Search

Few papers have been published to date regarding the stability of neural networks described by fractional difference operators. This paper makes a contribution to the topic by presenting a variable-order fractional discrete neural network model and by proving its Ulam–Hyers stability. In particular, two novel theorems are illustrated, one regarding the existence of the solution for the proposed variable-order network and the other regarding its Ulam–Hyers stability. Finally, numerical simulations of three-dimensional and two-dimensional variable-order fractional neural networks were carried out to highlight the effectiveness of the conceived theoretical approach. [ABSTRACT FROM AUTHOR]

The study of the chaotic dynamics in fractional-order discrete-time systems has received great attention in the past years. In this paper, we propose a new 2D fractional map with the simplest algebraic structure reported to date and with an infinite line of equilibrium. The conceived map possesses an interesting property not explored in literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of periodic, chaotic and hyper-chaotic attractors. Bifurcation diagrams, computation of the maximum Lyapunov exponents, phase plots and 0–1 test are reported, with the aim to analyse the dynamics of the 2D fractional map as well as to highlight the coexistence of initial-boosting chaotic and hyperchaotic attractors in commensurate and incommensurate order. Results show that the 2D fractional map has an infinite number of coexistence symmetrical chaotic and hyper-chaotic attractors. Finally, the complexity of the fractional-order map is investigated in detail via approximate entropy. [ABSTRACT FROM AUTHOR]

Given the recent advances regarding the studies of discrete fractional calculus, and the fact that the dynamics of discrete-time neural networks in fractional variable-order cases have not been sufficiently documented, herein, we consider a novel class of discrete-time fractional-order neural networks using discrete nabla operator of variable-order. An adequate criterion for the existence of the solution in addition to its uniqueness for such systems is provided with the use of Banach fixed point technique. Moreover, the uniform stability is investigated. We provide at the end two numerical simulations illustrating the relevance of the aforementioned results. [ABSTRACT FROM AUTHOR]

The Selkov system, which is typically employed to model glycolysis phenomena, unveils some rich dynamics and some other complex formations in biochemical reactions. In the present work, the synchronization problem of the glycolysis reaction-diffusion model is handled and examined. In addition, a novel convenient control law is designed in a linear form and, on the other hand, the stability of the associated error system is demonstrated through utilizing a suitable Lyapunov function. To illustrate the applicability of the proposed schemes, several numerical simulations are performed in one- and two-spatial dimensions. [ABSTRACT FROM AUTHOR]

This paper proceeds from the perspective that most strongly nonlinear oscillators of fractional-order do not enjoy exact analytical solutions. Undoubtedly, this is a good enough reason to employ one of the major recent approximate methods, namely an Optimal Homotopy Asymptotic Method (OHAM), to offer approximate analytic solutions for two strongly fractional-order nonlinear benchmark oscillatory problems, namely: the fractional-order Duffing-relativistic oscillator and the fractional-order stretched elastic wire oscillator (with a mass attached to its midpoint). In this work, a further modification has been proposed for such method and then carried out through establishing an optimal auxiliary linear operator, an auxiliary function, and an auxiliary control parameter. In view of the two aforesaid applications, it has been demonstrated that the OHAM is a reliable approach for controlling the convergence of approximate solutions and, hence, it is an effective tool for dealing with such problems. This assertion is completely confirmed by performing several graphical comparisons between the OHAM and the Homotopy Analysis Method (HAM). [ABSTRACT FROM AUTHOR]

In this paper, we introduce new three-point fractional formulas which represent three generalizations for the well-known classical three-point formulas; central, forward and backward formulas. This has enabled us to study the function's behavior according to different fractional-order values of α numerically. Accordingly, we then introduce a new methodology for Richardson extrapolation depending on the fractional central formula in order to obtain a high accuracy for the gained approximations. We compare the efficiency of the proposed methods by using tables and figures to show their reliability. [ABSTRACT FROM AUTHOR]

