Although some numerical methods of the fractional-order chaotic systems have been announced,high-precision numerical methods have always been the direction that researchers strive to pursue.Based on this problem,this ...Although some numerical methods of the fractional-order chaotic systems have been announced,high-precision numerical methods have always been the direction that researchers strive to pursue.Based on this problem,this paper introduces a high-precision numerical approach.Some complex dynamic behavior of fractional-order Lorenz chaotic systems are shown by using the present method.We observe some novel dynamic behavior in numerical experiments which are unlike any that have been previously discovered in numerical experiments or theoretical studies.We investigate the influence of α_(1),α_(2),α_(3) on the numerical solution of fractional-order Lorenz chaotic systems.The simulation results of integer order are in good agreement with those of othermethods.The simulation results of numerical experiments demonstrate the effectiveness of the present method.展开更多
This paper explores the existence of heteroclinic cycles and corresponding chaotic dynamics in a class of 3-dimensional two-zone piecewise affine systems. Moreover, the heteroclinic cycles connect two saddle foci and ...This paper explores the existence of heteroclinic cycles and corresponding chaotic dynamics in a class of 3-dimensional two-zone piecewise affine systems. Moreover, the heteroclinic cycles connect two saddle foci and intersect the switching manifold at two points and the switching manifold is composed of two perpendicular planes.展开更多
In this paper, a learning control approach is applied to the generalized projective synchronisation (GPS) of different chaotic systems with unknown periodically time-varying parameters. Using the Lyapunov--Krasovski...In this paper, a learning control approach is applied to the generalized projective synchronisation (GPS) of different chaotic systems with unknown periodically time-varying parameters. Using the Lyapunov--Krasovskii functional stability theory, a differential-difference mixed parametric learning law and an adaptive learning control law are constructed to make the states of two different chaotic systems asymptotically synchronised. The scheme is successfully applied to the generalized projective synchronisation between the Lorenz system and Chen system. Moreover, numerical simulations results are used to verify the effectiveness of the proposed scheme.展开更多
In this paper, a very simple synchronization method is presented for a class of fractional-order chaotic systems only via feedback control. The synchronization technique, based on the stability theory of fractional-or...In this paper, a very simple synchronization method is presented for a class of fractional-order chaotic systems only via feedback control. The synchronization technique, based on the stability theory of fractional-order systems, is simple and theoretically rigorous.展开更多
A controller is designed to realize the synchronization between chaotic systems with different orders. The structure of the controller, the error equations and the Lyapunov functions are determined based on stability ...A controller is designed to realize the synchronization between chaotic systems with different orders. The structure of the controller, the error equations and the Lyapunov functions are determined based on stability theory. Hyperchaotic Chen system and Rossler system are taken for example to demonstrate the method to be effective and feasible. Simulation results show that all the state wriables of Rossler system can be synchronized with those of hyperchaotic Chen system by using only one controller, and the error signals approach zero smoothly and quickly.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
In this paper, some novel sufficient conditions for asymptotic stability of impulsive control systems are presented by comparison systems. The results are used to obtain the conditions under which the chaotic systems ...In this paper, some novel sufficient conditions for asymptotic stability of impulsive control systems are presented by comparison systems. The results are used to obtain the conditions under which the chaotic systems can be asymptotically controlled to the origin via impulsive control. Compared with some existing results, our results are more relaxed in the sense that the Lyapunov function is required to be nonincreasing only along a subsequence of switchings. Moreover, a larger upper bound of impulsive intervals for stabilization and synchronization is obtained.展开更多
In this paper, a modified impulsive control scheme is proposed to realize the complete synchronization of fractional order hyperchaotic systems. By constructing a suitable response system, an integral order synchroniz...In this paper, a modified impulsive control scheme is proposed to realize the complete synchronization of fractional order hyperchaotic systems. By constructing a suitable response system, an integral order synchronization error system is obtained. Based on the theory of Lyapunov stability and the impulsive differential equations, some effective sufficient conditions are derived to guarantee the asymptotical stability of the synchronization error system. In particular, some simpler and more convenient conditions are derived by taking the fixed impulsive distances and control gains. Compared with the existing results, the main results in this paper are practical and rigorous. Simulation results show the effectiveness and the feasibility of the proposed impulsive control method.展开更多
