This paper introduces a new four-dimensional （4D） hyperchaotic system, which has only two quadratic nonlinearity parameters but with a complex topological structure. Some complicated dynamical properties are then in...This paper introduces a new four-dimensional （4D） hyperchaotic system, which has only two quadratic nonlinearity parameters but with a complex topological structure. Some complicated dynamical properties are then investigated in detail by using bifurcations, Poincare mapping, LE spectra. Furthermore, a simple fourth-order electronic circuit is designed for hardware implementation of the 4D hyperchaotic attractors. In particular, a remarkable fractional-order circuit diagram is designed for physically verifying the hyperchaotic attractors existing not only in the integer-order system but also in the fractional-order system with an order as low as 3.6.展开更多
In order to better identify the parameters of the fractional-order system,a modified particle swarm optimization(MPSO)algorithm based on an improved Tent mapping is proposed.The MPSO algorithm is validated with eight ...In order to better identify the parameters of the fractional-order system,a modified particle swarm optimization(MPSO)algorithm based on an improved Tent mapping is proposed.The MPSO algorithm is validated with eight classical test functions,and compared with the POS algorithm with adaptive time varying accelerators(ACPSO),the genetic algorithm(GA),a d the improved PSO algorithm with passive congregation(IPSO).Based on the systems with known model structures a d unknown model structures,the proposed algorithm is adopted to identify two typical fractional-order models.The results of parameter identification show that the application of average value of position information is beneficial to making f 11 use of the information exchange among individuals and speeds up the global searching speed.By introducing the uniformity and ergodicity of Tent mapping,the MPSO avoids the extreme v^ue of position information,so as not to fall into the local optimal value.In brief the MPSOalgorithm is an effective a d useful method with a fast convergence rate and high accuracy.展开更多
A challenging topic in nonlinear dynamics concerns the study of fractional-order systems without equilibrium points.In particular, no paper has been published to date regarding the presence of hyperchaos in these syst...A challenging topic in nonlinear dynamics concerns the study of fractional-order systems without equilibrium points.In particular, no paper has been published to date regarding the presence of hyperchaos in these systems. This paper aims to bridge the gap by introducing a new example of fractional-order hyperchaotic system without equilibrium points. The conducted analysis shows that hyperchaos exists in the proposed system when its order is as low as 3.84. Moreover, an interesting application of hyperchaotic synchronization to the considered fractional-order system is provided.展开更多
This study focuses on a graphical approach to determine the robust stabilizing regions of fractional-order PIλ（proportional integration） controllers for fractional-order systems with time-delays. By D-decomposition...This study focuses on a graphical approach to determine the robust stabilizing regions of fractional-order PIλ（proportional integration） controllers for fractional-order systems with time-delays. By D-decomposition technique, the existence conditions and calculating method of the real root boundary（RRB） curves, complex root boundary（CRB） curves and infinite root boundary（IRB）lines are investigated for a given stability degree. The robust stabilizing regions in terms of the RRB curves, CRB curves and IRB lines are identified by the proposed criteria in this paper. Finally, two illustrative examples are given to verify the effectiveness of this graphical approach for different stability degrees.展开更多
In this paper, the fractional-order Genesio-Tesi system showing chaotic behaviours is introduced, and the corresponding one in an integer-order form is studied intensively. Based on the harmonic balance principle, whi...In this paper, the fractional-order Genesio-Tesi system showing chaotic behaviours is introduced, and the corresponding one in an integer-order form is studied intensively. Based on the harmonic balance principle, which is widely used in the frequency analysis of nonlinear control systems, a theoretical approach is used to investigate the conditions of system parameters under which this fractional-order system can give rise to a chaotic attractor. Finally, the numerical simulation is used to verify the validity of the theoretical results.展开更多
