found that when the proper parameters are selected in the local activity region, the cantilever beam can exhibit different types of chaotic motion. The local activity theory provides a theoretical basis for the parameter selection of the chaotic motion in the cantilever beam. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
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12 pages, 312 KiB  
Article
Existence Results of Global Solutions for a Coupled Implicit Riemann-Liouville Fractional Integral Equation via the Vector Kuratowski Measure of Noncompactness
by Noura Laksaci, Ahmed Boudaoui, Wasfi Shatanawi and Taqi A. M. Shatnawi
Fractal Fract. 2022, 6(3), 130; https://doi.org/10.3390/fractalfract6030130 - 24 Feb 2022
Cited by 5 | Viewed by 1790
Abstract
The main goal of this study is to demonstrate an existence result of a coupled implicit Riemann-Liouville fractional integral equation. First, we prove a new fixed point theorem in spaces with an extended norm structure. That theorem generalized Darbo’s theorem associated with the [...] Read more.
The main goal of this study is to demonstrate an existence result of a coupled implicit Riemann-Liouville fractional integral equation. First, we prove a new fixed point theorem in spaces with an extended norm structure. That theorem generalized Darbo’s theorem associated with the vector Kuratowski measure of noncompactness. Second, we employ our obtained fixed point theorem to investigate the existence of solutions to the coupled implicit fractional integral equation on the generalized Banach space C([0,1],R)×C([0,1],R). Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
24 pages, 2664 KiB  
Article
A New Numerical Scheme for Time Fractional Diffusive SEAIR Model with Non-Linear Incidence Rate: An Application to Computational Biology
by Yasir Nawaz, Muhammad Shoaib Arif and Wasfi Shatanawi
Fractal Fract. 2022, 6(2), 78; https://doi.org/10.3390/fractalfract6020078 - 31 Jan 2022
Cited by 12 | Viewed by 2215
Abstract
In this paper, we propose a modified fractional diffusive SEAIR epidemic model with a nonlinear incidence rate. A constructed model of fractional partial differential equations (PDEs) is more general than the corresponding model of fractional ordinary differential equations (ODEs). The Caputo fractional derivative [...] Read more.
In this paper, we propose a modified fractional diffusive SEAIR epidemic model with a nonlinear incidence rate. A constructed model of fractional partial differential equations (PDEs) is more general than the corresponding model of fractional ordinary differential equations (ODEs). The Caputo fractional derivative is considered. Linear stability analysis of the disease-free equilibrium state of the epidemic model (ODEs) is presented by employing Routh–Hurwitz stability criteria. In order to solve this model, a fractional numerical scheme is proposed. The proposed scheme can be used to find conditions for obtaining positive solutions for diffusive epidemic models. The stability of the scheme is given, and convergence conditions are found for the system of the linearized diffusive fractional epidemic model. In addition to this, the deficiencies of accuracy and consistency in the nonstandard finite difference method are also underlined by comparing the results with the standard fractional scheme and the MATLAB built-in solver pdepe. The proposed scheme shows an advantage over the fractional nonstandard finite difference method in terms of accuracy. In addition, numerical results are supplied to evaluate the proposed scheme’s performance. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
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18 pages, 339 KiB  
Article
Some Double Generalized Weighted Fractional Integral Inequalities Associated with Monotone Chebyshev Functionals
by Gauhar Rahman, Saud Fahad Aldosary, Muhammad Samraiz and Kottakkaran Sooppy Nisar
Fractal Fract. 2021, 5(4), 275; https://doi.org/10.3390/fractalfract5040275 - 15 Dec 2021
Cited by 2 | Viewed by 1858
Abstract
In this manuscript, we study the unified integrals recently defined by Rahman et al. and present some new double generalized weighted type fractional integral inequalities associated with increasing, positive, monotone and measurable function F. Also, we establish some new double-weighted inequalities, which [...] Read more.