Fractional-order systems have proved to be accurate in describing the spread of the COVID-19 pandemic by virtue of their capability to include the memory effects into the system dynamics. This manuscript presents a novel fractional discrete-time COVID-19 model that includes the number of vaccinated individuals as an additional state variable in the system equations. The paper shows that the proposed compartment model, described by difference equations, has two fixed points, i.e., a disease-free fixed point and an epidemic fixed point. A new theorem is proven which highlights that the pandemic disappears when an inequality involving the percentage of the population in quarantine is satisfied. Finally, numerical simulations are carried out to show that the proposed incommensurate fractional-order model is effective in describing the spread of the COVID-19 pandemic. [ABSTRACT FROM AUTHOR]

This work investigates the dynamics of discrete reaction-diffusion Gierer-Meinhardt system as mathematical models of biological pattern formation. We study the system's local asymptotic behaviour with and without the diffusion once developing the discrete integer variant of the well-known Gierer-Meinhardt model and proving that the model has a unique equilibrium. The requirements for the steady-state solution's local and global stability are found with the help of relevant approaches and the Lyapunov technique. Two large biological models and simulations are used throughout the work to validate the utility of the suggested technique. [ABSTRACT FROM AUTHOR]

This paper makes a significant contribution by focusing on estimating the coefficients of a sample of non-linear time series, a subject well-established in the statistical literature, using bilinear time series. Specifically, this study delves into a subset of bilinear models where Generalized Autoregressive Conditional Heteroscedastic (GARCH) models serve as the white noise component. The methodology involves applying the Klimko–Nilsen theorem, which plays a crucial role in extracting the asymptotic behavior of the estimators. In this context, the Generalized Autoregressive Conditional Heteroscedastic model of order (1,1) noted that the GARCH (1,1) model is defined as the white noise for the coefficients of the example models. Notably, this GARCH model satisfies the condition of having time-varying coefficients. This study meticulously outlines the essential stationarity conditions required for these models. The estimation of coefficients is accomplished by applying the least squares method. One of the key contributions lies in utilizing the fundamental theorem of Klimko and Nilsen, to prove the asymptotic behavior of the estimators, particularly how they vary with changes in the sample size. This paper illuminates the impact of estimators and their approximations based on varying sample sizes. Extending our study to include the estimation of bilinear models alongside GARCH and GARCH symmetric coefficients adds depth to our analysis and provides valuable insights into modeling financial time series data. Furthermore, this study sheds light on the influence of the GARCH white noise trace on the estimation of model coefficients. The results establish a clear connection between the model characteristics and the nature of the white noise, contributing to a more profound understanding of the relationship between these elements. [ABSTRACT FROM AUTHOR]

The aim of this research is to demonstrate the existence and the uniqueness of the weak solution for a semilinear fractional parabolic problem with the special case of the second integral boundary condition. For this aim, we split the proof into two parts; to study the main linear problem part, we used the variable separation method, and concerning the semilinear problem part, we apply an iterative method and a priori estimate for the study of the weak solution. [ABSTRACT FROM AUTHOR]

To follow up on the progress made on exploring the stability investigation of linear commensurate Fractional-order Difference Systems (FoDSs), such topic of its extended version that appears with incommensurate orders is discussed and examined in this work. Some simple applicable conditions for judging the stability of these systems are reported as novel results. These results are formulated by converting the linear incommensurate FoDS into another equivalent system consists of fractional-order difference equations of Volterra convolution-type as well as by using some properties of the Z-transform method. All results of this work are verified numerically by illustrating some examples that deal with the stability of solutions of such systems. [ABSTRACT FROM AUTHOR]

The aim of this research is to demonstrate the existence and the uniqueness of the weak solution for a semilinear fractional parabolic problem with the special case of the second integral boundary condition. For this aim, we split the proof into two parts; to study the main linear problem part, we used the variable separation method, and concerning the semilinear problem part, we apply an iterative method and a priori estimate for the study of the weak solution. [ABSTRACT FROM AUTHOR]

The solvability of the fractional partial differential equation with integral overdetermination condition for an inverse problem is investigated in this paper. We analyze the direct problem solution by using the "energy inequality" method. Using the fixed point technique, the existence and uniqueness of the solution of the inverse problem on the data are established. [ABSTRACT FROM AUTHOR]

