This paper proposes a novel approach for generating a multi-scroll chaotic system. Together with the theoretical design and numerical simulations, three different types of attractor are available, governed by construc...This paper proposes a novel approach for generating a multi-scroll chaotic system. Together with the theoretical design and numerical simulations, three different types of attractor are available, governed by constructing triangular wave, sawtooth wave and hysteresis sequence. The presented new multi-scroll chaotic system is different from the classical multi-scroll chaotic Chua system in dimensionless state equations, nonlinear functions and maximum Lyapunov exponents. In addition, the basic dynamical behaviours, including equilibrium points, eigenvalues, eigenvectors, eigenplanes, bifurcation diagrams and Lyapunov exponents, are further investigated. The success of the design is illustrated by both numerical simulations and circuit experiments.展开更多
Dynamical variables of coupled nonlinear oscillators can exhibit different synchronization patterns depending on the designed coupling scheme. In this paper, a non-fragile linear feedback control strategy with multipl...Dynamical variables of coupled nonlinear oscillators can exhibit different synchronization patterns depending on the designed coupling scheme. In this paper, a non-fragile linear feedback control strategy with multiplicative controller gain uncertainties is proposed for realizing the mixed-synchronization of Chua's circuits connected in a drive-response configuration. In particular, in the mixed-synchronization regime, different state variables of the response system can evolve into complete synchronization, anti-synchronization and even amplitude death simultaneously with the drive variables for an appropriate choice of scaling matrix. Using Lyapunov stability theory, we derive some sufficient criteria for achieving global mixed-synchronization. It is shown that the desired non-fragile state feedback controller can be constructed by solving a set of linear matrix inequalities (LMIs). Numerical simulations are also provided to demonstrate the effectiveness of the proposed control approach.展开更多
This paper proposed a method of generating two attractors in a novel grid multi-scroll chaotic system. Based on a newly generated three-dimensional system, a two-attractor grid multi-scroll attractor system can be gen...This paper proposed a method of generating two attractors in a novel grid multi-scroll chaotic system. Based on a newly generated three-dimensional system, a two-attractor grid multi-scroll attractor system can be generated by adding two triangular waves and a sign function. Some basic dynamical properties, such as equilibrium points, bifurcations, and phase diagrams, were studied. Furthermore, the system was experimentally confirmed by an electronic circuit. The circuit simulation results and numerical simulation results verified the feasibility of this method.展开更多
In this paper a new simple multi-scroll chaotic generator is studied. The characteristic of this new multi-scroll chaotic generator is that it is easy to generate different number of scroll chaotic attractors through ...In this paper a new simple multi-scroll chaotic generator is studied. The characteristic of this new multi-scroll chaotic generator is that it is easy to generate different number of scroll chaotic attractors through modifying the nature number n after fixing the suitable system parameters and it does not need complex mathematical derivation. Various number of scroll chaotic attractors are illustrated not only by computer simulation but also by the realization of an electronic circuit experiment on Electronic Workbench (EWB).展开更多
Based on a new three-dimensional autonomous linear system and designing a specific form of saturated function series and a sign function with two variables of system, which are employed to increase saddle-focus equili...Based on a new three-dimensional autonomous linear system and designing a specific form of saturated function series and a sign function with two variables of system, which are employed to increase saddle-focus equilibrium points with index 2, a novel multi-scroll chaotic system is proposed and its typical dynamical characteristics including bifurcation diagram, Poincare map, and the stability of equilibrium points are analyzed. The hardware circuit is designed and the experimental results are presented for confirmation.展开更多