This paper addresses improvements in fractional order(FO)system performance.Although the classical proportional-integral-derivative(PID)-like fuzzy controller can provide adequate results for both transient and steady...This paper addresses improvements in fractional order(FO)system performance.Although the classical proportional-integral-derivative(PID)-like fuzzy controller can provide adequate results for both transient and steady-state responses in both linear and nonlinear systems,the FOPID fuzzy controller has been proven to provide better results.This high performance was obtained thanks to the combinative benefits of FO and fuzzy-logic techniques.This paper describes how the optimal gains and FO parameters of the FOPID controller were obtained by the use of a modern optimizer,social spider optimization,in order to improve the response of fractional dynamical systems.This group of systems had usually produced multimodal error surfaces/functions that occasionally had many variant local minima.The integral time of absolute error(ITAE)used in this study was the error function.The results showed that the strategy adopted produced superior performance regarding the lowest ITAE value.It reached a value of 88.22 while the best value obtained in previous work was 98.87.A further comparison between the current work and previous studies concerning transient-analysis factors of the model’s response showed that the strategy proposed was the only one that was able to produce fast rise time,low-percentage overshoot,and very small steady-state error.However,the other strategies were good for one factor,but not for the others.展开更多
In this paper, we propose a robust fractional-order proportional-integral(FOPI) observer for the synchronization of nonlinear fractional-order chaotic systems. The convergence of the observer is proved, and sufficient...In this paper, we propose a robust fractional-order proportional-integral(FOPI) observer for the synchronization of nonlinear fractional-order chaotic systems. The convergence of the observer is proved, and sufficient conditions are derived in terms of linear matrix inequalities(LMIs) approach by using an indirect Lyapunov method. The proposed FOPI observer is robust against Lipschitz additive nonlinear uncertainty. It is also compared to the fractional-order proportional(FOP) observer and its performance is illustrated through simulations done on the fractional-order chaotic Lorenz system.展开更多
This study explores a stable model order reduction method for fractional-order systems. Using the unsymmetric Lanczos algorithm, the reduced order system with a certain number of matched moments is generated. To obtai...This study explores a stable model order reduction method for fractional-order systems. Using the unsymmetric Lanczos algorithm, the reduced order system with a certain number of matched moments is generated. To obtain a stable reduced order system, the stable model order reduction procedure is discussed. By the revised operation on the tridiagonal matrix produced by the unsymmetric Lanczos algorithm, we propose a reduced order modeling method for a fractional-order system to achieve a satisfactory fitting effect with the original system by the matched moments in the frequency domain. Besides, the bound function of the order reduction error is offered. Two numerical examples are presented to illustrate the effectiveness of the proposed method.展开更多
We present the generalized forms of Parrondo's paradox existing in fractional-order nonlinear systems.The gener-alization is implemented by applying a parameter switching(PS)algorithm to the corresponding initial ...We present the generalized forms of Parrondo's paradox existing in fractional-order nonlinear systems.The gener-alization is implemented by applying a parameter switching(PS)algorithm to the corresponding initial value problems associated with the fractional-order nonlinear systems.The PS algorithm switches a system parameter within a specific set of N≥2 values when solving the system with some numerical integration method.It is proven that any attractor of the concerned system can be approximated numerically.By replacing the words"winning"and"loosing"in the classical Parrondo's paradox with"order"and"chaos",respectively,the PS algorithm leads to the generalized Parrondo's paradox:chaos_(1)+chaos_(2)+…+chaos_(N)=order and order_(1)+order_(2)+…+order_(N)=chaos.Finally,the concept is well demon-strated with the results based on the fractional-order Chen system.展开更多
Investigating the combined effects of mining damage and creep damage on slope stability is crucial,as it can comprehensively reveal the non-linear deformation characteristics of rock under their joint influence.This s...Investigating the combined effects of mining damage and creep damage on slope stability is crucial,as it can comprehensively reveal the non-linear deformation characteristics of rock under their joint influence.This study develops a fractional-order nonlinear creep constitutive model that incorporates the double damage effect and implements a non-linear creep subroutine for soft rock using the threedimensional finite difference method on the FLAC3D platform.Comparative analysis of the theoretical,numerical,and experimental results reveals that the fractional-order constitutive model,which incorporates the double damage effect,accurately reflects the distinct deformation stages of green mudstone during creep failure and effectively captures the non-linear deformation in the accelerated creep phase.The numerical results show a fitting accuracy exceeding 97%with the creep test curves,significantly outperforming the 61%accuracy of traditional creep models.展开更多
