The existence of consensus of a leader-following problem with Caputo fractional derivative

In this paper, consensus of a leader-following problem is investigated. The leader-following problem describes a dynamics of the leader and a number of agents. The trajectory of the leader is given. The dynamics of each agent depends on the leader trajectory and others agents' inputs. Here, the leader-following problem is modeled by a system of nonlinear equations with Caputo fractional derivative, which can be rewritten as a system of Volterra equations. The main tools in the investigation are the properties of the resolvent kernel corresponding to the Volterra equations, and Schauder fixed point theorem. By study of the existence of an asymptotically stable solution of a suitable Volterra system, the sufficient conditions for consensus of the leader-following problem are obtained.

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Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Mathematics ◽

10.3390/math9161866 ◽

2021 ◽

Vol 9 (16) ◽

pp. 1866

Author(s):

Mohamed Jleli ◽

Bessem Samet ◽

Calogero Vetro

Keyword(s):

Fractional Derivative ◽

Caputo Fractional Derivative ◽

A Priori Estimates ◽

Sufficient Conditions ◽

Global Solutions ◽

Higher Order ◽

Differential Inequalities ◽

Structure Of Solutions ◽

Nonlinear Capacity ◽

Fractional Differential

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

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Existence of Solutions for Anti-Periodic Fractional Differential Inclusions Involving ψ-Riesz-Caputo Fractional Derivative

Mathematics ◽

10.3390/math7070630 ◽

2019 ◽

Vol 7 (7) ◽

pp. 630

Author(s):

Dandan Yang ◽

Chuanzhi Bai

Keyword(s):

Fractional Derivative ◽

Caputo Fractional Derivative ◽

Differential Inclusions ◽

Existence Of Solutions ◽

Sufficient Conditions ◽

Fractional Differential Inclusions ◽

Nonlinear Alternative ◽

Contractive Map ◽

Fractional Differential ◽

Definition Of

In this paper, we investigate the existence of solutions for a class of anti-periodic fractional differential inclusions with ψ -Riesz-Caputo fractional derivative. A new definition of ψ -Riesz-Caputo fractional derivative of order α is proposed. By means of Contractive map theorem and nonlinear alternative for Kakutani maps, sufficient conditions for the existence of solutions to the fractional differential inclusions are given. We present two examples to illustrate our main results.

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Chaos in perturbed Lotka-Volterra systems

The ANZIAM Journal ◽

10.1017/s1446181100012025 ◽

2001 ◽

Vol 42 (3) ◽

pp. 399-412

Author(s):

J. R. Christie ◽

K. Gopalsamy ◽

Jibin Li

Keyword(s):

Sufficient Conditions ◽

Chaotic Behaviour ◽

Volterra Equations ◽

Unperturbed System ◽

Heteroclinic Cycle ◽

Volterra System ◽

Perturbed System ◽

Volterra Systems ◽

Perturbed Systems ◽

Special Case

AbstractLotka-Volterra systems have been used extensively in modelling population dynamics. In this paper, it is shown that chaotic behaviour in the sense of Smale can exist in timeperiodically perturbed systems of Lotka-Volterra equations. First, a slowly varying threedimensional perturbed Lotka-Volterra system is considered and the corresponding unperturbed system is shown to possess a heteroclinic cycle. By using Melnikov's method, sufficient conditions are obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Then a special case involving a reduction to a two-dimensional Lotka-Volterra system is examined, leading finally to an application with a model for the self-organisation of macromolecules.

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Analysis of Coupled System of Implicit Fractional Differential Equations Involving Katugampola–Caputo Fractional Derivative

Complexity ◽

10.1155/2020/9285686 ◽

2020 ◽

Vol 2020 ◽

pp. 1-11 ◽

Cited By ~ 3

Author(s):

Manzoor Ahmad ◽

Jiqiang Jiang ◽

Akbar Zada ◽

Syed Omar Shah ◽

Jiafa Xu

Keyword(s):

Fractional Derivative ◽

Caputo Fractional Derivative ◽

Existence And Uniqueness ◽

Fixed Point Theorems ◽

Sufficient Conditions ◽

Coupled System ◽

Uniqueness Of Solutions ◽

Fractional Differential System ◽

Fractional Differential ◽

Implicit Fractional Differential Equations

In this paper, we study the existence and uniqueness of solutions to implicit the coupled fractional differential system with the Katugampola–Caputo fractional derivative. Different fixed-point theorems are used to acquire the required results. Moreover, we derive some sufficient conditions to guarantee that the solutions to our considered system are Hyers–Ulam stable. We also provided an example that explains our results.