This paper considers the chaos synchronization of the modified Chua 's circuit with x|x| function. We firstly show that a couple of the modified Chua systems with different parameters and initial conditions can be...This paper considers the chaos synchronization of the modified Chua 's circuit with x|x| function. We firstly show that a couple of the modified Chua systems with different parameters and initial conditions can be synchronized using active control when the values of parameters both in drive system and response system are known aforehand.Furthermore, based on Lyapunov stability theory we propose an adaptive active control approach to make the states of two identical Chua systems with unknown constant parameters asymptotically synchronized. Moreover the designed controller is independent of those unknown parameters. Numerical simulations are given to validate the proposed synchronization approach.展开更多
Presents analytic criteria for the local activity theory in two-port cellularneural network (CNN) cells with four local state variables, and gives the application to a smoothedChua's circuit (SCC) CNN with two-por...Presents analytic criteria for the local activity theory in two-port cellularneural network (CNN) cells with four local state variables, and gives the application to a smoothedChua's circuit (SCC) CNN with two-port and I 5 S 15 arrays. The bifurcation diagrams of the SCC CNNshow that they are completely the same as those of an SCC CNN with one-port calculated earlier;which do not exist locally passive domain. The evolution of the patterns of the state variables ofthe SCC CNN is stimulated. Oscillatory patterns, chaotic patterns, or divergent patterns may emergeif the selected cell parameters are located in the locally active unstable domains but nearby theedge of chaos domain.展开更多
The ultimate proof of our understanding of nature and engineering systems is reflected in our ability to control them.Since fractional calculus is more universal, we bring attention to the controllability of fractiona...The ultimate proof of our understanding of nature and engineering systems is reflected in our ability to control them.Since fractional calculus is more universal, we bring attention to the controllability of fractional order systems. First,we extend the conventional controllability theorem to the fractional domain. Strictly mathematical analysis and proof are presented. Because Chua's circuit is a typical representative of nonlinear circuits, we study the controllability of the fractional order Chua's circuit in detail using the presented theorem. Numerical simulations and theoretical analysis are both presented, which are in agreement with each other.展开更多
This paper investigates the chaos synchronisation between two coupled chaotic Chua's circuits. The sufficient condition presented by linear matrix inequalities (LMIs) of global asymptotic synchronisation is attaine...This paper investigates the chaos synchronisation between two coupled chaotic Chua's circuits. The sufficient condition presented by linear matrix inequalities (LMIs) of global asymptotic synchronisation is attained based on piecewise quadratic Lyapunov functions. First, we obtain the piecewise linear differential inclusions (pwLDIs) model of synchronisation error dynamics, then we design a switching (piecewise-linear) feedback control law to stabilise it based on the piecewise quadratic Laypunov functions. Then we give some numerical simulations to demonstrate the effectiveness of our theoretical results.展开更多
Chua's circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua's circuit system with the smooth deg...Chua's circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua's circuit system with the smooth degree of 3 are studied. The parametric areas of stability are specified in detail. It is found that the bifurcation graphs of the su- percritical and irregular pitchfork bifurcations are similar to those of the piecewise-smooth continuous (PWSC) systems caused by piecewise smoothness. However, the bifurcation graph of the supercritical Hopf bifurcation is similar to those of smooth systems. There- fore, the boundary equilibrium bifurcations of the non-smooth systems with the smooth degree of 3 should receive more attention due to their special features.展开更多
The analytic criteria for the local activity theory in one-port cellularneural network (CNN) with five local state variables are presented. The application to a Hyper-chaossynchronization Chua's circuit (HCSCC) CN...The analytic criteria for the local activity theory in one-port cellularneural network (CNN) with five local state variables are presented. The application to a Hyper-chaossynchronization Chua's circuit (HCSCC) CNN with 1125 variables is studied. The bifurcation diagramsof the HCSCC CNN show that they are slightly different from the smoothed CNN with one or two portsand four state variables calculated earlier. The evolution of the patterns of the state variables ofthe HCSCC CNN is stimulated. Oscillatory patterns, chaotic patterns, convergent or divergentpatterns may emerge if the selected cell parameters are located in the locally active domains butnearby or in the edge of chaos domain.展开更多