Recently, hidden attractors with stable equilibria have received considerable attention in chaos theory and nonlinear dynamical systems. Based on discrete fractional calculus, this paper proposes a simple two-dimensional and three-dimensional fractional maps. Both fractional maps are chaotic and have a unique equilibrium point. Results show that the dynamics of the proposed fractional maps are sensitive to both initial conditions and fractional order. There are coexisting attractors which have been displayed in terms of bifurcation diagrams, phase portraits and a 0-1 test. Furthermore, control schemes are introduced to stabilize the chaotic trajectories of the two novel systems. [ABSTRACT FROM AUTHOR]

In this paper, we propose two new two-dimensional chaotic maps with closed curve fixed points. The chaotic behavior of the two maps is analyzed by the 0–1 test, and explored numerically using Lyapunov exponents and bifurcation diagrams. It has been found that chaos exists in both fractional maps. In addition, result shows that the proposed fractional maps shows the property of coexisting attractors. [ABSTRACT FROM AUTHOR]

In this paper, we propose three fractional chaotic maps based on the well known 3D Stefanski, Rössler, and Wang maps. The dynamics of the proposed fractional maps are investigated experimentally by means of phase portraits, bifurcation diagrams, and Lyapunov exponents. In addition, three control laws are introduced for these fractional maps and the convergence of the controlled states towards zero is guaranteed by means of the stability theory of linear fractional discrete systems. Furthermore, a combined synchronization scheme is introduced whereby the fractional Rössler map is considered as a drive system with the response system being a combination of the remaining two maps. Numerical results are presented throughout the paper to illustrate the findings. [ABSTRACT FROM AUTHOR]

Today, the entire world is witnessing an enormous upsurge in coronavirus pandemic (COVID-19 pandemic). Confronting such acute infectious disease, which has taken multiple victims around the world, requires all specialists in all fields to devote their efforts to seek effective treatment or even control its disseminate. In the light of this aspect, this work proposes two new fractional-order versions for one of the recently extended forms of the SEIR model. These two versions, which are established in view of two fractional-order differential operators, namely, the Caputo and the Caputo–Fabrizio operators, are numerically solved based on the Generalized Euler Method (GEM) that considers Caputo sense, and the Adams–Bashforth Method (ABM) that considers Caputo–Fabrizio sense. Several numerical results reveal the impact of the fractional-order values on the two established disease models, and the continuation of the COVID-19 pandemic outbreak to this moment. In the meantime, some novel results related to the stability analysis and the basic reproductive number are addressed for the proposed fractional-order Caputo COVID-19 model. For declining the total of individuals infected by such pandemic, a new compartment is added to the proposed model, namely the disease prevention compartment that includes the use of face masks, gloves and sterilizers. In view of such modification, it is turned out that the performed addition to the fractional-order Caputo COVID-19 model yields a significant improvement in reducing the risk of COVID-19 spreading. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to examine the inverse problem associated with determining the right-hand side of a nonlinear fractional parabolic equation. This equation is accompanied by an integral over-determination supplementary condition. With the use of the functional analysis method, we establish the continuity, existence and uniqueness based on the construction of the direct problem. Such a method relies on the density of the range of the operator established for the problem at hand coupled with the energy inequality scheme. This scheme, also referred to as the method of a priori estimates, allows us to derive the existence theorem from the solution of the given problem, starting with the uniqueness theorem. For the solvability of the inverse problem and its uniqueness, we establish certain suitable conditions, and to demonstrate the existence and uniqueness of its solution, we utilize the fixed point theorem. [ABSTRACT FROM AUTHOR]

Recently, many nonlinear systems have been proposed to introduce the population dynamics of COVID-19. In this paper, we extend different physical conditions of the growth by employing fractional calculus. We propose a new fractional-order version for one of recently forms of the SEIR model. This version, which is established in view of the Caputo fractional-order differential operator, is numerically solved based on the Generalized Euler Method (GEM). Several numerical results reveal the impact of the fractional-order values on the established disease model. To help make a decline in the total of individuals infected by such pandemic, a new compartment is added to the proposed model; namely, the disease prevention compartment that includes the use of face masks, gloves and sterilizers. In view of such modification, it turned out that the performed addition to the fractional-order COVID-19 model yields a significant improvement in reducing the risk of COVID-19 spread. [ABSTRACT FROM AUTHOR]