In this manuscript, we study the unified integrals recently defined by Rahman et al. and present some new double generalized weighted type fractional integral inequalities associated with increasing, positive, monotone and measurable function F. Also, we establish some new double-weighted inequalities, which are particular cases of the main result and are represented by corollaries. These inequalities are further refinement of all other inequalities associated with increasing, positive, monotone and measurable function existing in literature. The existing inequalities associated with increasing, positive, monotone and measurable function are also restored by applying specific conditions as given in Remarks. Many other types of fractional integral inequalities can be obtained by applying certain conditions on F and Ψ given in the literature. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
11 pages, 313 KiB  
Article
Existence and Uniqueness Results of Coupled Fractional-Order Differential Systems Involving Riemann–Liouville Derivative in the Space Wa+γ1,1(a,b)×Wa+γ2,1(a,b) with Perov’s Fixed Point Theorem
by Noura Laksaci, Ahmed Boudaoui, Kamaleldin Abodayeh, Wasfi Shatanawi and Taqi A. M. Shatnawi
Fractal Fract. 2021, 5(4), 217; https://doi.org/10.3390/fractalfract5040217 - 13 Nov 2021
Cited by 2 | Viewed by 1721
Abstract
This paper is devoted to studying the existence and uniqueness of a system of coupled fractional differential equations involving a Riemann–Liouville derivative in the Cartesian product of fractional Sobolev spaces [...] Read more.
This paper is devoted to studying the existence and uniqueness of a system of coupled fractional differential equations involving a Riemann–Liouville derivative in the Cartesian product of fractional Sobolev spaces E=Wa+γ1,1(a,b)×Wa+γ2,1(a,b). Our strategy is to endow the space E with a vector-valued norm and apply the Perov fixed point theorem. An example is given to show the usefulness of our main results. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
26 pages, 401 KiB  
Article
H-U-Type Stability and Numerical Solutions for a Nonlinear Model of the Coupled Systems of Navier BVPs via the Generalized Differential Transform Method
by Shahram Rezapour, Brahim Tellab, Chernet Tuge Deressa, Sina Etemad and Kamsing Nonlaopon
Fractal Fract. 2021, 5(4), 166; https://doi.org/10.3390/fractalfract5040166 - 13 Oct 2021
Cited by 31 | Viewed by 1997
Abstract
This paper is devoted to generalizing the standard system of Navier boundary value problems to a fractional system of coupled sequential Navier boundary value problems by using terms of the Caputo derivatives. In other words, for the first time, we design a multi-term [...] Read more.
This paper is devoted to generalizing the standard system of Navier boundary value problems to a fractional system of coupled sequential Navier boundary value problems by using terms of the Caputo derivatives. In other words, for the first time, we design a multi-term fractional coupled system of Navier equations under the fractional boundary conditions. The existence theory is studied regarding solutions of the given coupled sequential Navier boundary problems via the Krasnoselskii’s fixed-point theorem on two nonlinear operators. Moreover, the Banach contraction principle is applied to investigate the uniqueness of solution. We then focus on the Hyers–Ulam-type stability of its solution. Furthermore, the approximate solutions of the proposed coupled fractional sequential Navier system are obtained via the generalized differential transform method. Lastly, the results of this research are supported by giving simulated examples. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
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12 pages, 293 KiB  
Article
Existence of the Class of Nonlinear Hybrid Fractional Langevin Quantum Differential Equation with Dirichlet Boundary Conditions
by Nagamanickam Nagajothi, Vadivel Sadhasivam, Omar Bazighifan and Rami Ahmad El-Nabulsi
Fractal Fract. 2021, 5(4), 156; https://doi.org/10.3390/fractalfract5040156 - 8 Oct 2021
Cited by 7 | Viewed by 1469
Abstract
In this paper, we investigate the existence results for nonlinear fractional q-difference equations with two different fractional orders supplemented with the Dirichlet boundary conditions. Our main existence results are obtained by applying the contraction mapping principle and Krasnoselskii’s fixed point theorem. An illustrative [...] Read more.
In this paper, we investigate the existence results for nonlinear fractional q-difference equations with two different fractional orders supplemented with the Dirichlet boundary conditions. Our main existence results are obtained by applying the contraction mapping principle and Krasnoselskii’s fixed point theorem. An illustrative example is also discussed. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
14 pages, 320 KiB  
Article
An Existence Result for ψ-Hilfer Fractional Integro-Differential Hybrid Three-Point Boundary Value Problems
by Chanakarn Kiataramkul, Sotiris K. Ntouyas and Jessada Tariboon
Fractal Fract. 2021, 5(4), 136; https://doi.org/10.3390/fractalfract5040136 - 24 Sep 2021
Cited by 4 | Viewed by 1618
Abstract
In this research work, we study a new class of ψ-Hilfer hybrid fractional integro-differential boundary value problems with three-point boundary conditions. An existence result is established by using a generalization of Krasnosel’skiĭ’s fixed point theorem. An example illustrating the main result is [...] Read more.