In this paper, we propose a multidimensional version of recurrent least squares support vector machines (MDRLS- SVM) to solve the problem about the prediction of chaotic system. To acquire better prediction performa...In this paper, we propose a multidimensional version of recurrent least squares support vector machines (MDRLS- SVM) to solve the problem about the prediction of chaotic system. To acquire better prediction performance, the high-dimensional space, which provides more information on the system than the scalar time series, is first reconstructed utilizing Takens's embedding theorem. Then the MDRLS-SVM instead of traditional RLS-SVM is used in the high- dimensional space, and the prediction performance can be improved from the point of view of reconstructed embedding phase space. In addition, the MDRLS-SVM algorithm is analysed in the context of noise, and we also find that the MDRLS-SVM has lower sensitivity to noise than the RLS-SVM.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
In this paper, a practical impulsive lag synchronization scheme for different chaotic systems with parametric uncertainties is proposed. By virtue of the new definition of synchronization and the theory of impulsive d...In this paper, a practical impulsive lag synchronization scheme for different chaotic systems with parametric uncertainties is proposed. By virtue of the new definition of synchronization and the theory of impulsive differential equations, some new and less conservative sufficient conditions are established to guarantee that the error dynamics can converge to a predetermined level. The idea and approach developed in this paper can provide a more practical framework for the synchronization between identical and different chaotic systems in parameter perturbation circumstances. Simulation results finally demonstrate the effectiveness of the method.展开更多
The H∞ synchronization problem for a class of delayed chaotic systems with external disturbance is investigated. A novel delayed feedback controller is established under which the chaotic master and slave systems are...The H∞ synchronization problem for a class of delayed chaotic systems with external disturbance is investigated. A novel delayed feedback controller is established under which the chaotic master and slave systems are synchronized with a guaranteed H∞ performance. Based on the Lyapunov stability theory, a delay-dependent condition is derived and formulated in the form of linear matrix inequality (LMI). A numerical simulation is also presented to validate the effectiveness of the developed theoretical results.展开更多
The stability of impulsive fractional-order systems is discussed. A new synchronization criterion of fractional-order chaotic systems is proposed based on the stability theory of impulsive fractional-order systems. Th...The stability of impulsive fractional-order systems is discussed. A new synchronization criterion of fractional-order chaotic systems is proposed based on the stability theory of impulsive fractional-order systems. The synchronization criterion is suitable for the case of the order 0 〈 q ≤ 1. It is more general than those of the known results. Simulation results are given to show the effectiveness of the proposed synchronization criterion.展开更多
This paper presents the problem of generating four-wing (eight-wing) chaotic attractors. The adopted method consists in suitably coupling two (three) identical Lorenz systems. In analogy with the original Lorenz s...This paper presents the problem of generating four-wing (eight-wing) chaotic attractors. The adopted method consists in suitably coupling two (three) identical Lorenz systems. In analogy with the original Lorenz system, where the two wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four wings (eight wings) of these novel attractors axe located around the four (eight) equilibria with two (three) pairs of unstable complex-conjugate eigenvalues.展开更多
In this paper, an improved impulsive lag synchronization scheme for different chaotic systems with parametric uncertainties is proposed. Based on the new definition of synchronization with error bound and a novel impu...In this paper, an improved impulsive lag synchronization scheme for different chaotic systems with parametric uncertainties is proposed. Based on the new definition of synchronization with error bound and a novel impulsive control scheme (the so-called dual-stage impulsive control), some new and less conservative sufficient conditions are established to guarantee that the error dynamics can converge to a predetermined level, which is more reasonable and rigorous than the existing results. In particular, some simpler and more convenient conditions are derived by taking the same impulsive distances and control gains. Finally, some numerical simulations for the Lorenz system and the Chen system are given to demonstrate the effectiveness and feasibility of the proposed method.展开更多
Relerrlng to contlnuous-Ume claaotlc systems, tills paper presents a new projective syncnromzatlon scheme, wnlcn enables each drive system state to be synchronized with a linear combination of response system states f...Relerrlng to contlnuous-Ume claaotlc systems, tills paper presents a new projective syncnromzatlon scheme, wnlcn enables each drive system state to be synchronized with a linear combination of response system states for any arbitrary scaling matrix. The proposed method, based on a structural condition related to the uncontrollable eigenvalues of the error system, can be applied to a wide class of continuous-time chaotic (hyperchaotic) systems and represents a general framework that includes any type of synchronization defined to date. An example involving a hyperchaotic oscillator is reported, with the aim of showing how a response system attractor is arbitrarily shaped using a scalar synchronizing signal only. Finally, it is shown that the recently introduced dislocated synchronization can be readily achieved using the conceived scheme.展开更多
This paper is concerned with the existence of adaptive generalized synchronization(GS) of two chaotic systems.An adaptive control is designed based on a Lyapunov approach.By using modified system approach,some suffici...This paper is concerned with the existence of adaptive generalized synchronization(GS) of two chaotic systems.An adaptive control is designed based on a Lyapunov approach.By using modified system approach,some sufficient conditions for the existence of first two types of adaptive GS inertial manifolds are established.Finally,some numerical simulations are provided to illustrate the theoretical results.展开更多
