This paper investigates the function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems using the stability theory of fractional-order systems. The function projectiv...This paper investigates the function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems using the stability theory of fractional-order systems. The function projective synchronization between three-dimensional （3D） integer-order Lorenz chaotic system and 3D fractional-order Chen chaotic system are presented to demonstrate the effectiveness of the proposed scheme.展开更多
In this paper, coexistence and local Mittag–Leffler stability of fractional-order recurrent neural networks with discontinuous activation functions are addressed. Because of the discontinuity of the activation functi...In this paper, coexistence and local Mittag–Leffler stability of fractional-order recurrent neural networks with discontinuous activation functions are addressed. Because of the discontinuity of the activation function, Filippov solution of the neural network is defined. Based on Brouwer's fixed point theorem and definition of Mittag–Leffler stability, sufficient criteria are established to ensure the existence of (2k + 3)~n (k ≥ 1) equilibrium points, among which (k + 2)~n equilibrium points are locally Mittag–Leffler stable. Compared with the existing results, the derived results cover local Mittag–Leffler stability of both fractional-order and integral-order recurrent neural networks. Meanwhile discontinuous networks might have higher storage capacity than the continuous ones. Two numerical examples are elaborated to substantiate the effective of the theoretical results.展开更多
The finite-time Mittag-Leffler synchronization is investigated for fractional-order delayed memristive neural networks(FDMNN)with parameters uncertainty and discontinuous activation functions.The relevant results are ...The finite-time Mittag-Leffler synchronization is investigated for fractional-order delayed memristive neural networks(FDMNN)with parameters uncertainty and discontinuous activation functions.The relevant results are obtained under the framework of Filippov for such systems.Firstly,the novel feedback controller,which includes the discontinuous functions and time delays,is proposed to investigate such systems.Secondly,the conditions on finite-time Mittag-Leffler synchronization of FDMNN are established according to the properties of fractional-order calculus and inequality analysis technique.At the same time,the upper bound of the settling time for Mittag-Leffler synchronization is accurately estimated.In addition,by selecting the appropriate parameters of the designed controller and utilizing the comparison theorem for fractional-order systems,the global asymptotic synchronization is achieved as a corollary.Finally,a numerical example is given to indicate the correctness of the obtained conclusions.展开更多
We present the joint probability density function(PDF) between the bucket signals and reference signals in thermal light ghost imaging, by regarding these signals as stochastic variables. The joint PDF allows us to ex...We present the joint probability density function(PDF) between the bucket signals and reference signals in thermal light ghost imaging, by regarding these signals as stochastic variables. The joint PDF allows us to examine the fractional-order moments of the bucket and the reference signals, in which the correlation orders are fractional numbers,other than positive integers in previous studies. The experimental results show that various images can be reconstructed from fractional-order moments. Negative(positive) ghost images are obtained with negative(positive) orders of the bucket signals. The visibility and peak signal-to-noise ratios of the diverse ghost images depend greatly on the fractional orders.展开更多
The external stability of fractional-order continuous linear control systems described by both fractional-order state space representation and fractional-order transfer function is mainly investigated in this paper. I...The external stability of fractional-order continuous linear control systems described by both fractional-order state space representation and fractional-order transfer function is mainly investigated in this paper. In terms of Lyapunov’s stability theory and the stability analysis of the integer-order linear control systems, the definitions of external stability for fractional-order control systems are presented. By using the theorems of the Mittag-Leffler function in two parameters, the necessary and sufficient conditions of external stability are directly derived. The illustrative examples and simulation results are also given.展开更多
This paper focuses on synchronization of fractionalorder complex dynamical networks with decentralized adaptive coupling.Based on local information among neighboring nodes,two fractional-order decentralized adaptive s...This paper focuses on synchronization of fractionalorder complex dynamical networks with decentralized adaptive coupling.Based on local information among neighboring nodes,two fractional-order decentralized adaptive strategies are designed to tune all or only a small fraction of the coupling gains respectively.By constructing quadratic Lyapunov functions and utilizing fractional inequality techniques,Mittag-Leffler function,and Laplace transform,two sufficient conditions are derived for reaching network synchronization by using the proposed adaptive laws.Finally,two numerical examples are given to verify the theoretical results.展开更多