Breast cancer’s heterogeneous progression demands innovative tools for accurate prediction.We present a hybrid framework that integrates machine learning(ML)and fractional-order dynamics to predict tumor growth acros...Breast cancer’s heterogeneous progression demands innovative tools for accurate prediction.We present a hybrid framework that integrates machine learning(ML)and fractional-order dynamics to predict tumor growth across diagnostic and temporal scales.On the Wisconsin Diagnostic Breast Cancer dataset,seven ML algorithms were evaluated,with deep neural networks(DNNs)achieving the highest accuracy(97.72%).Key morphological features(area,radius,texture,and concavity)were identified as top malignancy predictors,aligning with clinical intuition.Beyond static classification,we developed a fractional-order dynamical model using Caputo derivatives to capture memory-driven tumor progression.The model revealed clinically interpretable patterns:lower fractional orders correlated with prolonged aggressive growth,while higher orders indicated rapid stabilization,mimicking indolent subtypes.Theoretical analyses were rigorously proven,and numerical simulations closely fit clinical data.The framework’s clinical utility is demonstrated through an interactive graphics user interface(GUI)that integrates real-time risk assessment with growth trajectory simulations.展开更多
In this paper,a fractional-order kinematic model is utilized to capture the size-dependent static bending and free vibration responses of piezoelectric nanobeams.The general nonlocal strains in the Euler-Bernoulli pie...In this paper,a fractional-order kinematic model is utilized to capture the size-dependent static bending and free vibration responses of piezoelectric nanobeams.The general nonlocal strains in the Euler-Bernoulli piezoelectric beam are defined by a frame-invariant and dimensionally consistent Riesz-Caputo fractional-order derivatives.The strain energy,the work done by external loads,and the kinetic energy based on the fractional-order kinematic model are derived and expressed in explicit forms.The boundary conditions for the nonlocal Euler-Bernoulli beam are derived through variational principles.Furthermore,a finite element model for the fractional-order system is developed in order to obtain the numerical solutions to the integro-differential equations.The effects of the fractional order and the vibration order on the static bending and vibration responses of the Euler-Bernoulli piezoelectric beams are investigated numerically.The results from the present model are validated against the existing results in the literature,and it is demonstrated that they are theoretically consistent.Although this fractional finite element method(FEM)is presented in the context of a one-dimensional(1D)beam,it can be extended to higher dimensional fractional-order boundary value problems.展开更多
The collective dynamic of a fractional-order globally coupled system with time delays and fluctuating frequency is investigated.The power-law memory of the system is characterized using the Caputo fractional derivativ...The collective dynamic of a fractional-order globally coupled system with time delays and fluctuating frequency is investigated.The power-law memory of the system is characterized using the Caputo fractional derivative operator.Additionally,time delays in the potential field force and coupling force transmission are both considered.Firstly,based on the delay decoupling formula,combined with statistical mean method and the fractional-order Shapiro–Loginov formula,the“statistic synchronization”among particles is obtained,revealing the statistical equivalence between the mean field behavior of the system and the behavior of individual particles.Due to the existence of the coupling delay,the impact of the coupling force on synchronization exhibits non-monotonic,which is different from the previous monotonic effects.Then,two kinds of theoretical expression of output amplitude gains G and G are derived by time-delay decoupling formula and small delay approximation theorem,respectively.Compared to G,G is an exact theoretical solution,which means that G is not only more accurate in the region of small delay,but also applies to the region of large delay.Finally,the study of the output amplitude gain G and its resonance behavior are explored.Due to the presence of the potential field delay,a new resonance phenomenon termed“periodic resonance”is discovered,which arises from the periodic matching between the potential field delay and the driving frequency.This resonance phenomenon is analyzed qualitatively and quantitatively,uncovering undiscovered characteristics in previous studies.展开更多
This study introduces an enhanced adaptive fractional-order nonsingular terminal sliding mode controller(AFONTSMC)tailored for stabilizing a fully submerged hydrofoil craft(FSHC)under external disturbances,model uncer...This study introduces an enhanced adaptive fractional-order nonsingular terminal sliding mode controller(AFONTSMC)tailored for stabilizing a fully submerged hydrofoil craft(FSHC)under external disturbances,model uncertainties,and actuator saturation.A novel nonlinear disturbance observer modified by fractional-order calculus is proposed for flexible and less conservative estimation of lumped disturbances.An enhanced adaptive fractional-order nonsingular sliding mode scheme augmented by disturbance estimation is also introduced to improve disturbance rejection.This controller design only necessitates surpassing the estimation error rather than adhering strictly to the disturbance upper bound.Additionally,an adaptive fast-reaching law with a hyperbolic tangent function is incorporated to enhance the responsiveness and convergence rates of the controller,thereby reducing chattering.Furthermore,an auxiliary actuator compensator is developed to address saturation effects.The resultant closed system of the FSHC with the designed controller is globally asymptotically stable.展开更多