In this paper, the problem of synchronization of a class of spatiotemporal fractional-order partial differential systems is studied. Subject to homogeneous Neumann boundary conditions and using fractional Lyapunov approach, nonlinear and linear control schemes have been proposed to synchronize coupled general fractional reaction–diffusion systems. As a numerical application, we investigate complete synchronization behaviors of coupled fractional Lengyel–Epstein systems. [ABSTRACT FROM AUTHOR]

This paper studies the synchronisation of integer- and fractional-order discrete-time chaotic systems with different dimensions. Control laws are proposed for the full-state hybrid projective synchronisation (FSHPS) of a master–slave pair, where the difference equations of the master have an integer order while those of the slave have a fractional order. Moreover, inverse FSHPS laws are proposed for a fractional-order master and an integer-order slave. The Lyapunov stability theory of integer-order maps and the stability theory of linear fractional-order maps are utilised to establish the asymptotic stability of the zero equilibrium corresponding to the synchronisation error system. Numerical results are presented to verify the findings of the study. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fractional form of a new three-dimensional generalized Hénon map and study the existence of chaos and its control. Using bifurcation diagrams, phase portraits and Lyapunov exponents, we show that the general behavior of the proposed fractional map depends on the fractional order. We also present two control schemes for the proposed map, one that adaptively stabilizes the fractional map, and another to achieve the synchronization of the proposed fractional map. [ABSTRACT FROM AUTHOR]

A map without equilibrium has been proposed and studied in this paper. The proposed map has no fixed point and exhibits chaos. We have investigated its dynamics and shown its chaotic behavior using tools such as return map, bifurcation diagram and Lyapunov exponents' diagram. Entropy of this new map has been calculated. Using an open micro-controller platform, the map is implemented, and experimental observation is presented. In addition, two control schemes have been proposed to stabilize and synchronize the chaotic map. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a fractional order chaotic map corresponding to a recently developed standard map that exhibits a chaotic behavior with no fixed point. This is the first study to explore a fractional chaotic map without a fixed point. In our investigation, we use phase plots and bifurcation diagrams to examine the dynamics of the fractional map and assess the effect of varying the fractional order. We also use the approximate entropy measure to quantify the level of chaos in the fractional map. In addition, we propose a one-dimensional stabilization controller and establish its asymptotic convergence by means of the linearization method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the co-existence of different synchronization types for fractional-order discrete-time chaotic systems with different dimensions. In particular, we show that through appropriate nonlinear control, projective synchronization (PS), full state hybrid projective synchronization (FSHPS), and generalized synchronization (GS) can be achieved simultaneously. A second nonlinear control scheme is developed whereby inverse full state hybrid projective synchronization (IFSHPS) and inverse generalized synchronization (IGS) are shown to co-exist. Numerical examples are presented to confirm the findings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two approaches to the generalized synchronized synchronization and the inverse generalized synchronization of fractional discrete-time chaotic systems with non-identical dimensions. The convergence of the proposed approaches is established by means of recently developed stability theory. Numerical results are presented based on well-known maps in the literature. Two examples are considered: a 3D generalized synchronization and a 2D inverse generalized synchronization. [ABSTRACT FROM AUTHOR]

Social media occupies an important place in people's daily lives where users share various contents and topics such as thoughts, experiences, events and feelings. The massive use of social media has led to the generation of huge volumes of data. These data constitute a treasure trove, allowing the extraction of high volumes of relevant information particularly by involving deep learning techniques. Based on this context, various research studies have been carried out with the aim of studying the detection of mental disorders, notably depression and anxiety, through the analysis of data extracted from the Twitter platform. However, although these studies were able to achieve very satisfactory results, they nevertheless relied mainly on binary classification models by treating each mental disorder separately. Indeed, it would be better if we managed to develop systems capable of dealing with several mental disorders at the same time. To address this point, we propose a well-defined methodology involving the use of deep learning to develop effective multi-class models for detecting both depression and anxiety disorders through the analysis of tweets. The idea consists in testing a large number of deep learning models ranging from simple to hybrid variants to examine their strengths and weaknesses. Moreover, we involve the grid search technique to help find suitable values for the learning rate hyper-parameter due to its importance in training models. Our work is validated through several experiments and comparisons by considering various datasets and other binary classification models. The aim is to show the effectiveness of both the assumptions used to collect the data and the use of multi-class models rather than binary class models. Overall, the results obtained are satisfactory and very competitive compared to related works. [ABSTRACT FROM AUTHOR]

