Discrete Fractional Difference Research Articles

Robust sliding mode control of discrete fractional difference chaotic system

Robust sliding mode control of discrete fractional difference chaotic system

Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability

Abstract Spacecraft models that mimic the Planck satellite’s behaviour have produced information on cosmic microwave background radiation, assisting physicists in their understanding of the composition and expansion of the universe. For achieving the intended formation, a framework for a discrete fractional difference spacecraft model is constructed by the use of a discrete nabla operator of variable order containing the Mittag–Leffler kernel. The efficacy of the suggested framework is evaluated employing a numerical simulation of the concerning dynamic systems of motion while taking into account multiple considerations such as exterior disruptions, parameterized variations, time-varying feedback delays, and actuator defects. The implementation of the Banach fixed-point approach provides sufficient requirements for the presence of the solution as well as a distinctive feature for such mechanisms Furthermore, the consistent stability is examined. With the aid of discrete nabla operators, we monitor the qualitative behavioural patterns of spacecraft systems to provide justification for structure’s chaos. We acquire the fixed points of the proposed trajectory. At each fixed point, we calculate the eigenvalue of the spacecraft system’s Jacobian matrix and check for zones of instability. The outcomes exhibit a wide range of multifaceted behaviours resulting from the interaction with various fractional orders in the offered system. To maintain stability and synchronize the system, nonlinear controllers are additionally provided. The study highlights the technique’s vulnerability to fractional-order factors, resulting in exclusive, changing trends and equilibrium frameworks. Because of its diverse and convoluted behaviour, the spacecraft chaotic model is an intriguing and crucial subject for research.

On Hybrid and Non-Hybrid Discrete Fractional Difference Inclusion Problems for the Elastic Beam Equation

On Hybrid and Non-Hybrid Discrete Fractional Difference Inclusion Problems for the Elastic Beam Equation

Enhancing the trustworthiness of chaos and synchronization of chaotic satellite model: a practice of discrete fractional-order approaches

Accurate development of satellite maneuvers necessitates a broad orbital dynamical system and efficient nonlinear control techniques. For achieving the intended formation, a framework of a discrete fractional difference satellite model is constructed by the use of commensurate and non-commensurate orders for the control and synchronization of fractional-order chaotic satellite system. The efficacy of the suggested framework is evaluated employing a numerical simulation of the concerning dynamic systems of motion while taking into account multiple considerations such as Lyapunov exponent research, phase images and bifurcation schematics. With the aid of discrete nabla operators, we monitor the qualitative behavioural patterns of satellite systems in order to provide justification for the structure’s chaos. We acquire the fixed points of the proposed trajectory. At each fixed point, we calculate the eigenvalue of the satellite system’s Jacobian matrix and check for zones of instability. The outcomes exhibit a wide range of multifaceted behaviours resulting from the interaction with various fractional-orders in the offered system. Additionally, the sample entropy evaluation is employed in the research to determine complexities and endorse the existence of chaos. To maintain stability and synchronize the system, nonlinear controllers are additionally provided. The study highlights the technique’s vulnerability to fractional-order factors, resulting in exclusive, changing trends and equilibrium frameworks. Because of its diverse and convoluted behaviour, the satellite chaotic model is an intriguing and crucial subject for research.

On existence results of coupled pantograph discrete fractional order difference equations with numerical application

The goal of this article is to study the existence and unique solutions of coupled pantograph discrete fractional order difference equations. The Banach contraction principal method and fixed point theorems are used to prove the existence and uniqueness criterion of the suggested coupled system associated with a discrete fractional operator in the Caputo sense. In addition, stability analysis for the considered discrete pantograph fractional order coupled system is investigated by Hyers–Ulam theoretical approach. Finally, as an application, an example is presented to illustrate the obtained results.