This paper presents a general method of the generalized projective synchronization and the parameter identification between two different chaotic systems with unknown parameters. This approach is based on Lyapunov sta...This paper presents a general method of the generalized projective synchronization and the parameter identification between two different chaotic systems with unknown parameters. This approach is based on Lyapunov stability theory, and employs a combination of feedback control and adaptive control. With this method one can achieve the generalized projective synchronization and realize the parameter identifications between almost all chaotic (hyperchaotic) systems with unknown parameters. Numerical simulations results are presented to demonstrate the effectiveness of the method.展开更多
The projective reduced-order synchronization of two different chaotic systems with different orders is investigated based on the observer design in this paper.According to the observer theory,the reduced-order observe...The projective reduced-order synchronization of two different chaotic systems with different orders is investigated based on the observer design in this paper.According to the observer theory,the reduced-order observer is designed.The projective synchronization can be realized by choosing the transition matrix of the observer as a diagonal matrix.Further,the synchronization between hyperchaotic Chen system(fourth order)and Rssler system(third order)is taken as the example to demonstrate the effectiveness of the proposed observer.Numerical simulations confirm the effectiveness of the method.展开更多
The networked synchronization problem of a class of master-slave chaotic systems with time-varying communication topologies is investigated in this paper. Based on algebraic graph theory and matrix theory, a simple li...The networked synchronization problem of a class of master-slave chaotic systems with time-varying communication topologies is investigated in this paper. Based on algebraic graph theory and matrix theory, a simple linear state feedback controller is designed to synchronize the master chaotic system and the slave chaotic systems with a time- varying communication topology connection. The exponential stability of the closed-loop networked synchronization error system is guaranteed by applying Lyapunov stability theory. The derived novel criteria are in the form of linear matrix inequalities (LMIs), which are easy to examine and tremendously reduce the computation burden from the feedback matrices. This paper provides an alternative networked secure communication scheme which can be extended conveniently. An illustrative example is given to demonstrate the effectiveness of the proposed networked synchronization method.展开更多
基金supported by the Natural Science Foundation of Inner Mongolia[2021MS01009]Jining Normal University[JSJY2021040,Jsbsjj1704,jsky202145].
文摘Although some numerical methods of the fractional-order chaotic systems have been announced,high-precision numerical methods have always been the direction that researchers strive to pursue.Based on this problem,this paper introduces a high-precision numerical approach.Some complex dynamic behavior of fractional-order Lorenz chaotic systems are shown by using the present method.We observe some novel dynamic behavior in numerical experiments which are unlike any that have been previously discovered in numerical experiments or theoretical studies.We investigate the influence of α_(1),α_(2),α_(3) on the numerical solution of fractional-order Lorenz chaotic systems.The simulation results of integer order are in good agreement with those of othermethods.The simulation results of numerical experiments demonstrate the effectiveness of the present method.
文摘This paper explores the existence of heteroclinic cycles and corresponding chaotic dynamics in a class of 3-dimensional two-zone piecewise affine systems. Moreover, the heteroclinic cycles connect two saddle foci and intersect the switching manifold at two points and the switching manifold is composed of two perpendicular planes.
基金supported by the National Natural Science Foundation of China (Grant No. 60374015)
文摘In this paper, a learning control approach is applied to the generalized projective synchronisation (GPS) of different chaotic systems with unknown periodically time-varying parameters. Using the Lyapunov--Krasovskii functional stability theory, a differential-difference mixed parametric learning law and an adaptive learning control law are constructed to make the states of two different chaotic systems asymptotically synchronised. The scheme is successfully applied to the generalized projective synchronisation between the Lorenz system and Chen system. Moreover, numerical simulations results are used to verify the effectiveness of the proposed scheme.
文摘In this paper, a very simple synchronization method is presented for a class of fractional-order chaotic systems only via feedback control. The synchronization technique, based on the stability theory of fractional-order systems, is simple and theoretically rigorous.
基金Project supported by the National Natural Science Foundation of China (Grant No 20373021) and Natural Science Foundation of Liaoning Province (Grant No 20052151).
文摘A controller is designed to realize the synchronization between chaotic systems with different orders. The structure of the controller, the error equations and the Lyapunov functions are determined based on stability theory. Hyperchaotic Chen system and Rossler system are taken for example to demonstrate the method to be effective and feasible. Simulation results show that all the state wriables of Rossler system can be synchronized with those of hyperchaotic Chen system by using only one controller, and the error signals approach zero smoothly and quickly.