Because they are useful for both enabling numerical simulations and containing well-defined physical phenomena, discrete fractional reaction–diffusion models have attracted a great deal of interest from academics. Within the family of fractional reaction–diffusion models, a discrete form is examined in detail in this study. Furthermore, we investigate the complex synchronization dynamics of a suggested discrete master–slave reaction–diffusion system using the accuracy of linear control techniques combined with a fractional discrete Lyapunov approach. This study's deviation from the behavior of equivalents with integer orders makes it very fascinating. Like the non-local nature inherent in Caputo fractional derivatives, it creates a memory Lyapunov function that is closely linked to the historical background of the system. The investigation provides a strong basis to the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is interested in studying the one-phase Stefan problem. For this purpose, we use the nonlinear sentinel method, which relies typically on the approximate controllability and the Fanchel-Rockafellar duality of the minimization problem, to prove the existence and uniqueness of a solution to this problem. In particular, our research focuses on the application of the nonlinear sentinel method to the single-phase Stefan problem. This approach aids in identifying an unspecified boundary section within the domain undergoing a liquid-solid phase transition. We track the evolution of the temperature profile in the liquid-solid material and the corresponding movement of its interface over time. Eventually, the local convergence used for the iterative numerical scheme is demonstrated. [ABSTRACT FROM AUTHOR]

The Lozi map is well-known and has been studied in various researches. By combining three research trends (discrete map, memristor and fractional calculus) we investigate a fractional memristive Lozi map in this work. Firstly the Grunwald–Letnikov fractional difference operator is used to introduce the new fractional map with no equilibrium point. Then, the coexistence of several chaotic hidden attractors is shown, along with the coexistence of a number of bifurcations, depending on the values of the initial conditions. We found attractive dynamics and characteristics of this fractional Lozi map. The realization with hardware platform illustrates the map's feasibility. [ABSTRACT FROM AUTHOR]

This study considers the problem of control and synchronization between fractional‐order and integer‐order, N‐components reaction‐diffusion systems with nonidentical coefficients and different nonlinear parts. The control scheme is designed using the Lyapunov direct method. The results are exemplified by two significant biochemical models, namely, the fractional‐order Lengyel‐Epstein model and the Gray‐Scott model. To illustrate the effectiveness of the proposed scheme, numerical simulations are performed in one and two space dimensions using Homotopy Analysis Method (HAM). [ABSTRACT FROM AUTHOR]

Some endeavors have been recently dedicated to explore the dynamic properties of the fractional-order discrete-time chaotic systems. To date, attention has been mainly focused on fractional-order discrete-time systems with "self-excited attractors." This paper makes a contribution to the topic of fractional-order discrete-time systems with "hidden attractors" by presenting a new 2-dimensional discrete-time system without equilibrium points. The conceived system possesses an interesting property not explored in the literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of chaotic attractors. Bifurcation diagrams, computation of the largest Lyapunov exponents, phase plots, and the 0-1 test method are reported, with the aim to analyze the dynamics of the system, as well as to highlight the coexistence of chaotic attractors. Finally, an entropy algorithm is used to measure the complexity of the proposed system. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fractional map based on the integer-order unified map. The chaotic behavior of the proposed map is analyzed by means of bifurcations plots, and experimental bounds are placed on the parameters and fractional order. Different control laws are proposed to force the states to zero asymptotically and to achieve the complete synchronization of a pair of fractional unified maps with identical or nonidentical parameters. Numerical results are used throughout the paper to illustrate the findings. [ABSTRACT FROM AUTHOR]