Complexity analysis and discrete fractional difference implementation of the Hindmarsh–Rose neuron system

Nowadays, multistability assessment in discrete nonlinear fractional difference systems is an emotive topic. This paper introduces a novel discrete, nonequilibrium, memristor-based Hindmarsh–Rose neuron (HRN) with the Caputo fractional difference scheme. Furthermore, in a setting involving commensurate and incommensurate scenarios, the complex structure of the proposed discrete fractional model, which includes its multistability, concealed chaos and hyperchaotic attractor, is examined using a wide range of computational approaches, such as Lyapunov exponents, phase portraits, bifurcation illustrations, and the 0–1 evaluation emergence of synchronization. These evolving properties imply that the fractional discrete HRN has an undetected multistability. Ultimately, an elaborate investigation is performed to confirm the existence of unpredictability employing approximation entropy (ApEn) and the ℂ0 measurements. It is demonstrated that whenever the immediate synchronization indicates that it happens, the neurons’ interactions modify, recurring in decreasing fractional orders. Furthermore, minimizing the derivative order increases the incidence of explosions in the synchronization manifold, which is opposite to the behaviour of one nerve cell.

Discrete fuzzy complex-valued function and complex fuzzy Caputo fractional difference equations

In this paper, we introduce a notion of complex compact intervals on which Minkowski addition and scalar multiplication operators are well defined to form the space of complex fuzzy numbers whose α-cuts are complex intervals. Some basic fuzzy partial ordering relations among the complex numbers with “≤C”, the complex intervals with “≪”, the complex fuzzy numbers with “⪯C˜” and the discrete fuzzy complex-valued functions with ≲ are studied. Based on these fundamental properties, we introduce a concept of Caputo fractional sum and difference for the discrete fuzzy complex-valued functions and establish some of their basic results. Moreover, by introducing the complex monotone iterative technique combined with the method of upper and lower solutions, the existence of extremal solutions for discrete complex fuzzy Caputo fractional difference equations is investigated. In each section, several examples are presented to show the feasibility of our obtained results.

Predictive dynamical modeling and stability of the equilibria in a discrete fractional difference COVID-19 epidemic model

The SARSCoV-2 virus, also known as the coronavirus-2, is the consequence of COVID-19, a severe acute respiratory syndrome. Droplets from an infectious individual are how the pathogen is transmitted from one individual to another and occasionally, these particles can contain toxic textures that could also serve as an entry point for the pathogen. We formed a discrete fractional-order COVID-19 framework for this investigation using information and inferences from Thailand. To combat the illnesses, the region has implemented mandatory vaccination, interpersonal stratification and mask distribution programs. As a result, we divided the vulnerable people into two groups: those who support the initiatives and those who do not take the influence regulations seriously. We analyze endemic problems and common data while demonstrating the threshold evolution defined by the fundamental reproductive quantity R0. Employing the mean general interval, we have evaluated the configuration value systems in our framework. Such a framework has been shown to be adaptable to changing pathogen populations over time. The Picard Lindelöf technique is applied to determine the existence-uniqueness of the solution for the proposed scheme. In light of the relationship between the R0 and the consistency of the fixed points in this framework, several theoretical conclusions are made. Numerous numerical simulations are conducted to validate the outcome.

Existence and uniqueness of non-periodic solutions to boundary value problems for discrete fractional difference equations with uncertainty

In this study, we introduce discrete interval fractional difference equations subject to non-periodic boundary conditions. We obtain two types of solutions for these problems under the generalized Hukuhara difference. Then, we present the explicit expressions of each type of solutions for the convenience of applicability. Further, we investigate the necessary and sufficient conditions which guarantee the existence and uniqueness of a non-periodic solution of each specific type by applying the Banach contraction mapping principle. Finally, we provide an example in order to illustrate the main result.