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 low...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.展开更多
In this paper, the leader-following tracking problem of fractional-order multi-agent systems is addressed. The dynamics of each agent may be heterogeneous and has unknown nonlinearities. By assumptions that the intera...In this paper, the leader-following tracking problem of fractional-order multi-agent systems is addressed. The dynamics of each agent may be heterogeneous and has unknown nonlinearities. By assumptions that the interaction topology is undirected and connected and the unknown nonlinear uncertain dynamics can be parameterized by a neural network, an adaptive learning law is proposed to deal with unknown nonlinear dynamics, based on which a kind of cooperative tracking protocols are constructed. The feedback gain matrix is obtained to solve an algebraic Riccati equation. To construct the fully distributed cooperative tracking protocols, the adaptive law is also adopted to adjust the coupling weight. With the developed control laws,we can prove that all signals in the closed-loop systems are guaranteed to be uniformly ultimately bounded. Finally, a simple simulation example is provided to illustrate the established result.展开更多
In this paper, the synchronization in a unified fractional-order chaotic system is investigated by two methods. One is the frequency-domain method that is analysed by using the Laplace transform theory. The other is t...In this paper, the synchronization in a unified fractional-order chaotic system is investigated by two methods. One is the frequency-domain method that is analysed by using the Laplace transform theory. The other is the time-domain method that is analysed by using the Lyapunov stability theory. Finally, the numerical simulations are used-to illustrate the effectiveness of the proposed synchronization methods.展开更多
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
A no-chattering sliding mode control strategy for a class of fractional-order chaotic systems is proposed in this paper. First, the sliding mode control law is derived to stabilize the states of the commensurate fract...A no-chattering sliding mode control strategy for a class of fractional-order chaotic systems is proposed in this paper. First, the sliding mode control law is derived to stabilize the states of the commensurate fractional-order chaotic system and the non-commensurate fractional-order chaotic system, respectively. The designed control scheme guarantees the asymptotical stability of an uncertain fractional-order chaotic system. Simulation results are given for several fractional-order chaotic examples to illustrate the effectiveness of the proposed scheme.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
文摘This paper introduces a new four-dimensional （4D） hyperchaotic system, which has only two quadratic nonlinearity parameters but with a complex topological structure. Some complicated dynamical properties are then investigated in detail by using bifurcations, Poincare mapping, LE spectra. Furthermore, a simple fourth-order electronic circuit is designed for hardware implementation of the 4D hyperchaotic attractors. In particular, a remarkable fractional-order circuit diagram is designed for physically verifying the hyperchaotic attractors existing not only in the integer-order system but also in the fractional-order system with an order as low as 3.6.
基金The National Natural Science Foundation of China(No.61374153,61473138,61374133)the Natural Science Foundation of Jiangsu Province(No.BK20151130)+1 种基金Six Talent Peaks Project in Jiangsu Province(No.2015-DZXX-011)China Scholarship Council Fund(No.201606845005)
文摘In order to better identify the parameters of the fractional-order system,a modified particle swarm optimization(MPSO)algorithm based on an improved Tent mapping is proposed.The MPSO algorithm is validated with eight classical test functions,and compared with the POS algorithm with adaptive time varying accelerators(ACPSO),the genetic algorithm(GA),a d the improved PSO algorithm with passive congregation(IPSO).Based on the systems with known model structures a d unknown model structures,the proposed algorithm is adopted to identify two typical fractional-order models.The results of parameter identification show that the application of average value of position information is beneficial to making f 11 use of the information exchange among individuals and speeds up the global searching speed.By introducing the uniformity and ergodicity of Tent mapping,the MPSO avoids the extreme v^ue of position information,so as not to fall into the local optimal value.In brief the MPSOalgorithm is an effective a d useful method with a fast convergence rate and high accuracy.
