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Dynamics of Lu et al.’s fractional-order system with the order α=0.95, in context of x, y, and z directions.
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In this paper, we consider a class of fractional-order systems described by the Caputo derivative. The behaviors of the dynamics of this particular class of fractional-order systems will be proposed and experienced by a numerical scheme to obtain the phase portraits. Before that, we will provide the conditions under which the considered fractional-...
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Citations
... It is essential to emphasize that fractional-order differentiators have two primary operators, the Riemann-Liouville derivative operator and the Caputo derivative operator, which are both methods that relate to the local derivative of a function [11]. The Caputo differentiator has been shown to be superior to the Riemann-Liouville differentiator for a number of real-world applications [12,13], despite the divergent views of many mathematicians [14,15]. This is because it can be used with certain supposed initial conditions once taking the derivatives of fractional-order [16,17]. ...
... In this regard, we assume that we want to find an approximate value for J α a f (x) when α = 0.85 and α = 1 over the interval [0, 1] by choosing subintervals' number as n = 8. Observe that the values of J α a f (x) when α = 0.85 and α = 1 that could be obtained using Ortigueira et al. [23] or formula (13) can be viewed in Table 2. It should be noticed here that the last two approximated values generated in Tables 3 and 4 using the proposed composite fractional Trapezoidal rule with Romberg integration are very close to the values of J α a f (x) obtained using formula (13) reported in Table 2. ...
... Observe that the values of J α a f (x) when α = 0.85 and α = 1 that could be obtained using Ortigueira et al. [23] or formula (13) can be viewed in Table 2. It should be noticed here that the last two approximated values generated in Tables 3 and 4 using the proposed composite fractional Trapezoidal rule with Romberg integration are very close to the values of J α a f (x) obtained using formula (13) reported in Table 2. In particular, one can find the absolute values of the errors gained from the two approaches to be as 0.011257 and 0.002916 for α = 0.85 and α = 1, respectively. ...
The aim of this research is to demonstrate a novel scheme for approximating the Riemann-Liouville fractional integral operator. This would be achieved by first establishing a fractional-order version of the 2-point Trapezoidal rule and then by proposing another fractional-order version of the (n + 1)-composite Trapezoidal rule. In particular, the so-called divided-difference formula is typically employed to derive the 2-point Trapezoidal rule, which has accordingly been used to derive a more accurate fractional-order formula called the (n + 1)-composite Trapezoidal rule. Additionally, in order to increase the accuracy of the proposed approximations by reducing the true errors, we incorporate the so-called Romberg integration, which is an extrapolation formula of the Trapezoidal rule for integration, into our proposed approaches. Several numerical examples are provided and compared with a modern definition of the Riemann-Liouville fractional integral operator to illustrate the efficacy of our scheme.
... Eventually, the validity of the theoretical results was verified by three numerical simulations. In addition, some of the issues discussed in [36][37][38] (fractional-order chaotic or hyperchaotic systems, synchronous ...
... Eventually, the validity of the theoretical results was verified by three numerical simulations. In addition, some of the issues discussed in [36][37][38] (fractional-order chaotic or hyperchaotic systems, synchronous communication of fractional-order chaotic systems, and event-triggered impulsive chaotic synchronization of fractional-order systems) are also interesting and will be further considered in future work. ...
This paper studies the asymptotic stability of fractional-order neural networks (FONNs) with time delay utilizing a sampled-data controller. Firstly, a novel class of Lyapunov–Krasovskii functions (LKFs) is established, in which time delay and fractional-order information are fully taken into account. Secondly, by combining with the fractional-order Leibniz–Newton formula, LKFs, and other analysis techniques, some less conservative stability criteria that depend on time delay and fractional-order information are given in terms of linear matrix inequalities (LMIs). In the meantime, the sampled-data controller gain is developed under a larger sampling interval. Last, the proposed criteria are shown to be valid and less conservative than the existing ones using three numerical examples.
... Despite the fact that there is no clear and universal definition for continuous chaos, it is generally accepted to be an aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions [Strogatz, 1994]. The Lyapunov exponent method is the most used instrument for determining chaos in both integer and fractional order differential systems [Ahmad & Sprott, 2003;Danca & Kuznetsov, 2018;Danca, 2018;Sene & Ndiaye, 2020]. The bounds of Lyapunov exponents for classical continuous-time systems, Riemann-Liouville fractional differential systems, and Caputo fractional differential systems were discussed in [Li & Xia, 2004;Li et al., 2010]. ...
