A Cauchy Problem for Some Fractional q-Difference Equations with Nonlocal Conditions

In this paper, we discussed the problem of nonlocal value for nonlinear fractional q-difference equation. The classical tools of fixed point theorems such as Krasnoselskii’s theorem and Banach’s contraction principle are used. At the end of the manuscript, we have an example that illustrates the key findings.

Chaotic dynamics of the fractional-order Ikeda delay system and its synchronization

In this paper we numerically investigate the chaotic behaviours of the fractional-order Ikeda delay system. The results show that chaos exists in the fractional-order Ikeda delay system with order less than 1. The lowest order for chaos to be a, ble to appear in this system is found to be 0.1. Master-slave synchronization of chaotic fractional-order Ikeda delay systems with linear coupling is also studied.

EXACT CONTROLLABILITY AND CONTINUOUS DEPENDENCE OF FRACTIONAL NEUTRAL INTEGRO-DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

In the present paper,with the help of the resolvent operator and some analytic methods,the exact controllability and continuous dependence are investigated for a fractional neutral integro-differential equations with state-dependent delay.As an application,we also give one example to demonstrate our results.

Robust Stability and Stabilization of Discrete Singular Systems with Interval Time-varying Delay and Linear Fractional Uncertainty

The robust stability and robust stabilization problems for discrete singular systems with interval time-varying delay and linear fractional uncertainty are discussed. A new delay-dependent criterion is established for the nominal discrete singular delay systems to be regular, causal and stable by employing the linear matrix inequality （LMI） approach. It is shown that the newly proposed criterion can provide less conservative results than some existing ones. Then, with this criterion, the problems of robust stability and robust stabilization for uncertain discrete singular delay systems are solved, and the delay-dependent LMI conditions are obtained. Finally, numerical examples are given to illustrate the effectiveness of the proposed approach.

A Fractional-Order Phase-Locked Loop with Time-Delay and Its Hopf Bifurcation

A fractional-order phase-locked loop(PLL)with a time-delay is firstly proposed on the basis of the fact that a capacitor has memory.The existence of Hopf bifurcation of the fractional-order PLL with a time-delay is investigated by studying the root location of the characteristic equation,and the bifurcated periodic solution and its stability are studied simply by using"pseudo-oscillator analysis".The results are checked by numerical simulation.It is found that the fractional-order PLL with a time-delay reduces the locking time,and it minimizes the amplitude of the bifurcated periodic solution when the order is properly chosen.

Dynamics Analysis of Fractional Order Memristive Time Delay Chaotic System and Circuit Implementation

The integer order memristive time delay chaotic system has attracted much attention and has been well discussed.However,the fractional order system is closer to the real system.In this paper,a nonlinear time delay chaotic circuit based on fractional order memristive system was proposed.Some dynamical properties,including equilibrium points,stability,bifurcation,and Lyapunov exponent of the oscillator,were investigated in detail by theoretical analyses and simulations.Moreover,the nonlinear phenomena of coexisting bifurcation and attractor was found.The phenomenon shows that the state of this oscilator was highly sensitive to its initial value,which is called coexistent oscillation in this paper.Finally,the results of the system circuit simulation accomplished by Multisim were perfectly consistent with theoretical analyses and numerical simulation.

A FRACTIONAL NONLINEAR EVOLUTIONARY DELAY SYSTEM DRIVEN BY A HEMI-VARIATIONAL INEQUALITY IN BANACH SPACES

This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space.Utilizing the KKM theorem,a result concerned with the upper semicontinuity and measurability of the solution set of a hemivariational inequality is established.By using a fixed point theorem for a condensing setvalued map,the nonemptiness and compactness of the set of mild solutions are also obtained for such a system under mild conditions.Finally,an example is presented to illustrate our main results.

Hidden Attractors in a Delayed Memristive Differential System with Fractional Order and Chaos Synchronization

As an important research branch,memristor has attracted a range of scholars to study the property of memristive chaotic systems.Additionally,time⁃delayed systems are considered a significant and newly⁃developing field in modern research.By combining memristor and time⁃delay,a delayed memristive differential system with fractional order is proposed in this paper,which can generate hidden attractors.First,we discussed the dynamics of the proposed system where the parameter was set as the bifurcation parameter,and showed that with the increase of the parameter,the system generated rich chaotic phenomena such as bifurcation,chaos,and hypherchaos.Then we derived adequate and appropriate stability criteria to guarantee the system to achieve synchronization.Lastly,examples were provided to analyze and confirm the influence of parameter a,fractional order q,and time delayτon chaos synchronization.The simulation results confirm that the chaotic synchronization is affected by a,q andτ.

Stabilization of Uncertain Fractional Memristor Chaotic Time⁃Delay System Based on Fractional Order Sliding Mode Control

Based on Lyapunov theorem and sliding mode control scheme,the chaos control of fractional memristor chaotic time⁃delay system was studied.In order to stabilize the system,a fractional sliding mode control method for fractional time⁃delay system was proposed.In addition,Lyapunov stability theorem was used to analyze the control scheme theoretically,which guaranteed the stability of commensurate and non⁃commensurate order systems with or without uncertainties and disturbances.Furthermore,to illustrate the feasibility of controller,the conditions for designing the controller parameters were derived.Finally,the simulation results presented the effectiveness of the designed strategy.

Finite-Time Stability for Nonlinear Fractional Differential Equations with Time Delay

The finite-time stability and the finite-time contractive stability of solutions for nonlinear fractional differential equations with bounded delay are investigated. The derivative of Lyapunov function along solutions of the considered system is defined in terms of the Caputo fractional Dini derivative. Based on the Lyapunov-Razumikhin method, several sufficient criteria are established to guarantee the finite-time stability and the finite-time contractive stability of solutions for the related systems. An example is provided to illustrate the effectiveness of the obtained results.