Using fractional difference equations to describe fractional and variable-order maps, this manuscript discusses the dynamics of the discrete 4D sinusoidal feedback sine iterative chaotic map with infinite collapse (ICMIC) modulation map (SF-SIMM) with fractional-order. Also, it presents a novel variable-order version of SF-SIMM and discusses their chaotic dynamic behavior by employing a distinct function for the variable fractional-order. To establish the existence of chaos in the suggested discrete SF-SIMM, some numerical methods such as phase plots, bifurcation and largest Lyapunov exponent diagrams,  C 0  complexity and 0–1 test are utilized. After that, two different control schemes are used for the conceived discrete system. The states are stabilized and asymptotically forced towards zero by the first controller. The second controller is used to synchronize a pair of maps with non–identical parameters. Finally, MATLAB simulations will be executed to confirm the results provided. [ABSTRACT FROM AUTHOR]

In this study, we expand a 2D sine map via adding the discrete memristor to introduce a new 3D fractional-order sine-based memristor map. Under commensurate and incommensurate orders, we conduct an extensive exploration and analysis of its nonlinear dynamic behaviors, employing diverse numerical techniques, such as analyzing Lyapunov exponents, visualizing phase portraits, and plotting bifurcation diagrams. The results emphasize the sine-based memristor map's sensitivity to fractional-order parameters, resulting in the emergence of distinct and diverse dynamic patterns. In addition, we employ the sample entropy ( S a m p E n ) method and C 0 complexity to quantitatively measure complexity, and we also utilize the 0–1 test to validate the presence of chaos in the proposed fractional-order sine-based memristor map. Finally, MATLAB simulations are be executed to confirm the results provided. [ABSTRACT FROM AUTHOR]

This paper considers the dynamical properties of a space and time discrete fractional reaction–diffusion epidemic model, introducing a novel generalized incidence rate. The linear stability of the equilibrium solutions of the considered discrete fractional reaction–diffusion model has been carried out, and a global asymptotic stability analysis has been undertaken. We conducted a global stability analysis using a specialized Lyapunov function that captures the system's historical data, distinguishing it from the integer-order version. This approach significantly advanced our comprehension of the complex stability properties within discrete fractional reaction–diffusion epidemic models. To substantiate the theoretical underpinnings, this paper is accompanied by numerical examples. These examples serve a dual purpose: not only do they validate the theoretical findings, but they also provide illustrations of the practical implications of the proposed discrete fractional system. [ABSTRACT FROM AUTHOR]

This paper introduces a novel fractional Ikeda-based memristor map and investigates its non-linear dynamics under commensurate and incommensurate orders using various numerical techniques, including Lyapunov exponent analysis, phase portraits, and bifurcation diagrams. The results reveal diverse and complex system behaviors arising from the interplay of different fractional orders in the proposed map. Furthermore, the study employs the sample entropy test to quantify complexity and validate the presence of chaos. Non-linear controllers are also presented to stabilize and synchronize the model. The research emphasizes the system's sensitivity to the fractional order parameters, leading to distinct dynamic patterns and stability regimes. The memristor-based chaotic map exhibits rich and intricate behavior, making it an interesting and important area of research. [ABSTRACT FROM AUTHOR]

In this paper, we present an innovative 3D fractional Hénon-based memristor map and conduct an extensive exploration and analysis of its dynamic behaviors under commensurate and incommensurate orders. The study employs diverse numerical techniques, such as visualizing phase portraits, analyzing Lyapunov exponents, plotting bifurcation diagrams, and applying the sample entropy test to assess the complexity and validate the chaotic characteristics. However, since the proposed fractional map has no fixed points, the outcomes reveal that the map can exhibit a wide range of hidden dynamical behaviors. This phenomenon significantly augments the complexity of the fractal structure inherent to the chaotic attractors. Moreover, we introduce nonlinear controllers designed for stabilizing and synchronizing the proposed fractional Hénon-based memristor map. The research emphasizes the system's sensitivity to fractional-order parameters, resulting in the emergence of distinct dynamic patterns. The memristor-based chaotic map exhibits rich and intricate behavior, making it a captivating and significant area of investigation. [ABSTRACT FROM AUTHOR]