文摘A challenging topic in nonlinear dynamics concerns the study of fractional-order systems without equilibrium points.In particular, no paper has been published to date regarding the presence of hyperchaos in these systems. This paper aims to bridge the gap by introducing a new example of fractional-order hyperchaotic system without equilibrium points. The conducted analysis shows that hyperchaos exists in the proposed system when its order is as low as 3.84. Moreover, an interesting application of hyperchaotic synchronization to the considered fractional-order system is provided.
基金supported by National Natural Science Foundation of China(No.61304094)
文摘This study focuses on a graphical approach to determine the robust stabilizing regions of fractional-order PIλ（proportional integration） controllers for fractional-order systems with time-delays. By D-decomposition technique, the existence conditions and calculating method of the real root boundary（RRB） curves, complex root boundary（CRB） curves and infinite root boundary（IRB）lines are investigated for a given stability degree. The robust stabilizing regions in terms of the RRB curves, CRB curves and IRB lines are identified by the proposed criteria in this paper. Finally, two illustrative examples are given to verify the effectiveness of this graphical approach for different stability degrees.
文摘In this paper, the fractional-order Genesio-Tesi system showing chaotic behaviours is introduced, and the corresponding one in an integer-order form is studied intensively. Based on the harmonic balance principle, which is widely used in the frequency analysis of nonlinear control systems, a theoretical approach is used to investigate the conditions of system parameters under which this fractional-order system can give rise to a chaotic attractor. Finally, the numerical simulation is used to verify the validity of the theoretical results.
文摘This paper addresses improvements in fractional order(FO)system performance.Although the classical proportional-integral-derivative(PID)-like fuzzy controller can provide adequate results for both transient and steady-state responses in both linear and nonlinear systems,the FOPID fuzzy controller has been proven to provide better results.This high performance was obtained thanks to the combinative benefits of FO and fuzzy-logic techniques.This paper describes how the optimal gains and FO parameters of the FOPID controller were obtained by the use of a modern optimizer,social spider optimization,in order to improve the response of fractional dynamical systems.This group of systems had usually produced multimodal error surfaces/functions that occasionally had many variant local minima.The integral time of absolute error(ITAE)used in this study was the error function.The results showed that the strategy adopted produced superior performance regarding the lowest ITAE value.It reached a value of 88.22 while the best value obtained in previous work was 98.87.A further comparison between the current work and previous studies concerning transient-analysis factors of the model’s response showed that the strategy proposed was the only one that was able to produce fast rise time,low-percentage overshoot,and very small steady-state error.However,the other strategies were good for one factor,but not for the others.
基金supported by King Abdullah University of Science and Technology (KAUST),KSA
文摘In this paper, we propose a robust fractional-order proportional-integral(FOPI) observer for the synchronization of nonlinear fractional-order chaotic systems. The convergence of the observer is proved, and sufficient conditions are derived in terms of linear matrix inequalities(LMIs) approach by using an indirect Lyapunov method. The proposed FOPI observer is robust against Lipschitz additive nonlinear uncertainty. It is also compared to the fractional-order proportional(FOP) observer and its performance is illustrated through simulations done on the fractional-order chaotic Lorenz system.
基金supported by the National Natural Science Foundation of China(61304094,61673198,61773187)the Natural Science Foundation of Liaoning Province,China(20180520009)
文摘This study explores a stable model order reduction method for fractional-order systems. Using the unsymmetric Lanczos algorithm, the reduced order system with a certain number of matched moments is generated. To obtain a stable reduced order system, the stable model order reduction procedure is discussed. By the revised operation on the tridiagonal matrix produced by the unsymmetric Lanczos algorithm, we propose a reduced order modeling method for a fractional-order system to achieve a satisfactory fitting effect with the original system by the matched moments in the frequency domain. Besides, the bound function of the order reduction error is offered. Two numerical examples are presented to illustrate the effectiveness of the proposed method.