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 National Natural Science Foundation of China(Grant Nos.10926066 and 11026182)the Natural Science Foundation of Zhejiang Province,China(Grant No.Y6100007)+3 种基金the Zhejiang Educational Committee,China(Grant No.Y200805720)the Natural Science Foundation of Jiangsu Province,China(Grant No.BK2010408)the Innovation Fund of Basic Scientific Research Operating Expenses,China(Grant No.3207010501)the Alexander von Humboldt Foundation of Germany
文摘In this paper, some novel sufficient conditions for asymptotic stability of impulsive control systems are presented by comparison systems. The results are used to obtain the conditions under which the chaotic systems can be asymptotically controlled to the origin via impulsive control. Compared with some existing results, our results are more relaxed in the sense that the Lyapunov function is required to be nonincreasing only along a subsequence of switchings. Moreover, a larger upper bound of impulsive intervals for stabilization and synchronization is obtained.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 50830202 and 51073179)the Natural Science Foundation of Chongqing,China (Grant No. CSTC 2010BB2238)+2 种基金the Doctoral Program of Higher Education Foundation of Institutions of China (Grant Nos. 20090191110011 and 20100191120025)the Natural Science Foundation for Postdoctoral Scientists of China (Grant Nos. 20100470813 and 20100480043)the Fundamental Research Funds for the Central Universities(Grant Nos. CDJZR11 12 00 03 and CDJZR11 12 88 01)
文摘In this paper, a modified impulsive control scheme is proposed to realize the complete synchronization of fractional order hyperchaotic systems. By constructing a suitable response system, an integral order synchronization error system is obtained. Based on the theory of Lyapunov stability and the impulsive differential equations, some effective sufficient conditions are derived to guarantee the asymptotical stability of the synchronization error system. In particular, some simpler and more convenient conditions are derived by taking the fixed impulsive distances and control gains. Compared with the existing results, the main results in this paper are practical and rigorous. Simulation results show the effectiveness and the feasibility of the proposed impulsive control method.
基金Project supported by the National Natural Science Foundation of China (Grant No 90207012).
文摘In this paper, we propose a multidimensional version of recurrent least squares support vector machines (MDRLS- SVM) to solve the problem about the prediction of chaotic system. To acquire better prediction performance, the high-dimensional space, which provides more information on the system than the scalar time series, is first reconstructed utilizing Takens's embedding theorem. Then the MDRLS-SVM instead of traditional RLS-SVM is used in the high- dimensional space, and the prediction performance can be improved from the point of view of reconstructed embedding phase space. In addition, the MDRLS-SVM algorithm is analysed in the context of noise, and we also find that the MDRLS-SVM has lower sensitivity to noise than the RLS-SVM.
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.
基金Project supported by the National Natural Science foundation of China (Grant Nos 60534010, 60572070, 60774048 and 60728307)the Program for Changjiang Scholars and Innovative Research Team in University (Grant No 60521003) the National High Technology Research and Development Program of China (Grant No 2006AA04Z183)
文摘In this paper, a practical impulsive lag synchronization scheme for different chaotic systems with parametric uncertainties is proposed. By virtue of the new definition of synchronization and the theory of impulsive differential equations, some new and less conservative sufficient conditions are established to guarantee that the error dynamics can converge to a predetermined level. The idea and approach developed in this paper can provide a more practical framework for the synchronization between identical and different chaotic systems in parameter perturbation circumstances. Simulation results finally demonstrate the effectiveness of the method.
基金supported by National Natural Science Foundation of China (No.60674092)
文摘The H∞ synchronization problem for a class of delayed chaotic systems with external disturbance is investigated. A novel delayed feedback controller is established under which the chaotic master and slave systems are synchronized with a guaranteed H∞ performance. Based on the Lyapunov stability theory, a delay-dependent condition is derived and formulated in the form of linear matrix inequality (LMI). A numerical simulation is also presented to validate the effectiveness of the developed theoretical results.
基金supported by Scientific Research Foundation of Huaiyin Institute of Technology (Grant No. HGA1102)
文摘The stability of impulsive fractional-order systems is discussed. A new synchronization criterion of fractional-order chaotic systems is proposed based on the stability theory of impulsive fractional-order systems. The synchronization criterion is suitable for the case of the order 0 〈 q ≤ 1. It is more general than those of the known results. Simulation results are given to show the effectiveness of the proposed synchronization criterion.