The purpose of this study is to design a fractional-order super-twisting sliding-mode controller for a class of nonlinear fractionalorder systems.The proposed method has the following advantages:(1)Lyapunov stability ...The purpose of this study is to design a fractional-order super-twisting sliding-mode controller for a class of nonlinear fractionalorder systems.The proposed method has the following advantages:(1)Lyapunov stability of the overall closed-loop system,(2)output tracking error’s convergence to zero,(3)robustness against external uncertainties and disturbances,and(4)reduction of the chattering phenomenon.To investigate the performance of the method,the proposed controller is applied to an autonomous underwater robot and Lorenz chaotic system.Finally,a simulation is performed to verify the potential of the proposed method.展开更多
In this paper we present a new version of Chen＇s system： a piecewise linear （PWL） Chert system of fractional-order. Via a sigmoid-like function, the discontinuous system is transformed into a continuous system. By...In this paper we present a new version of Chen＇s system： a piecewise linear （PWL） Chert system of fractional-order. Via a sigmoid-like function, the discontinuous system is transformed into a continuous system. By numerical simulations, we reveal chaotic behaviors and also multistability, i.e., the existence of small pararheter windows where, for some fixed bifurcation parameter and depending on initial conditions, coexistence of stable attractors and chaotic attractors is possible. Moreover, we show that by using an algorithm to switch the bifurcation parameter, the stable attractors can be numerically approximated.展开更多
The dynamic behavior of a viscoelastic high-order shear microbeam is studied based on a new constitutive model which incorporates size effects and viscoelasticity simultaneously.The size effects are modeled by the non...The dynamic behavior of a viscoelastic high-order shear microbeam is studied based on a new constitutive model which incorporates size effects and viscoelasticity simultaneously.The size effects are modeled by the nonlocal gradient elasticity,while viscoelastic effects are modeled by fractional-order derivatives.The constitutive relation and the equations of motion are both differential equations with fractional-order derivatives.Based on the Laplace transform and inverse transform,the analytical solution of the dynamic response under a step load is obtained in terms of the Mittag–Leffler function.In order to verify the reliability of the analytical solution,a comparison with the numerical solution is also provided.Based on the numerical results,the effects of the nonlocal parameter,strain gradient parameter,fractional-order parameter,and viscosity coefficient on the dynamic response of the viscoelastic microbeam are discussed.It is found that the influences of the fractional order and the coefficient of viscosity on the dynamic response of the microbeam are very different,although both are related to the viscoelastic behavior.展开更多
In this paper,we formulate and analyze a new fractional-order Logistic model with feedback control,which is different from a recognized mathematical model proposed in our very recent work.Asymptotic stability of the p...In this paper,we formulate and analyze a new fractional-order Logistic model with feedback control,which is different from a recognized mathematical model proposed in our very recent work.Asymptotic stability of the proposed model and its numerical solutions are studied rigorously.By using the Lyapunov direct method for fractional dynamical systems and a suitable Lyapunov function,we show that a unique positive equilibrium point of the new model is asymptotically stable.As an important consequence of this,we obtain a new mathematical model in which the feedback control variables only change the position of the unique positive equilibrium point of the original model but retain its asymptotic stability.Furthermore,we construct unconditionally positive nonstandard finite difference(NSFD)schemes for the proposed model using the Mickens’methodology.It is worth noting that the constructed NSFD schemes not only preserve the positivity but also provide reliable numerical solutions that correctly reflect the dynamics of the new fractional-order model.Finally,we report some numerical examples to support and illustrate the theoretical results.The results indicate that there is a good agreement between the theoretical results and numerical ones.展开更多
Though vaccination protects individuals against many infectious diseases,such protection does not always last forever since a few vaccinated individuals could lose their lifelong immunity and eventually become infecte...Though vaccination protects individuals against many infectious diseases,such protection does not always last forever since a few vaccinated individuals could lose their lifelong immunity and eventually become infected.This study,therefore,determines the effects of imperfect vaccination and memory index on the spread of diseases through the Caputo fractional-order SIRV(Susceptible-Infected-Recovered-Vaccinated)epidemic model.Vital properties of the new model including the conditions for the existence of a unique solution determined through the fixed-point theory and the conditions for the existence of a positive solution of the model obtained via the Mittag-Leffler function along with the Laplace transformation-are thoroughly studied.Consequently,our simulation results report that an increase in the imperfect vaccination force increases the population of infected individuals.For the memory effect,the higher“memory”the epidemic system has of past states(which corresponds to decreasing values of fractionalorder parameter),the greater the peaks and magnitudes of infection shaping the epidemiological system dynamics.展开更多
文摘This paper investigates the function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems using the stability theory of fractional-order systems. The function projective synchronization between three-dimensional （3D） integer-order Lorenz chaotic system and 3D fractional-order Chen chaotic system are presented to demonstrate the effectiveness of the proposed scheme.