Mandelbrot set and Julia sets of fractional order

In this paper, the fractional-order Mandelbrot and Julia sets in the sense of q-th Caputo-like discrete fractional differences, for $$q\in (0,1)$$ , are introduced and several properties are analytically and numerically studied. Some intriguing properties of the fractional models are revealed. Thus, for $$q\uparrow 1$$ , contrary to expectations, it is not obtained the known shape of the Mandelbrot of integer order, but for $$q\downarrow 0$$ . Also, we conjecture that for $$q\downarrow 0$$ , the fractional-order Mandelbrot set is similar to the integer-order Mandelbrot set, while for $$q\downarrow 0$$ and $$c=0$$ , one of the underlying fractional-order Julia sets is similar to the integer-order Mandelbrot set. In support of our conjecture, several extensive numerical experiments were done. To draw the Mandelbrot and Julia sets of fractional order, the numerical integral of the underlying initial values problem of fractional order is used, while to draw the sets, the escape-time algorithm adapted for the fractional-order case is used. The algorithm is presented as pseudocode.

THEORETICAL AND NUMERICAL COMPUTATIONS OF CONVEXITY ANALYSIS FOR FRACTIONAL DIFFERENCES USING LOWER BOUNDEDNESS

This study focuses on the analytical and numerical solutions of the convexity analysis for fractional differences with exponential and Mittag-Leffler kernels involving negative and nonnegative lower bounds. In the analytical part of the paper, we will give a new formula for [Formula: see text] of the discrete fractional differences, which can be useful to obtain the convexity results. The correlation between the nonnegativity and negativity of both of the discrete fractional differences, [Formula: see text] with the convexity of the functions will be examined. In light of the main lemmas, we will define the two decreasing subsets of [Formula: see text], namely [Formula: see text] and [Formula: see text]. The decrease of these sets enables us to obtain the relationship between the negative lower bound of [Formula: see text] and the convexity of the function on a finite time set given by [Formula: see text] for some [Formula: see text] Besides, the numerical part of the paper is dedicated to examine the validity of the sets [Formula: see text] and [Formula: see text] in certain regions of the solutions for different values of [Formula: see text] and [Formula: see text]. For this reason, we will illustrate the domain of the solutions by means of several figures in which the validity of the main theorems are explained.

Short-memory discrete fractional difference equation wind turbine model and its inferential control of a chaotic permanent magnet synchronous transformer in time-scale analysis

<abstract><p>The aerodynamics analysis has grown in relevance for wind energy projects; this mechanism is focused on elucidating aerodynamic characteristics to maximize accuracy and practicability via the modelling of chaos in a wind turbine system's permanent magnet synchronous generator using short-memory methodologies. Fractional derivatives have memory impacts and are widely used in numerous practical contexts. Even so, they also require a significant amount of storage capacity and have inefficient operations. We suggested a novel approach to investigating the fractional-order operator's Lyapunov candidate that would do away with the challenging task of determining the indication of the Lyapunov first derivative. Next, a short-memory fractional modelling strategy is presented, followed by short-memory fractional derivatives. Meanwhile, we demonstrate the dynamics of chaotic systems using the Lyapunov function. Predictor-corrector methods are used to provide analytical results. It is suggested to use system dynamics to reduce chaotic behaviour and stabilize operation; the benefit of such a framework is that it can only be used for one state of the hybrid power system. The key variables and characteristics, i.e., the modulation index, pitch angle, drag coefficients, power coefficient, air density, rotor angular speed and short-memory fractional differential equations are also evaluated via numerical simulations to enhance signal strength.</p></abstract>

Modified Fractional Difference Operators Defined Using Mittag-Leffler Kernels

The discrete fractional operators of Riemann–Liouville and Liouville–Caputo are omnipresent due to the singularity of the kernels. Therefore, convexity analysis of discrete fractional differences of these types plays a vital role in maintaining the safe operation of kernels and symmetry of discrete delta and nabla distribution. In their discrete version, the generalized or modified forms of various operators of fractional calculus are becoming increasingly important from the viewpoints of both pure and applied mathematical sciences. In this paper, we present the discrete version of the recently modified fractional calculus operator with the Mittag-Leffler-type kernel. Here, in this article, the expressions of both the discrete nabla derivative and its counterpart nabla integral are obtained. Some applications and illustrative examples are given to support the theoretical results.