The aim of this work is to describe the dynamics of a discrete fractional-order reaction–diffusion FitzHugh–Nagumo model. We established acceptable requirements for the local asymptotic stability of the system's unique equilibrium. Moreover, we employed a Lyapunov functional to show that the constant equilibrium solution is globally asymptotically stable. Furthermore, numerical simulations are shown to clarify and exemplify the theoretical results. [ABSTRACT FROM AUTHOR]

Discrete fractional models with reaction-diffusion have gained significance in the scientific field in recent years, not only due to the need for numerical simulation but also due to the stated biological processes. In this paper, we investigate the problem of synchronization-control in a fractional discrete nonlinear bacterial culture reaction-diffusion model using the Caputo h-difference operator and a second-order central difference scheme and an L1 finite difference scheme after deriving the discrete fractional version of the well-known Degn–Harrison system and Lengyel–Epstein system. Using appropriate techniques and the direct Lyapunov method, the conditions for full synchronization are determined.Furthermore, this research shows that the L1 finite difference scheme and the second-order central difference scheme may successfully retain the properties of the related continuous system. The conclusions are proven throughout the paper using two major biological models, and numerical simulations are carried out to demonstrate the practical use of the recommended technique. [ABSTRACT FROM AUTHOR]

Variable-order fractional discrete calculus is a new and unexplored part of calculus that provides extraordinary capabilities for simulating multidisciplinary processes. Recognizing this incredible potential, the scientific community has been researching variable-order fractional discrete calculus applications to the modeling of engineering and physical systems. This research makes a contribution to the topic by describing and establishing the first generalized discrete fractional variable order Gronwall inequality that we employ to examine the finite time stability of nonlinear Nabla fractional variable-order discrete neural networks. This is followed by a specific version of a generalized variable-order fractional discrete Gronwall inequality described using discrete Mittag–Leffler functions. A specific version of a generalized variable-order fractional discrete Gronwall inequality represented using discrete Mittag–Leffler functions is shown. As an application, utilizing the contracting mapping principle and inequality approaches, sufficient conditions are developed to assure the existence, uniqueness, and finite-time stability of the equilibrium point of the suggested neural networks. Numerical examples, as well as simulations, are provided to show how the key findings can be applied. [ABSTRACT FROM AUTHOR]

The study of the blow-up phenomenon for fractional reaction–diffusion problems is generally deemed of great importance in dealing with several situations that impact our daily lives, and it is applied in many areas such as finance and economics. In this article, we expand on some previous blow-up results for the explicit values and numerical simulation of finite-time blow-up solutions for a semilinear fractional partial differential problem involving a positive power of the solution. We show the behavior solution of the fractional problem, and the numerical solution of the finite-time blow-up solution is also considered. Finally, some illustrative examples and comparisons with the classical problem with integer order are presented, and the validity of the results is demonstrated. [ABSTRACT FROM AUTHOR]

In the last few years, reaction–diffusion models associated with discrete fractional calculus have risen in prominence in scientific fields, not just due to the requirement for numerical simulation but also due to the described biological phenomena. This work investigates a discrete equivalent of the fractional reaction–diffusion glycolysis model. The discrete fractional calculus tool is introduced to the discrete modeling of diffusion problems in the Caputo-like delta sense, and a fractional discretization diffusion model is described. The local stability of the equilibrium points in the proposed discrete system is examined. We additionally investigate the global stability of the equilibrium point by developing a Lyapunov function. Furthermore, this study indicates that the L1 finite difference scheme and the second-order central difference scheme can successfully preserve the characteristics of the associated continuous system. Finally, an equivalent summation representing the model's numerical formula is shown. The diffusion concentration is further investigated for different fractional orders, and examples with simulations are presented to corroborate the theoretical findings. [ABSTRACT FROM AUTHOR]