Asymptotic Analysis of Linear and Interval Linear Fractional-Order Neutral Delay Differential Systems Described by the Caputo-Fabrizio Derivative

Asymptotic stability of linear and interval linear fractional-order neutral delay differential systems described by the Caputo-Fabrizio (CF) fractional derivatives is investigated. Using Laplace transform, a novel characteristic equation is derived. Stability criteria are established based on an algebraic approach and norm-based criteria are also presented. It is shown that asymptotic stability is ensured for linear fractional-order neutral delay differential systems provided that the underlying stability criterion holds for any delay parameter. In addition, sufficient conditions are derived to ensure the asymptotic stability of interval linear fractional order neutral delay differential systems. Examples are provided to illustrate the effectiveness and applicability of the theoretical results.

Studying the intermittent stable theorem and the synchronization of a delayed fractional nonlinear system

In this paper, an intermittent synchronizing delayed fractional nonlinear system is studied. We propose a novel intermittent stable theorem for the delayed fractional system and derive a new synchronization criterion for delayed fractional systems by means of fractional stable theorem and the differential inequality method. Intermittent synchronizing fractional delayed Newton-Leipnik system is taken as an illustrative example and numerical simulation of this example is presented to show the feasibility and effectiveness of the proposed theorem.

Analytical and Numerical Solutions of Riesz Space Fractional Advection-Dispersion Equations with Delay

In this paper,we propose numerical methods for the Riesz space fractional advection-dispersion equations with delay(RFADED).We utilize the fractional backward differential formulas method of second order(FBDF2)and weighted shifted Grünwald difference(WSGD)operators to approximate the Riesz fractional derivative and present the finite difference method for the RFADED.Firstly,the FBDF2 and the shifted Grünwald methods are introduced.Secondly,based on the FBDF2 method and the WSGD operators,the finite difference method is applied to the problem.We also show that our numerical schemes are conditionally stable and convergent with the accuracy of O(+h2)and O(2+h2)respectively.Thirdly we find the analytical solution for RFDED in terms Mittag-Leffler type functions.Finally,some numerical examples are given to show the efficacy of the numerical methods and the results are found to be in complete agreement with the analytical solution.

Exact Solutions and Finite Time Stability of Linear Conformable Fractional Systems with Pure Delay

We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay.By using new conformable delayed matrix functions and the method of variation,we obtain a representation of their solutions.As an application,we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayedmatrix functions.The obtained results are new,and they extend and improve some existing ones.Finally,an example is presented to illustrate the validity of our theoretical results.

Stability of Neutral Fractional Neural Networks with Delay

This paper studies stability of neutral fractional neural networks with delay. By introducing the definition of norm and using the uniform stability, the sufficient condition for uniform stability of neutral fractional neural networks with delay is obtained.

THE NONEMPTINESS AND COMPACTNESS OF MILD SOLUTION SETS FOR RIEMANN-LIOUVILLE FRACTIONAL DELAY DIFFERENTIAL VARIATIONAL INEQUALITIES

This paper investigates the nonemptiness and compactness of the mild solution set for a class of Riemann-Liouville fractional delay differential variational inequalities,which are formulated by a Riemann-Liouville fractional delay evolution equation and a variational inequality.Our approach is based on the resolvent technique and a generalization of strongly continuous semigroups combined with Schauder's fixed point theorem.

Fixed Point Theorem and Fractional Differential Equations with Multiple Delays Related with Chaos Neuron Models

In this paper, we show a fixed point theorem which deduces to both of Lou’s fixed point theorem and de Pascale and de Pascale’s fixed point theorem. Moreover, our result can be applied to show the existence and uniqueness of solutions for fractional differential equations with multiple delays. Using the theorem, we discuss the fractional chaos neuron model.

Some New Delay Integral Inequalities Based on Modified Riemann-Liouville Fractional Derivative and Their Applications

By using the properties of modified Riemann-Liouville fractional derivative, some new delay integral inequalities have been studied. First, we offered explicit bounds for the unknown functions, then we applied the results to the research concerning the boundness, uniqueness and continuous dependence on the initial for solutions to certain fractional differential equations.

Stabilizing controller design for nonlinear fractional order systems with time varying delays

To deal with stabilizing of nonlinear affine fractional order systems subject to time varying delays,two methods for finding an appropriate pseudo state feedback controller are discussed.In the first method,using the Mittag-Lefler function,Laplace transform and Gronwall inequality,a linear stabilizing controller is derived,which uses the fractional order of the delayed system and the upper bound of system nonlinear functions.In the second method,at first a sufficient stability condition for the delayed system is given in the form of a simple linear matrix inequality(LMI)which can easily be solved.Then,on the basis of this result,a stabilizing pseudo-state feedback controller is designed in which the controller gain matrix is easily computed by solving an LMI in terms of delay bounds.Simulation results show the effectiveness of the proposed methods.

Chaos in the Fractional Order Logistic Delay System

In this paper, we study the chaotic behaviors in a fractional order logistic delay system. We find that chaos exists in the fractional order logistic delay system with an order being less than 1. In addition, we numerically simulate the continuances of the chaotic behaviors in the logistic delay system with orders from 0.1 to 0.9. The lowest order we find to have chaos in this system is 0.1. Then we further investigate two methods in controlling the fractional order chaotic logistic delay system based on feedback. Finally, we investigate a lag synchronization scheme in this system. Numerical simulations show the effectiveness and feasibility of our approach.