基金Project supported by the Natural Science Foundation of Zhejiang Province,China(Grant Nos.LY18F030023,LY17F030016,and LY18F020028)the National Natural Science Foundation of China(Grant Nos.61503338,61502422,and 61773348)
文摘In this paper, coexistence and local Mittag–Leffler stability of fractional-order recurrent neural networks with discontinuous activation functions are addressed. Because of the discontinuity of the activation function, Filippov solution of the neural network is defined. Based on Brouwer's fixed point theorem and definition of Mittag–Leffler stability, sufficient criteria are established to ensure the existence of (2k + 3)~n (k ≥ 1) equilibrium points, among which (k + 2)~n equilibrium points are locally Mittag–Leffler stable. Compared with the existing results, the derived results cover local Mittag–Leffler stability of both fractional-order and integral-order recurrent neural networks. Meanwhile discontinuous networks might have higher storage capacity than the continuous ones. Two numerical examples are elaborated to substantiate the effective of the theoretical results.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61703312 and 61703313)。
文摘The finite-time Mittag-Leffler synchronization is investigated for fractional-order delayed memristive neural networks(FDMNN)with parameters uncertainty and discontinuous activation functions.The relevant results are obtained under the framework of Filippov for such systems.Firstly,the novel feedback controller,which includes the discontinuous functions and time delays,is proposed to investigate such systems.Secondly,the conditions on finite-time Mittag-Leffler synchronization of FDMNN are established according to the properties of fractional-order calculus and inequality analysis technique.At the same time,the upper bound of the settling time for Mittag-Leffler synchronization is accurately estimated.In addition,by selecting the appropriate parameters of the designed controller and utilizing the comparison theorem for fractional-order systems,the global asymptotic synchronization is achieved as a corollary.Finally,a numerical example is given to indicate the correctness of the obtained conclusions.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11674273,11304016,and 11204062)
文摘We present the joint probability density function(PDF) between the bucket signals and reference signals in thermal light ghost imaging, by regarding these signals as stochastic variables. The joint PDF allows us to examine the fractional-order moments of the bucket and the reference signals, in which the correlation orders are fractional numbers,other than positive integers in previous studies. The experimental results show that various images can be reconstructed from fractional-order moments. Negative(positive) ghost images are obtained with negative(positive) orders of the bucket signals. The visibility and peak signal-to-noise ratios of the diverse ghost images depend greatly on the fractional orders.
文摘The external stability of fractional-order continuous linear control systems described by both fractional-order state space representation and fractional-order transfer function is mainly investigated in this paper. In terms of Lyapunov’s stability theory and the stability analysis of the integer-order linear control systems, the definitions of external stability for fractional-order control systems are presented. By using the theorems of the Mittag-Leffler function in two parameters, the necessary and sufficient conditions of external stability are directly derived. The illustrative examples and simulation results are also given.
基金supported by the"Chunhui Plan"Cooperative Research for Ministry of Education(Z2016133)the Open Research Fund of Key Laboratory of Automobile Engineering(Xihua University)+3 种基金Sichuan Province(szjj2016-017)the National Natural Science Foundation of China(51177137)the Scientific Research Foundation of the Education Department of Sichuan Province(16ZB0163)the China Scholarship Council
文摘This paper focuses on synchronization of fractionalorder complex dynamical networks with decentralized adaptive coupling.Based on local information among neighboring nodes,two fractional-order decentralized adaptive strategies are designed to tune all or only a small fraction of the coupling gains respectively.By constructing quadratic Lyapunov functions and utilizing fractional inequality techniques,Mittag-Leffler function,and Laplace transform,two sufficient conditions are derived for reaching network synchronization by using the proposed adaptive laws.Finally,two numerical examples are given to verify the theoretical results.
