Adel Ouannas

Larbi Ben M'hidi University of Oum El Bouaghi · Department of Mathematics and Computer Science

The Ueda oscillator is one of the most popular and studied nonlinear oscilla-tors. Whenever subjected to external periodic excitation, it exhibits a fascinating array of nonlinear behaviors, including chaos. This research introduces a novel fractional discrete Ueda system based on Υ-th Caputo fractional difference and thoroughly investigates its ch...

This research aims to explore the complex stability of Fractional Variable Order Discrete Time Systems by introducing new stability criteria. To achieve this, we utilize the properties of Volterra convolution-type systems and the Z-transform methodology. We validate these criteria through practical numerical experiments, showing their usefulness in...

The purpose of this article is to investigate a semilinear nonlocal problem with a 2 nd-type integral condition applied in a specific category of nonlinear equations of parabolic type. The linear problem is analyzed using the Fadeo-Galarkin approach, and the primary objective of the study is to determine whether the weak solution is unique and exis...

This paper delves into a comprehensive analysis of a generalized impulsive discrete reaction–diffusion system under periodic boundary conditions. It investigates the behavior of reactant concentrations through a model governed by partial differential equations (PDEs) incorporating both diffusion mechanisms and nonlinear interactions. By employing f...

According to recent research, discrete-time fractional-order models have greater potential to investigate behaviors, and chaotic maps with fractional derivative values exhibit rich dynamics. This manuscript studies the dynamics of a new fractional chaotic map-based three functions. We analyze the behaviors in commensurate and incommensurate orders,...

Ionic diffusion across cytomembranes plays a critical role in both biological and chemical systems. This paper reexamines the FitzHugh-Nagumo reaction-diffusion system, specifically incorporating the influence of diffusion on the system's dynamics. We focus on the system's finite-time stability, demonstrating that it achieves and maintains equilibr...

The objective of this study is to explore synchronizationa spects of discrete fractional neural networks, encompassing both constant and variable orders. By employing nonlinear feedback control techniques, we establish a sufficient criterion to ensure the synchronization of discrete fractional neural networks with constant orders. Moreover, under c...

In this paper, we present an investigation into the stability of equilibrium points and synchronization within a finite time frame for fractional-order Lengyel-Epstein reaction-diffusion systems. Initially, we utilize Lyapunov theory and multiple criteria to examine the finite-time stability of equilibrium points. Following this analysis, we design...

Memristors special qualities and potential to completely transform computing and memory technology have made them indispensable parts of many applications , which have attracted a lot of interest in complex dynamics. The manuscript focuses on an exhaustive exploration of the chaotic dynamical behaviors exhibited by memristor-based discrete systems...

This paper investigates the existence of the solution for one of the most important fractional partial differential problems called fractional reaction-diffusion system. In particular, with the use of combining the compact semigroup methods and some L^1-estimates, we prove the global existence of the solution for the fractional reaction-diffusion s...

The dynamic analysis of financial systems is a developing field that combines mathematics and economics to understand and explain fluctuations in financial markets. This paper introduces a new three-dimensional (3D) fractional financial map and we dissect its nonlinear dynamics system under commensurate and incommensurate orders. As such, we evalua...

This paper presents a new third-order symmetric difference equation transformed into a 3D discrete symmetric map. The nonlinear dynamics and symmetry of the proposed map are analyzed with two initial conditions for exploring the sensitivity of the map and highlighting the influence of the map parameters on its behaviors, thus comparing the findings...

Due to the very complex algebraic structure of hyperchaotic models, it is often difficult to determine their limits. Using Lyapunov's stability theory and optimization methods, we study the limits of a new 4D hyperchaotic Lorenz model. Based on the results obtained, we study complete chaotic synchronization. Finally, to demonstrate the effectivenes...

This study aims to address the topic of finite-time synchronization within a specific subset of fractional-order Degn–Harrison reaction–diffusion systems. To achieve this goal, we begin with the introduction of a novel lemma specific for finite-time stability analysis. Diverging from existing criteria, this lemma represents a significant extension...

In this research paper, we delve into the analysis of a generalized discrete reaction-diffusion system. Our study begins with the discretization of a generalized reaction-diffusion model, achieved through second-order and 𝐿1-difference approximations. We explore the local stability of its unique solution, both in the absence and presence of the dif...

This paper introduces a new symmetric fractional-order discrete system. The dynamics and symmetry of the suggested model are studied under two initial conditions, mainly a comparison of the commensurate order and incommensurate order maps, which highlights their effect on symmetry-breaking bifurcations. In addition, a theoretical analysis examines...

The aim of this paper is to explore finite-time synchronization in a specific subset of fractional-order epidemic reaction-diffusion systems. Initially, we introduce a new lemma for finite-time stability, which extends existing criteria and builds upon previous discoveries. Following this, we design effective state-dependent linear controllers. By...

As mathematical models of biological pattern generation, this study investigates the dynamics of the fractional discrete Gierer-Meinhardt reaction-diffusion system. After deriving the discrete non-integer fractional variant of the Gierer-Meinhardt system and establishing that the system has a unique equilibrium, we analyze the system's local asympt...

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...

This paper introduces and explores the dynamics of a novel three-dimensional (3D) fractional map with hidden dynamics. The map is constructed through the integration of a discrete sinusoidal memristive into a discrete Duffing map. Moreover, a mathematical operator, namely, a fractional variable-order Caputo-like difference operator, is employed to...

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) mode...

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 uniq...

In this paper, we purpose to investigate the inverse problem of a superlinear parabolic Dirichlet equation with a supplementary integral over determination condition. In this connection, we use the energy inequality for the solvability of direct problem and the fixed point technique for the inverse problem. More particularly, the present paper is d...