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Green’s functions, positive solutions, and a Lyapunov inequality for a caputo fractional-derivative boundary value problem

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2019-0041 ◽

2019 ◽

Vol 22 (3) ◽

pp. 750-766 ◽

Cited By ~ 3

Author(s):

Xiangyun Meng ◽

Martin Stynes

Keyword(s):

Boundary Value Problem ◽

Fractional Derivative ◽

Boundary Problem ◽

Caputo Fractional Derivative ◽

Caputo Derivative ◽

Nonlinear Boundary ◽

Boundary Value ◽

Sufficient Conditions ◽

Lyapunov Inequality ◽

Nonlinear Boundary Problem

Abstract We consider a nonlinear boundary problem whose highest-order derivative is a Caputo derivative of order α with 1 < α < 2. Properties of its associated Green’s function are derived. These properties enable us to deduce sufficient conditions for the existence of a positive solution to the boundary value problem and to prove a Lyapunov inequality for the problem. Our results sharpen and extend earlier results of other authors.

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Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

Axioms ◽

10.3390/axioms10040322 ◽

2021 ◽

Vol 10 (4) ◽

pp. 322

Author(s):

Ricardo Almeida ◽

Ravi P. Agarwal ◽

Snezhana Hristova ◽

Donal O’Regan

Keyword(s):

Fractional Derivative ◽

Caputo Fractional Derivative ◽

Lyapunov Functions ◽

Fractional Derivatives ◽

Sufficient Conditions ◽

Hopfield Neural Network ◽

Quadratic Lyapunov Functions ◽

Fractional Differential ◽

The Stability ◽

Theoretical Results

A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results.

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Boundary Value Problems for Fractional Differential Equations

Georgian Mathematical Journal ◽

10.1515/gmj.2009.401 ◽

2009 ◽

Vol 16 (3) ◽

pp. 401-411 ◽

Cited By ~ 5

Author(s):

Ravi P. Agarwal ◽

Mouffak Benchohra ◽

Samira Hamani

Keyword(s):

Differential Equations ◽

Fractional Derivative ◽

Boundary Value Problems ◽

Fractional Differential Equations ◽

Caputo Fractional Derivative ◽

Existence Of Solutions ◽

Boundary Value ◽

Sufficient Conditions ◽

Fractional Differential

Abstract The sufficient conditions are established for the existence of solutions for a class of boundary value problems for fractional differential equations involving the Caputo fractional derivative.

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Asymptotic Stability for Some Types of Nonlinear Fractional Order Differential-Algebraic Control Systems

Iraqi Journal of Science ◽

10.24996/ijs.2021.62.2.28 ◽

2021 ◽

pp. 623-638

Author(s):

Sameer Qasim Hasan

Keyword(s):

Feedback Control ◽

Asymptotic Stability ◽

Control Systems ◽

Fractional Order ◽

Sufficient Conditions ◽

Stable Solution ◽

Asymptotically Stable ◽

Fractional Differential ◽

Necessary And Sufficient ◽

Asymptotically Stable Solution

The aim of this paper is to study the asymptotically stable solution of nonlinear single and multi fractional differential-algebraic control systems, involving feedback control inputs, by an effective approach that depends on necessary and sufficient conditions.

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A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

Mathematics ◽

10.3390/math9090979 ◽

2021 ◽

Vol 9 (9) ◽

pp. 979

Author(s):

Sandeep Kumar ◽

Rajesh K. Pandey ◽

H. M. Srivastava ◽

G. N. Singh

Keyword(s):

Differential Equation ◽

Fractional Derivative ◽

Collocation Method ◽

Caputo Fractional Derivative ◽

Fractional Derivatives ◽

Special Cases ◽

Collocation Approach ◽

Linear And Nonlinear ◽

Simulation Results ◽

Theoretical Results

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

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Caputo Fractional Derivative and Quantum-Like Coherence

Entropy ◽

10.3390/e23020211 ◽

2021 ◽

Vol 23 (2) ◽

pp. 211

Author(s):

Garland Culbreth ◽

Mauro Bologna ◽

Bruce J. West ◽

Paolo Grigolini

Keyword(s):

Diffusion Equation ◽

Fractional Derivative ◽

Anomalous Diffusion ◽

Caputo Fractional Derivative ◽

Quantum Coherence ◽

Time Derivative ◽

The Other ◽

Diffusion Equations ◽

Self Organization ◽

Ordinary Time

We study two forms of anomalous diffusion, one equivalent to replacing the ordinary time derivative of the standard diffusion equation with the Caputo fractional derivative, and the other equivalent to replacing the time independent diffusion coefficient of the standard diffusion equation with a monotonic time dependence. We discuss the joint use of these prescriptions, with a phenomenological method and a theoretical projection method, leading to two apparently different diffusion equations. We prove that the two diffusion equations are equivalent and design a time series that corresponds to the anomalous diffusion equation proposed. We discuss these results in the framework of the growing interest in fractional derivatives and the emergence of cognition in nature. We conclude that the Caputo fractional derivative is a signature of the connection between cognition and self-organization, a form of cognition emergence different from the other source of anomalous diffusion, which is closely related to quantum coherence. We propose a criterion to detect the action of self-organization even in the presence of significant quantum coherence. We argue that statistical analysis of data using diffusion entropy should help the analysis of physiological processes hosting both forms of deviation from ordinary scaling.

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