This paper focuses on the bounds of the Lyapunov exponents for fractional differential systems, where the fractional derivatives are Riemann–Liouville and Caputo fractional derivatives with the exponential kernel. First, the essential properties of fractional integral and derivatives with the exponential kernel are given. Then the continuous dependence of solutions on the initial value problems of some particular parameters is studied. On these bases, the bounds of Lyapunov exponents are estimated. Finally, the theoretical results are illustrated by numerical simulations.
... Before the motivations and the works described in this paper, we give a brief literature review concerning modeling chaotic systems with and without fractional-order derivative. Recently, Sene et al. introduce several fractional chaotic systems via Caputo derivative and study their phase *Corresponding Author portraits, bifurcations diagrams, Lyapunov exponents, sensitivity to the initial conditions, and influence of the fractional orders in obtaining new types of attractors, see in the following investigations [11,14,[16][17][18]. In [13], In the fractional context, Pacheco et al. introduce and investigate fractional chaotic systems with hidden and selfexcited attractors. ...
This paper introduces the properties of a fractional-order chaotic system described by the Caputo derivative. The impact of the fractional-order derivative has been focused on. The phase portraits in different orders are obtained with the aids of the proposed numerical discretization, including the discretization of the Riemann-Liouville fractional integral. The stability analysis has been used to help us to delimit the chaotic region. In other words, the region where the order of the Caputo derivative involves and where the presented system in this paper is chaotic. The nature of the chaos has been established using the Lyapunov exponents in the fractional context. The schematic circuit of the proposed fractional-order chaotic system has been presented and simulated in via Mutltisim. The results obtained via Multisim simulation of the chaotic circuit are in good agreement with the results with Matlab simulations. That provided the fractional operators can be applied in real- worlds applications as modeling electrical circuits. The presence of coexisting attractors for particular values of the parameters of the presented fractional-order chaotic model has been studied.
... Banach space X. en, any contraction M: Π ⟶ Π has a unique fixed point [31,32]. ...
Memristor is a nonlinear and memory element that has a future of replacing resistors for nonlinear circuit computation. It exhibits complex properties such as chaos and hyperchaos. A five-dimensional memristor-based circuit in the context of a nonlocal and nonsingular fractional derivative is considered for analysis. The Banach fixed point theorem and contraction principle are utilized to verify the existence and uniqueness of the solution of the five-dimensional system. A numerical method developed by Toufik and Atangana is used to get approximate solutions of the system. Local stability analysis is examined using the Matignon fractional-order stability criteria, and it is shown that the trivial equilibrium point is unstable. The Lyapunov exponents for different fractional orders exposed that the nature of the five-dimensional fractional-order system is hyperchaotic. Bifurcation diagrams are obtained by varying the fractional order and two of the parameters in the model. It is shown using phase-space portraits and time-series orbit figures that the system is sensitive to derivative order change, parameter change, and small initial condition change. Master-slave synchronization of the hyperchaotic system was established, the error analysis was made, and the simulation results of the synchronized systems revealed a strong correlation among themselves.
... e theory of differential equations of arbitrary order has been recently proved to be an important tool for modeling many physical phenomena. For more details, refer to [4][5][6][7][8][9]. e fractional integro-differential equations have been recently used as effective tools in the modeling of many phenomena in various fields of applied sciences and engineering such as acoustic control, signal processing, electrochemistry, viscoelasticity, polymer physics, electromagnetics, optics, medicine, economics, chemical engineering, chaotic dynamics, and statistical physics (see [10][11][12][13][14][15][16]). ...
In this paper, we study a Volterra–Fredholm integro-differential equation. The considered problem involves the fractional Caputo derivatives under some conditions on the order. We prove an existence and uniqueness analytic result by application of the Banach principle. Then, another result that deals with the existence of at least one solution is delivered, and some sufficient conditions for this result are established by means of the fixed point theorem of Schaefer. Ulam stability of the solution is discussed before including an example to illustrate the results of the proposal.