Monotonicity Results for Nabla Riemann–Liouville Fractional Differences

Positivity analysis is used with some basic conditions to analyse monotonicity across all discrete fractional disciplines. This article addresses the monotonicity of the discrete nabla fractional differences of the Riemann–Liouville type by considering the positivity of ∇b0RLθg(z) combined with a condition on g(b0+2), g(b0+3) and g(b0+4), successively. The article ends with a relationship between the discrete nabla fractional and integer differences of the Riemann–Liouville type, which serves to show the monotonicity of the discrete fractional difference ∇b0RLθg(z).

Positivity analysis for the discrete delta fractional differences of the Riemann-Liouville and Liouville-Caputo types

<abstract><p>In this article, we investigate some new positivity and negativity results for some families of discrete delta fractional difference operators. A basic result is an identity which will prove to be a useful tool for establishing the main results. Our first main result considers the positivity and negativity of the discrete delta fractional difference operator of the Riemann-Liouville type under two main conditions. Similar results are then obtained for the discrete delta fractional difference operator of the Liouville-Caputo type. Finally, we provide a specific example in which the chosen function becomes nonincreasing on a time set.</p></abstract>

Positivity and monotonicity results for discrete fractional operators involving the exponential kernel.

This work deals with the construction and analysis of convexity and nabla positivity for discrete fractional models that includes singular (exponential) kernel. The discrete fractional differences are considered in the sense of Riemann and Liouville, and the υ1-monotonicity formula is employed as our initial result to obtain the mixed order and composite results. The nabla positivity is discussed in detail for increasing discrete operators. Moreover, two examples with the specific values of the orders and starting points are considered to demonstrate the applicability and accuracy of our main results.

Analysis of convexity results for discrete fractional nabla operators

We investigate the relationship between the sign of the discrete fractional difference (∇aνf)(t) and the convexity of the function f in the case where 2<ν<3. We focus on the scenario in which (∇aνf)(t) satisfies a negative lower bound. Our analytical results are complemented with numerical simulations, the latter providing insight beyond what we are able to produce analytically.

Explicit Representation of Discrete Fractional Resolvent Families in Banach Spaces

In this paper we introduce a discrete fractional resolvent family {Sα,βn}n∈ℕ0} generated by a closed linear operator in a Banach space X for a given α, β > 0. Moreover, we study its main properties and, as a consequence, we obtain a method to study the existence and uniqueness of the solutions to discrete fractional difference equations in a Banach space

The discrete fractional order difference applied to an epidemic model with indirect transmission

A discrete fractional order model is proposed to analyze the behaviour of an epidemic process with indirect transmission. This model is based on a discrete version of the Grunwald-Letnikov fractional derivative operator. Some properties of this operator are shown and used to derive a truncated version of the operator, which is used to propose a model with short-term memory. Based on the biological meaning of the problem, some bounds have been obtained to assure the nonnegativity of the model solution. The (α,k)-Basic Reproduction Number has been introduced and used to analyze the stability of the solution around its equilibrium points. Moreover, the influence of the fractional order, α, and the memory steps, k, on the behaviour of the solution has been analyzed. Finally, the results obtained have been clarified by means of numerical simulations of a model for the evolution of an infection by Salmonella in a hens farm.

Monotonicity results for sequential fractional differences of mixed orders with negative lower bound

We investigate the relationship between the sign of the discrete fractional sequential difference and the monotonicity of the function in the case where the orders of the fractional difference are mixed – i.e. but with . We consider the special case in which the bound for some and where , is satisfied. Even though the fractional difference is negative, we demonstrate that there still exists a relationship between this quantity and the monotonicity of f. We augment our analytical results with numerical simulations.