In this paper, our objective is to investigate the unique solvability and the weak controllability of the fractional degenerate and singular problem. The energy inequality method is gives a sufficient conditions for the existence and the uniqueness of the strong solution of our problem. This problem is ill-posed in the sense of Hadamard. To address...

In the topic of discrete systems, fractional-order discrete-time neural networks stand out with their utilization of non-integer order difference operators. Despite their potential significance in various applications, the topic of network stability has remained relatively unexplored. This paper seeks to address this gap by introducing a novel netw...

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 b...

The aim of this paper is to investigate the existence and uniqueness of the strong solution for the linear time-fractional partial differential equation with purely integral conditions. The aimed investigation is demonstrated based on the so-called energy inequality method and the density of the operator generated by the considered problem. To do s...

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...

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 investig...

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...

In this chapter, we investigate the stabilization and synchronization of different fractional chaotic maps described by the Caputo h-difference operator using linear control laws. First, we proposed different chaotic maps and presented some simple linear control laws intended to control and synchronize the dynamics of the maps. In addition, using a...

In this work, we will discuss the stability of discrete incommensurate fractional order discrete systems, and then we will apply it in the stabilization of two chaotic systems, the predator-prey system and the cancer growth system.

Fractional-order discrete-time neural networks are a type of discrete system characterized by non-integer order difference operators. Because the dynamics of these networks are required for their practical uses, this chapter concentrates on the dynamics of fractional discrete-time neural networks. First, a class of discrete-time fractional-order ne...

In this paper, we present and discuss a class of semi-linear heat equation with nonlinear nonlocal conditions of second type. The existence and uniqueness of weak solution of the presented problem are investigated in view of the linearisation method. Besides, a well study on the generalized Fujuta problem is also presented. Several graphical compar...

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...

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, vis...

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...

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...

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 interpl...

The solvability of the semilinear parabolic problem with integral overde-termination condition for an inverse problem is investigated in this work. Accordingly , we solve the generated direct problem by using the so-called "energy inequality" method and then the inverse problem is handled with the use of the fixed point technique .

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...

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 phy...

In this article, we are devoted to study the problem of the existence, uniqueness and positivity of the global solutions of the 3 × 3 reaction-diffusion systems with the total mass of the components with time. We also suppose that the nonlinear reaction term has a critical growth with respect to the gradient. The technique that we used to prove the...

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 simula...

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....

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-dif...

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....

This paper presents a multistable discrete memristor that is based on the discretization of a continuous-time model. It has been observed that the discrete memristor model is capable of preserving the characteristics of the continuous memristor model. Furthermore, a three-dimensional memristor discrete-time FitzHugh–Nagumo model is constructed by i...

This paper describes a new four-dimensional fractional discrete neural network with electromagnetic radiation model. In addition, the non-linear dynamics of the suggested model are examined, within the framework of commensurate, incommensurate and variable orders, through different numerical techniques such as Lyapunov exponent, phase portraits, bi...

A novel two-dimensional fractional discrete Hopfield neural network is presented in this study, which is based on discrete fractional calculus. This network incorporates both constant and variable orders, and its behavior is examined using phase plots, time evolution, bifurcation, Lyapunov exponents, and complexity analysis. Compared to integer and...

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 a...

In this work, we are concerned by the problem of identification of noisy terms which arise in singular problem as for remote sensing problems, and which are modeled by a linear singular parabolic equation. For the reason of missing some data that could be arisen when using the traditional sentinel method, the later will be changed by a new sentinel...

Variable-order Fractional discrete neural networks are dynamical systems represented by noninteger order difference equations where the fractional order varies throughout discrete time. Despite the fact that the dynamics of these networks are required for their effective applications, very few articles have been published on the subject. This study...

From the perspective of the fact that confirms all statistics on epidemics can be classified as discrete, we aim in this paper to provide a new discrete-time version of a recent SEIR mathematical model. In other words, a new nabla fractional-order discrete-time system associated with the SEIR model is investigated in terms of its stability analysis...

The global stability of solutions for a discrete-time globally dispersed reaction-diffusion SEI epidemic model with individual immigration is investigated in this work. The global stability is addressed using the Lyapunov functional after giving a discrete form of the reaction-diffusion SEI epidemic model. As in the continuous case, the unique stea...

This paper aims to present the explosion phenomena for a special semi-parabolic problem with a classical Neumann condition where we are interested in the finite time to blow up by using the energy method. A new theoretical result is provided with its proof.KeywordsParabolic equationNonlinear equationsFinite-time blow-up of solution

Reaction–diffusion systems have a broad variety of applications, particularly in biology, and it is well known that fractional calculus has been successfully used with this type of system. However, analyzing these systems using discrete fractional calculus is novel and requires significant research in a diversity of disciplines. Thus, in this paper...

Fractional-order discrete-time neural networks are a type of discrete systems characterized by non-integer order difference operators. Despite the fact that the stability of these networks is required for their effective applications, only a few studies on the subject have been presented. The purpose of this research is to contribute to the topic o...

This paper is devoted to the study of linear nonlocal problems Dirichlet condition and Neumann condition modeling integration condition a second class of a class of linear reaction-diffusion equations. We show the existence and uniqueness of weak solutions to problems Fadeo-Galarkin method developed to circumvent the resulting complexities due to t...

In this paper, we examine a nonlinear hyperbolic equation with a nonlinear integral condition. In particular, we prove the existence and the uniqueness of the linear problem by the Fadeo Galerkin method, and by applying an iterative process to some significant results obtained for the linear problem, the existence and the uniqueness of the weak sol...