Based on the α-β bifurcation curves and the special characteristics of chaotic spectrum of chua’s circuit, this paper presents here a method for designing a Chua’s circuit which approximately satisfy specified spe...Based on the α-β bifurcation curves and the special characteristics of chaotic spectrum of chua’s circuit, this paper presents here a method for designing a Chua’s circuit which approximately satisfy specified spectrum distribution range.展开更多
Least-square support vector machines(LS-SVM) are applied for learning the chaotic behavior of Chua's circuit.The system is divided into three multiple-input single-output(MISO) structures and the LS-SVM are train...Least-square support vector machines(LS-SVM) are applied for learning the chaotic behavior of Chua's circuit.The system is divided into three multiple-input single-output(MISO) structures and the LS-SVM are trained individually.Comparing with classical approaches,the proposed one reduces the structural complexity and the selection of parameters is avoided.Some parameters of the attractor are used to compare the chaotic behavior of the reconstructed and the original systems for model validation.Results show that the LS-SVM combined with the MISO can be trained to identify the underlying link among Chua's circuit state variables,and exhibit the chaotic attractors under the autonomous working mode.展开更多
The eigenvalue space of the canonical four-dimensional Chua's circuit which can realize every eigenvalue for fourdimensional system is studied in this paper. First, the analytical relations between the circuit parame...The eigenvalue space of the canonical four-dimensional Chua's circuit which can realize every eigenvalue for fourdimensional system is studied in this paper. First, the analytical relations between the circuit parameters and the eigenvalues of the system are established, and therefore all the circuit parameters can be determined explicitly by any given set of eigenvalues. Then, the eigenvalue space of the circuit is investigated in two cases by the nonlinear elements used. According to the types of the eigenvalues, some novel hyperchaotic attractors are presented. Further, the dynamic behaviours of the circuit are studied by the bifurcation diagrams and the Lyapunov spectra of the eigenvalues.展开更多
This paper proposes a new simple autonomous chaotic system which can generate multi-scroll chaotic attractors. The characteristic of this new multi-scroll chaotic system is that the 4n + 2rn + 4-scroll chaotic attra...This paper proposes a new simple autonomous chaotic system which can generate multi-scroll chaotic attractors. The characteristic of this new multi-scroll chaotic system is that the 4n + 2rn + 4-scroll chaotic attractors are generated easily with n and m vaxying under n ≤ m. Various number of scroll chaotic attractors are illustrated not only by computer simulation but also by the realization of an electronic circuit experiment on EWB (Electronics Workbench).展开更多
A new method is presented to generate two-directional (2D) grid multi-scroll chaotic attractors via a specific form of the sine function and sign function series, which are applied to increase saddle points of index 2...A new method is presented to generate two-directional (2D) grid multi-scroll chaotic attractors via a specific form of the sine function and sign function series, which are applied to increase saddle points of index 2. The scroll number in the x-direction is modified easily through changing the thresholds of the specific form of the sine function, while the scroll number in the y-direction is controlled by the sign function series. Some basic dynamical properties, such as equilibrium points, bifurcation diagram, phase portraits, and Lyapunov exponents spectrum are studied. Furthermore, the electronic circuit of the system is designed and its simulation results are given by Multisim 10.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos 60572073 and 60871025)the Natural Science Foundation of Guangdong Province,China (Grant Nos 8151009001000060,5001818 and 8351009001000002)
文摘This paper proposes a novel approach for generating a multi-scroll chaotic system. Together with the theoretical design and numerical simulations, three different types of attractor are available, governed by constructing triangular wave, sawtooth wave and hysteresis sequence. The presented new multi-scroll chaotic system is different from the classical multi-scroll chaotic Chua system in dimensionless state equations, nonlinear functions and maximum Lyapunov exponents. In addition, the basic dynamical behaviours, including equilibrium points, eigenvalues, eigenvectors, eigenplanes, bifurcation diagrams and Lyapunov exponents, are further investigated. The success of the design is illustrated by both numerical simulations and circuit experiments.