文摘The purpose of this study is to design a fractional-order super-twisting sliding-mode controller for a class of nonlinear fractionalorder systems.The proposed method has the following advantages:(1)Lyapunov stability of the overall closed-loop system,(2)output tracking error’s convergence to zero,(3)robustness against external uncertainties and disturbances,and(4)reduction of the chattering phenomenon.To investigate the performance of the method,the proposed controller is applied to an autonomous underwater robot and Lorenz chaotic system.Finally,a simulation is performed to verify the potential of the proposed method.
基金funded by the European Regional Development Funding via RISC projectby CPER Region Haute Normandie France,the Australian Research Council via a Future Fellowship(FT110100896)Discovery Project(DP140100203)
文摘In this paper we present a new version of Chen＇s system： a piecewise linear （PWL） Chert system of fractional-order. Via a sigmoid-like function, the discontinuous system is transformed into a continuous system. By numerical simulations, we reveal chaotic behaviors and also multistability, i.e., the existence of small pararheter windows where, for some fixed bifurcation parameter and depending on initial conditions, coexistence of stable attractors and chaotic attractors is possible. Moreover, we show that by using an algorithm to switch the bifurcation parameter, the stable attractors can be numerically approximated.
基金supported by the National Natural Science Foundation of China(Grant Nos.12072022,11872105,and 11911530176)the Fundamental Research Funds for the Central Universities(FRF-BR-18-008B,FRF-TW-2018-005).
文摘The dynamic behavior of a viscoelastic high-order shear microbeam is studied based on a new constitutive model which incorporates size effects and viscoelasticity simultaneously.The size effects are modeled by the nonlocal gradient elasticity,while viscoelastic effects are modeled by fractional-order derivatives.The constitutive relation and the equations of motion are both differential equations with fractional-order derivatives.Based on the Laplace transform and inverse transform,the analytical solution of the dynamic response under a step load is obtained in terms of the Mittag–Leffler function.In order to verify the reliability of the analytical solution,a comparison with the numerical solution is also provided.Based on the numerical results,the effects of the nonlocal parameter,strain gradient parameter,fractional-order parameter,and viscosity coefficient on the dynamic response of the viscoelastic microbeam are discussed.It is found that the influences of the fractional order and the coefficient of viscosity on the dynamic response of the microbeam are very different,although both are related to the viscoelastic behavior.
文摘In this paper,we formulate and analyze a new fractional-order Logistic model with feedback control,which is different from a recognized mathematical model proposed in our very recent work.Asymptotic stability of the proposed model and its numerical solutions are studied rigorously.By using the Lyapunov direct method for fractional dynamical systems and a suitable Lyapunov function,we show that a unique positive equilibrium point of the new model is asymptotically stable.As an important consequence of this,we obtain a new mathematical model in which the feedback control variables only change the position of the unique positive equilibrium point of the original model but retain its asymptotic stability.Furthermore,we construct unconditionally positive nonstandard finite difference(NSFD)schemes for the proposed model using the Mickens’methodology.It is worth noting that the constructed NSFD schemes not only preserve the positivity but also provide reliable numerical solutions that correctly reflect the dynamics of the new fractional-order model.Finally,we report some numerical examples to support and illustrate the theoretical results.The results indicate that there is a good agreement between the theoretical results and numerical ones.
基金support from the Fundamental Research Grant Scheme with Project Code:FRGS/1/2022/STG06/USM/02/1 by the Ministry of Higher Education,Malaysia(MOHE).
文摘Though vaccination protects individuals against many infectious diseases,such protection does not always last forever since a few vaccinated individuals could lose their lifelong immunity and eventually become infected.This study,therefore,determines the effects of imperfect vaccination and memory index on the spread of diseases through the Caputo fractional-order SIRV(Susceptible-Infected-Recovered-Vaccinated)epidemic model.Vital properties of the new model including the conditions for the existence of a unique solution determined through the fixed-point theory and the conditions for the existence of a positive solution of the model obtained via the Mittag-Leffler function along with the Laplace transformation-are thoroughly studied.Consequently,our simulation results report that an increase in the imperfect vaccination force increases the population of infected individuals.For the memory effect,the higher“memory”the epidemic system has of past states(which corresponds to decreasing values of fractionalorder parameter),the greater the peaks and magnitudes of infection shaping the epidemiological system dynamics.