In this research work, we study a new class of ψ-Hilfer hybrid fractional integro-differential boundary value problems with three-point boundary conditions. An existence result is established by using a generalization of Krasnosel’skiĭ’s fixed point theorem. An example illustrating the main result is also constructed. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
17 pages, 342 KiB  
Article
A Note on Existence of Mild Solutions for Second-Order Neutral Integro-Differential Evolution Equations with State-Dependent Delay
by Shahram Rezapour, Hernán R. Henríquez, Velusamy Vijayakumar, Kottakkaran Sooppy Nisar and Anurag Shukla
Fractal Fract. 2021, 5(3), 126; https://doi.org/10.3390/fractalfract5030126 - 17 Sep 2021
Cited by 16 | Viewed by 2739
Abstract
This article is mainly devoted to the study of the existence of solutions for second-order abstract non-autonomous integro-differential evolution equations with infinite state-dependent delay. In the first part, we are concerned with second-order abstract non-autonomous integro-differential retarded functional differential equations with infinite state-dependent [...] Read more.
This article is mainly devoted to the study of the existence of solutions for second-order abstract non-autonomous integro-differential evolution equations with infinite state-dependent delay. In the first part, we are concerned with second-order abstract non-autonomous integro-differential retarded functional differential equations with infinite state-dependent delay. In the second part, we extend our results to study the second-order abstract neutral integro-differential evolution equations with state-dependent delay. Our results are established using properties of the resolvent operator corresponding to the second-order abstract non-autonomous integro-differential equation and fixed point theorems. Finally, an application is presented to illustrate the theory obtained. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
20 pages, 949 KiB  
Article
Qualitative Study on Solutions of a Hadamard Variable Order Boundary Problem via the Ulam–Hyers–Rassias Stability
by Amar Benkerrouche, Mohammed Said Souid, Sina Etemad, Ali Hakem, Praveen Agarwal, Shahram Rezapour, Sotiris K. Ntouyas and Jessada Tariboon
Fractal Fract. 2021, 5(3), 108; https://doi.org/10.3390/fractalfract5030108 - 2 Sep 2021
Cited by 27 | Viewed by 2196
Abstract
In this paper, the existence, uniqueness and stability of solutions to a boundary value problem of nonlinear FDEs of variable order are established. To do this, we first investigate some aspects of variable order operators of Hadamard type. Then, with the help of [...] Read more.
In this paper, the existence, uniqueness and stability of solutions to a boundary value problem of nonlinear FDEs of variable order are established. To do this, we first investigate some aspects of variable order operators of Hadamard type. Then, with the help of the generalized intervals and piecewise constant functions, we convert the variable order Hadamard FBVP to an equivalent standard Hadamard BVP of the fractional constant order. Further, two fixed point theorems due to Schauder and Banach are used and, finally, the Ulam–Hyers–Rassias stability of the given variable order Hadamard FBVP is examined. These results are supported with the aid of a comprehensive example. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
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14 pages, 298 KiB  
Article
Synchronization Analysis of Multiple Integral Inequalities Driven by Steklov Operator
by Wedad Albalawi and Zareen A. Khan
Fractal Fract. 2021, 5(3), 97; https://doi.org/10.3390/fractalfract5030097 - 14 Aug 2021
Cited by 1 | Viewed by 1628
Abstract
We construct a subclass of Copson’s integral inequality in this article. In order to achieve this goal, we attempt to use the Steklov operator for generalizing different inequalities of the Copson type relevant to the situations ρ>1 as well as [...] Read more.
We construct a subclass of Copson’s integral inequality in this article. In order to achieve this goal, we attempt to use the Steklov operator for generalizing different inequalities of the Copson type relevant to the situations ρ>1 as well as ρ<1. We demonstrate the inequalities with the guidance of basic comparison, Holder’s inequality, and the integration by parts approach. Moreover, some new variations of Hardy’s integral inequality are also presented with the utilization of Steklov operator. We also formulate many remarks and two examples to show the novelty and authenticity of our results. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)

Review

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49 pages, 537 KiB  
Review
Oscillation Results for Solutions of Fractional-Order Differential Equations
by Jehad Alzabut, Ravi P. Agarwal, Said R. Grace and Jagan M. Jonnalagadda
Fractal Fract. 2022, 6(9), 466; https://doi.org/10.3390/fractalfract6090466 - 25 Aug 2022
Cited by 6 | Viewed by 1247
Abstract
This survey paper is devoted to succinctly reviewing the recent progress in the field of oscillation theory for linear and nonlinear fractional differential equations. The paper provides a fundamental background for all interested researchers who would like to contribute to this topic. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
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