基金Project supported by the Foundation for Distinguished Young Talents in Higher Education of Guangdong Province of China(Grant No. LYM10074)the Natural Science Foundation of Guangdong Province,China (Grant No. 9451042001004076)
文摘Dynamical variables of coupled nonlinear oscillators can exhibit different synchronization patterns depending on the designed coupling scheme. In this paper, a non-fragile linear feedback control strategy with multiplicative controller gain uncertainties is proposed for realizing the mixed-synchronization of Chua's circuits connected in a drive-response configuration. In particular, in the mixed-synchronization regime, different state variables of the response system can evolve into complete synchronization, anti-synchronization and even amplitude death simultaneously with the drive variables for an appropriate choice of scaling matrix. Using Lyapunov stability theory, we derive some sufficient criteria for achieving global mixed-synchronization. It is shown that the desired non-fragile state feedback controller can be constructed by solving a set of linear matrix inequalities (LMIs). Numerical simulations are also provided to demonstrate the effectiveness of the proposed control approach.
基金Project supported by the National Natural Science Foundation of China (Grant No. 60972069)the Science and Technology Foundation of the Education Department of Chongqing (Grant No. KJ090513)
文摘This paper proposed a method of generating two attractors in a novel grid multi-scroll chaotic system. Based on a newly generated three-dimensional system, a two-attractor grid multi-scroll attractor system can be generated by adding two triangular waves and a sign function. Some basic dynamical properties, such as equilibrium points, bifurcations, and phase diagrams, were studied. Furthermore, the system was experimentally confirmed by an electronic circuit. The circuit simulation results and numerical simulation results verified the feasibility of this method.
文摘In this paper a new simple multi-scroll chaotic generator is studied. The characteristic of this new multi-scroll chaotic generator is that it is easy to generate different number of scroll chaotic attractors through modifying the nature number n after fixing the suitable system parameters and it does not need complex mathematical derivation. Various number of scroll chaotic attractors are illustrated not only by computer simulation but also by the realization of an electronic circuit experiment on Electronic Workbench (EWB).
基金Project supported by the National Natural Science Foundation of China(Grant Nos.51377124 and 51521065)the Foundation for the Author of National Excellent Doctoral Dissertation,China(Grant No.201337)the New Star of Youth Science and Technology of Shaanxi Province,China(Grant No.2016KJXX-40)
文摘Based on a new three-dimensional autonomous linear system and designing a specific form of saturated function series and a sign function with two variables of system, which are employed to increase saddle-focus equilibrium points with index 2, a novel multi-scroll chaotic system is proposed and its typical dynamical characteristics including bifurcation diagram, Poincare map, and the stability of equilibrium points are analyzed. The hardware circuit is designed and the experimental results are presented for confirmation.
文摘This paper considers the chaos synchronization of the modified Chua 's circuit with x|x| function. We firstly show that a couple of the modified Chua systems with different parameters and initial conditions can be synchronized using active control when the values of parameters both in drive system and response system are known aforehand.Furthermore, based on Lyapunov stability theory we propose an adaptive active control approach to make the states of two identical Chua systems with unknown constant parameters asymptotically synchronized. Moreover the designed controller is independent of those unknown parameters. Numerical simulations are given to validate the proposed synchronization approach.
基金This project is jointly supported by the National Nature Science Foundation of China (Grant No. 60074034) and the Foundation for
文摘Presents analytic criteria for the local activity theory in two-port cellularneural network (CNN) cells with four local state variables, and gives the application to a smoothedChua's circuit (SCC) CNN with two-port and I 5 S 15 arrays. The bifurcation diagrams of the SCC CNNshow that they are completely the same as those of an SCC CNN with one-port calculated earlier;which do not exist locally passive domain. The evolution of the patterns of the state variables ofthe SCC CNN is stimulated. Oscillatory patterns, chaotic patterns, or divergent patterns may emergeif the selected cell parameters are located in the locally active unstable domains but nearby theedge of chaos domain.