文摘This paper presents the problem of generating four-wing (eight-wing) chaotic attractors. The adopted method consists in suitably coupling two (three) identical Lorenz systems. In analogy with the original Lorenz system, where the two wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four wings (eight wings) of these novel attractors axe located around the four (eight) equilibria with two (three) pairs of unstable complex-conjugate eigenvalues.
基金supported by the National Natural Science Foundation of China (Grant Nos 60534010,60774048,60728307,60804006 and 60521003)the National High Technology Research and Development Program of China (Grant No 2006AA04Z183)+2 种基金Liaoning Provincial Natural Science Foundation of China (Grant No 20062018)State Key Development Program for Basic research of China (Grant No 2009CB320601)111 Project,China (Grant No B08015)
文摘In this paper, an improved impulsive lag synchronization scheme for different chaotic systems with parametric uncertainties is proposed. Based on the new definition of synchronization with error bound and a novel impulsive control scheme (the so-called dual-stage impulsive control), some new and less conservative sufficient conditions are established to guarantee that the error dynamics can converge to a predetermined level, which is more reasonable and rigorous than the existing results. In particular, some simpler and more convenient conditions are derived by taking the same impulsive distances and control gains. Finally, some numerical simulations for the Lorenz system and the Chen system are given to demonstrate the effectiveness and feasibility of the proposed method.
文摘Relerrlng to contlnuous-Ume claaotlc systems, tills paper presents a new projective syncnromzatlon scheme, wnlcn enables each drive system state to be synchronized with a linear combination of response system states for any arbitrary scaling matrix. The proposed method, based on a structural condition related to the uncontrollable eigenvalues of the error system, can be applied to a wide class of continuous-time chaotic (hyperchaotic) systems and represents a general framework that includes any type of synchronization defined to date. An example involving a hyperchaotic oscillator is reported, with the aim of showing how a response system attractor is arbitrarily shaped using a scalar synchronizing signal only. Finally, it is shown that the recently introduced dislocated synchronization can be readily achieved using the conceived scheme.
文摘This paper is concerned with the existence of adaptive generalized synchronization(GS) of two chaotic systems.An adaptive control is designed based on a Lyapunov approach.By using modified system approach,some sufficient conditions for the existence of first two types of adaptive GS inertial manifolds are established.Finally,some numerical simulations are provided to illustrate the theoretical results.
基金Project supported by the Natural Science Foundation of Hebei Province, China (Grant No A2006000128)
文摘This paper presents a general method of the generalized projective synchronization and the parameter identification between two different chaotic systems with unknown parameters. This approach is based on Lyapunov stability theory, and employs a combination of feedback control and adaptive control. With this method one can achieve the generalized projective synchronization and realize the parameter identifications between almost all chaotic (hyperchaotic) systems with unknown parameters. Numerical simulations results are presented to demonstrate the effectiveness of the method.
基金Sponsored by the National Natural Science Foundation of China(Grant No.50877007)the Fundamental Research Funds for the Central Universities(Grant No.DUT10LK12)
文摘The projective reduced-order synchronization of two different chaotic systems with different orders is investigated based on the observer design in this paper.According to the observer theory,the reduced-order observer is designed.The projective synchronization can be realized by choosing the transition matrix of the observer as a diagonal matrix.Further,the synchronization between hyperchaotic Chen system(fourth order)and Rssler system(third order)is taken as the example to demonstrate the effectiveness of the proposed observer.Numerical simulations confirm the effectiveness of the method.
基金supported by the National Natural Science Foundation of China (Grant Nos. 60904046, 60972164, 60974071, and 60804006)the Special Fund for Basic Scientific Research of Central Colleges, Northeastern University, China (Grant No. 090604005)+2 种基金the Science and Technology Program of Shenyang (Grant No. F11-264-1-70)the Program for Liaoning Excellent Talents in University (Grant No. LJQ2011137)the Program for Liaoning Innovative Research Team in University (Grant No. LT2011019)
文摘The networked synchronization problem of a class of master-slave chaotic systems with time-varying communication topologies is investigated in this paper. Based on algebraic graph theory and matrix theory, a simple linear state feedback controller is designed to synchronize the master chaotic system and the slave chaotic systems with a time- varying communication topology connection. The exponential stability of the closed-loop networked synchronization error system is guaranteed by applying Lyapunov stability theory. The derived novel criteria are in the form of linear matrix inequalities (LMIs), which are easy to examine and tremendously reduce the computation burden from the feedback matrices. This paper provides an alternative networked secure communication scheme which can be extended conveniently. An illustrative example is given to demonstrate the effectiveness of the proposed networked synchronization method.