文摘We present the generalized forms of Parrondo's paradox existing in fractional-order nonlinear systems.The gener-alization is implemented by applying a parameter switching(PS)algorithm to the corresponding initial value problems associated with the fractional-order nonlinear systems.The PS algorithm switches a system parameter within a specific set of N≥2 values when solving the system with some numerical integration method.It is proven that any attractor of the concerned system can be approximated numerically.By replacing the words"winning"and"loosing"in the classical Parrondo's paradox with"order"and"chaos",respectively,the PS algorithm leads to the generalized Parrondo's paradox:chaos_(1)+chaos_(2)+…+chaos_(N)=order and order_(1)+order_(2)+…+order_(N)=chaos.Finally,the concept is well demon-strated with the results based on the fractional-order Chen system.
基金support from the National Natural Science Foundation of China(No.52308316)the Scientific Research Foundation of Weifang University(Grant No.2024BS42)+2 种基金China Postdoctoral Science Foundation(No.2022M721885)the Key Laboratory of Rock Mechanics and Geohazards of Zhejiang Province(No.ZJRMG-2022-01)supported by Open Research Fund of State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences(NO.SKLGME023017).
文摘Investigating the combined effects of mining damage and creep damage on slope stability is crucial,as it can comprehensively reveal the non-linear deformation characteristics of rock under their joint influence.This study develops a fractional-order nonlinear creep constitutive model that incorporates the double damage effect and implements a non-linear creep subroutine for soft rock using the threedimensional finite difference method on the FLAC3D platform.Comparative analysis of the theoretical,numerical,and experimental results reveals that the fractional-order constitutive model,which incorporates the double damage effect,accurately reflects the distinct deformation stages of green mudstone during creep failure and effectively captures the non-linear deformation in the accelerated creep phase.The numerical results show a fitting accuracy exceeding 97%with the creep test curves,significantly outperforming the 61%accuracy of traditional creep models.
文摘Breast cancer’s heterogeneous progression demands innovative tools for accurate prediction.We present a hybrid framework that integrates machine learning(ML)and fractional-order dynamics to predict tumor growth across diagnostic and temporal scales.On the Wisconsin Diagnostic Breast Cancer dataset,seven ML algorithms were evaluated,with deep neural networks(DNNs)achieving the highest accuracy(97.72%).Key morphological features(area,radius,texture,and concavity)were identified as top malignancy predictors,aligning with clinical intuition.Beyond static classification,we developed a fractional-order dynamical model using Caputo derivatives to capture memory-driven tumor progression.The model revealed clinically interpretable patterns:lower fractional orders correlated with prolonged aggressive growth,while higher orders indicated rapid stabilization,mimicking indolent subtypes.Theoretical analyses were rigorously proven,and numerical simulations closely fit clinical data.The framework’s clinical utility is demonstrated through an interactive graphics user interface(GUI)that integrates real-time risk assessment with growth trajectory simulations.
基金Project supported by the National Natural Science Foundation of China(No.12172169)。
文摘In this paper,a fractional-order kinematic model is utilized to capture the size-dependent static bending and free vibration responses of piezoelectric nanobeams.The general nonlocal strains in the Euler-Bernoulli piezoelectric beam are defined by a frame-invariant and dimensionally consistent Riesz-Caputo fractional-order derivatives.The strain energy,the work done by external loads,and the kinetic energy based on the fractional-order kinematic model are derived and expressed in explicit forms.The boundary conditions for the nonlocal Euler-Bernoulli beam are derived through variational principles.Furthermore,a finite element model for the fractional-order system is developed in order to obtain the numerical solutions to the integro-differential equations.The effects of the fractional order and the vibration order on the static bending and vibration responses of the Euler-Bernoulli piezoelectric beams are investigated numerically.The results from the present model are validated against the existing results in the literature,and it is demonstrated that they are theoretically consistent.Although this fractional finite element method(FEM)is presented in the context of a one-dimensional(1D)beam,it can be extended to higher dimensional fractional-order boundary value problems.
基金supported by the Natural Science Foundation of Sichuan Province,China(Youth Science Foundation)(Grant No.2022NSFSC1952).