The paper introduces a novel two-dimensional fractional discrete-time predator–prey Leslie–Gower model with an Allee effect on the predator population. The model's nonlinear dynamics are explored using various numerical techniques, including phase portraits, bifurcations and maximum Lyapunov exponent, with consideration given to both commensurate and incommensurate fractional orders. These techniques reveal that the fractional-order predator–prey Leslie–Gower model exhibits intricate and diverse dynamical characteristics, including stable trajectories, periodic motion, and chaotic attractors, which are affected by the variance of the system parameters, the commensurate fractional order, and the incommensurate fractional order. Finally, we employ the 0–1 method, the approximate entropy test and the  C 0  algorithm to measure complexity and confirm chaos in the proposed system. [ABSTRACT FROM AUTHOR]

In this paper, we prove the common fixed point theorems for weakly compatible mappings in non-Archimedean Menger PM-spaces, that use the common limit range property. In addition, we give some examples of these results. Then we will extend our main result to four finite families of self-mappings by the notion of pairwise commuting. Finally, we will give applications for our main theorem. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate certain type superlinear nonlocal problems that are a class of parabolic equations with the second-type integral condition. We use the Fadeo-Galarkin method to establish the existence of the weak solution and we prove the uniqueness of this solution for the problem by using an a priori estimate. In addition, we study the theoretical blow-up solution and perform several numerical simulations of finite-time blow-up of a particular example of the main problem. [ABSTRACT FROM AUTHOR]

This paper describes a new fractional predator–prey discrete system of the Leslie type. In addition, the non-linear dynamics of the suggested model are examined within the framework of commensurate and non-commensurate orders, using different numerical techniques such as Lyapunov exponent, phase portraits, and bifurcation diagrams. These behaviours imply that the fractional predator–prey discrete system of Leslie type has rich and complex dynamical properties that are influenced by commensurate and incommensurate orders. Moreover, the sample entropy test is carried out to measure the complexity and validate the presence of chaos. Finally, nonlinear controllers are illustrated to stabilize and synchronize the proposed model. [ABSTRACT FROM AUTHOR]

Owing to the COVID-19 pandemic, which broke out in December 2019 and is still disrupting human life across the world, attention has been recently focused on the study of epidemic mathematical models able to describe the spread of the disease. The number of people who have received vaccinations is a new state variable in the COVID-19 model that this paper introduces to further the discussion of the subject. The study demonstrates that the proposed compartment model, which is described by differential equations of integer order, has two fixed points, a disease-free fixed point and an endemic fixed point. The global stability of the disease-free fixed point is guaranteed by a new theorem that is proven. This implies the disappearance of the pandemic, provided that an inequality involving the vaccination rate is satisfied. Finally, simulation results are carried out, with the aim of highlighting the usefulness of the conceived COVID-19 compartment model. [ABSTRACT FROM AUTHOR]

Nowadays, a lot of research papers are concentrating on the diffusion dynamics of infectious diseases, especially the most recent one: COVID-19. The primary goal of this work is to explore the stability analysis of a new version of the S E I R model formulated with incommensurate fractional-order derivatives. In particular, several existence and uniqueness results of the solution of the proposed model are derived by means of the Picard–Lindelöf method. Several stability analysis results related to the disease-free equilibrium of the model are reported in light of computing the so-called basic reproduction number, as well as in view of utilising a certain Lyapunov function. In conclusion, various numerical simulations are performed to confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

Research in the field of dynamic behaviors in neural networks with variable-order differences is currently a thriving area, marked by various significant discoveries. However, when it comes to discrete-time neural networks featuring fractional variable-order nonlocal and nonsingular kernels, there has been limited exploration. This paper stands as one of the initial contributions to this subject, focusing primarily on the topics of stability and synchronization in finite-time within discrete neural networks. The research employs the nabla ABC variable-order difference operator, with a primary approach involving the investigation of a novel Gronwall inequality using the Atangana-Baleanu difference variable-order sum operator. This analysis leads to the development of a uniqueness theorem and a criterion for the stability in finite-time of variable-order discrete neural networks. Furthermore, the requirements stemming from this type of stability and the novel Gronwall inequality serve as the foundation for establishing the conditions necessary for achieving finite-time synchronization in these networks, employing a specific control using state feedback method. Finally, the study utilizes numerical solutions to validate the obtained results. [ABSTRACT FROM AUTHOR]