... Further, the classical IDEs cannot provide the result for non-integer values. In order to overcome the above limitation of integer-order derivatives, different types of fractionalorder operators were defined, some of the research can be found in the existing literature [8] and applications of these fractional-order operators are noted in [9], also Sene in [10,11] consider the Liu et al. [12] chaotic system with Caputo fractional derivative and show the Lyapunov exponent to characterize the nature of chaos and prove the dissipativity of the considered chaotic system. However, it is known that the classical form of the fractional derivatives with singular kernel may not be suitable to characterize the non-local dynamics in an appropriate manner. ...
Abstract This paper investigates a new model on coronavirus-19 disease (COVID-19), that is complex fractional SIR epidemic model with a nonstandard nonlinear incidence rate and a recovery, where derivative operator with Mittag-Leffler kernel in the Caputo sense (ABC). The model has two equilibrium points when the basic reproduction number R 0 > 1 $R_{0} > 1$ ; a disease-free equilibrium E 0 $E_{0}$ and a disease endemic equilibrium E 1 $E_{1}$ . The disease-free equilibrium stage is locally and globally asymptotically stable when the basic reproduction number R 0 < 1 $R_{0} 1 $R_{0} > 1$ . We also prove the existence and uniqueness of the solution for the Atangana–Baleanu SIR model by using a fixed-point method. Since the Atangana–Baleanu fractional derivative gives better precise results to the derivative with exponential kernel because of having fractional order, hence, it is a generalized form of the derivative with exponential kernel. The numerical simulations are explored for various values of the fractional order. Finally, the effect of the ABC fractional-order derivative on suspected and infected individuals carefully is examined and compared with the real data.
... In [29], Baskonus et al. propose active control to stabilize a fractional-order macroeconomic model using Lyapunov direct method. See more investigations related to fractional modeling of chaotic systems using Caputo derivative, bifurcation, and Lyapunov analysis in [30], modeling class of fractional-order chaotic or hyperchaotic system with Caputo derivative in [31], analysis of a four-dimensional hyperchaotic system described by Caputo-Liouville derivative in [32], modeling Chua's electrical circuit in the fractional context in [33], and modeling chaotic processes with Caputo fractional-order derivative in [34]. ...
This paper presents a modified chaotic system under the fractional operator with singularity. The aim of the present subject will be to focus on the influence of the new model’s parameters and its fractional order using the bifurcation diagrams and the Lyapunov exponents. The new fractional model will generate chaotic behaviors. The Lyapunov exponents’ theories in fractional context will be used for the characterization of the chaotic behaviors. In a fractional context, the phase portraits will be obtained with a predictor-corrector numerical scheme method. The details of the numerical scheme will be presented in this paper. The numerical scheme will be used to analyze all the properties addressed in this present paper. The Matignon criterion will also play a fundamental role in the local stability of the presented model’s equilibrium points. We will find a threshold under which the stability will be removed and the chaotic and hyperchaotic behaviors will be generated. An adaptative control will be proposed to correct the instability of the equilibrium points of the model. Sensitive to the initial conditions, we will analyze the influence of the initial conditions on our fractional chaotic system. The coexisting attractors will also be provided for illustrations of the influence of the initial conditions.
This paper explores the investigation of a Volterra-Fredholm integro-differential equation that incorporates Caputo fractional derivatives and adheres to specific order conditions. The study rigorously establishes both the existence and uniqueness of analytical solutions by applying the Banach principle. Additionally, it presents a unique outcome regarding the existence of at least one solution, supported by exacting conditions derived from the Krasnoselskii fixed point theorem. Furthermore, the paper encompasses neutral Volterra-Fredholm integro-differential equations, thus extending the applicability of the findings. Additionally, the paper explores the concept of Ulam stability for the obtained solutions, providing valuable insights into their long-term behavior. To emphasis the practical significance and reliability of the results, an illustrative example is included, effectively demonstrating the applicability of the theoretical discoveries.
The study of fractional variational problems with derivatives in the sense of Caputo is a recent subject in Newtonian fluids. In this article, the homotopy perturbation method (HPM), the variational iteration method (VIM), and the Akbari–Ganji method (AGM) were used to solve linear fractional differential equations order in fluid mechanics and fractional optimal control problems in the sense of Caputo. Achieving an exact solution to these equations has been one of the researchers’ goals; according to the research results, the proposed methods provide an exact solution. Comparing the results obtained by the proposed methods with those obtained by the Shehu method reveals that the present methods are very effective and convenient in applied fields. Also, the VIM and AGM methods are more proper than HPM in solving these equations. The solutions obtained by the proposed methods agree with the solutions available in the literature.