文摘The collective dynamic of a fractional-order globally coupled system with time delays and fluctuating frequency is investigated.The power-law memory of the system is characterized using the Caputo fractional derivative operator.Additionally,time delays in the potential field force and coupling force transmission are both considered.Firstly,based on the delay decoupling formula,combined with statistical mean method and the fractional-order Shapiro–Loginov formula,the“statistic synchronization”among particles is obtained,revealing the statistical equivalence between the mean field behavior of the system and the behavior of individual particles.Due to the existence of the coupling delay,the impact of the coupling force on synchronization exhibits non-monotonic,which is different from the previous monotonic effects.Then,two kinds of theoretical expression of output amplitude gains G and G are derived by time-delay decoupling formula and small delay approximation theorem,respectively.Compared to G,G is an exact theoretical solution,which means that G is not only more accurate in the region of small delay,but also applies to the region of large delay.Finally,the study of the output amplitude gain G and its resonance behavior are explored.Due to the presence of the potential field delay,a new resonance phenomenon termed“periodic resonance”is discovered,which arises from the periodic matching between the potential field delay and the driving frequency.This resonance phenomenon is analyzed qualitatively and quantitatively,uncovering undiscovered characteristics in previous studies.
基金Supported by Natural Science Basic Research Program of Shaanxi under Grant No.2023-JC-QN-0751,No.2023-JC-QN-0778Fundamental Research Funds for the Central Universities,CHD under Grant No.300102324102+1 种基金the National Natural Science Foundation of China under Grant Nos.72471035,52271313Fundamental Research Funds for the Central Universities under Grant No.XK2040021004025.
文摘This study introduces an enhanced adaptive fractional-order nonsingular terminal sliding mode controller(AFONTSMC)tailored for stabilizing a fully submerged hydrofoil craft(FSHC)under external disturbances,model uncertainties,and actuator saturation.A novel nonlinear disturbance observer modified by fractional-order calculus is proposed for flexible and less conservative estimation of lumped disturbances.An enhanced adaptive fractional-order nonsingular sliding mode scheme augmented by disturbance estimation is also introduced to improve disturbance rejection.This controller design only necessitates surpassing the estimation error rather than adhering strictly to the disturbance upper bound.Additionally,an adaptive fast-reaching law with a hyperbolic tangent function is incorporated to enhance the responsiveness and convergence rates of the controller,thereby reducing chattering.Furthermore,an auxiliary actuator compensator is developed to address saturation effects.The resultant closed system of the FSHC with the designed controller is globally asymptotically stable.
基金Project supported by the National Natural Science Foundation of China （Grant No 60404005）.
文摘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.
基金supported by the National Natural Science Foundation of China(61303211)Zhejiang Provincial Natural Science Foundation of China(LY17F030003,LY15F030009)
文摘In this paper, the leader-following tracking problem of fractional-order multi-agent systems is addressed. The dynamics of each agent may be heterogeneous and has unknown nonlinearities. By assumptions that the interaction topology is undirected and connected and the unknown nonlinear uncertain dynamics can be parameterized by a neural network, an adaptive learning law is proposed to deal with unknown nonlinear dynamics, based on which a kind of cooperative tracking protocols are constructed. The feedback gain matrix is obtained to solve an algebraic Riccati equation. To construct the fully distributed cooperative tracking protocols, the adaptive law is also adopted to adjust the coupling weight. With the developed control laws,we can prove that all signals in the closed-loop systems are guaranteed to be uniformly ultimately bounded. Finally, a simple simulation example is provided to illustrate the established result.
文摘In this paper, the synchronization in a unified fractional-order chaotic system is investigated by two methods. One is the frequency-domain method that is analysed by using the Laplace transform theory. The other is the time-domain method that is analysed by using the Lyapunov stability theory. Finally, the numerical simulations are used-to illustrate the effectiveness of the proposed synchronization methods.
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supported by the National Natural Science Foundation of China (Grant No. 51109180)the Personal Special Fund of Northwest Agriculture and Forestry University,China (Grant No. RCZX-2009-01)
文摘A no-chattering sliding mode control strategy for a class of fractional-order chaotic systems is proposed in this paper. First, the sliding mode control law is derived to stabilize the states of the commensurate fractional-order chaotic system and the non-commensurate fractional-order chaotic system, respectively. The designed control scheme guarantees the asymptotical stability of an uncertain fractional-order chaotic system. Simulation results are given for several fractional-order chaotic examples to illustrate the effectiveness of the proposed scheme.
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.