基金supported by the National Natural Science Foundation of China(Grant Nos.51109180 and 51479173)the Fundamental Research Funds for the Central Universities,China(Grant No.201304030577)+1 种基金the Northwest A&F University Foundation,China(Grant No.2013BSJJ095)the Scientific Research Foundation on Water Engineering of Shaanxi Province,China(Grant No.2013slkj-12)
文摘The ultimate proof of our understanding of nature and engineering systems is reflected in our ability to control them.Since fractional calculus is more universal, we bring attention to the controllability of fractional order systems. First,we extend the conventional controllability theorem to the fractional domain. Strictly mathematical analysis and proof are presented. Because Chua's circuit is a typical representative of nonlinear circuits, we study the controllability of the fractional order Chua's circuit in detail using the presented theorem. Numerical simulations and theoretical analysis are both presented, which are in agreement with each other.
基金Project partially supported by the grant from the Research Grants Council of the Hong Kong Special Administrative Region,China (Grant No. 101005)the National Natural Science Foundation of China (Grant No. 60904004)the Key Youth Science and Technology Foundation of University of Electronic Science and Technology of China (Grant No. L08010201JX0720)
文摘This paper investigates the chaos synchronisation between two coupled chaotic Chua's circuits. The sufficient condition presented by linear matrix inequalities (LMIs) of global asymptotic synchronisation is attained based on piecewise quadratic Lyapunov functions. First, we obtain the piecewise linear differential inclusions (pwLDIs) model of synchronisation error dynamics, then we design a switching (piecewise-linear) feedback control law to stabilise it based on the piecewise quadratic Laypunov functions. Then we give some numerical simulations to demonstrate the effectiveness of our theoretical results.
基金supported by the National Natural Science Foundation of China(Nos.U1204106,11372282,11272024,and 11371046)the National Basic Research Program of China(973 Program)(Nos.2012CB821200 and 2012CB821202)
文摘Chua's circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua's circuit system with the smooth degree of 3 are studied. The parametric areas of stability are specified in detail. It is found that the bifurcation graphs of the su- percritical and irregular pitchfork bifurcations are similar to those of the piecewise-smooth continuous (PWSC) systems caused by piecewise smoothness. However, the bifurcation graph of the supercritical Hopf bifurcation is similar to those of smooth systems. There- fore, the boundary equilibrium bifurcations of the non-smooth systems with the smooth degree of 3 should receive more attention due to their special features.
基金the National Natural Science Foundation of China (Grant No. 60074034) and the Foundation forUniversity Key Teacher by the Ministry of Education of China.
文摘The analytic criteria for the local activity theory in one-port cellularneural network (CNN) with five local state variables are presented. The application to a Hyper-chaossynchronization Chua's circuit (HCSCC) CNN with 1125 variables is studied. The bifurcation diagramsof the HCSCC CNN show that they are slightly different from the smoothed CNN with one or two portsand four state variables calculated earlier. The evolution of the patterns of the state variables ofthe HCSCC CNN is stimulated. Oscillatory patterns, chaotic patterns, convergent or divergentpatterns may emerge if the selected cell parameters are located in the locally active domains butnearby or in the edge of chaos domain.
文摘Based on the α-β bifurcation curves and the special characteristics of chaotic spectrum of chua’s circuit, this paper presents here a method for designing a Chua’s circuit which approximately satisfy specified spectrum distribution range.
基金Project supported by the National Natural Science Foundation of China (Grant No. 61072103)the Jiangxi Province Training Program for Younger Scientists
文摘Least-square support vector machines(LS-SVM) are applied for learning the chaotic behavior of Chua's circuit.The system is divided into three multiple-input single-output(MISO) structures and the LS-SVM are trained individually.Comparing with classical approaches,the proposed one reduces the structural complexity and the selection of parameters is avoided.Some parameters of the attractor are used to compare the chaotic behavior of the reconstructed and the original systems for model validation.Results show that the LS-SVM combined with the MISO can be trained to identify the underlying link among Chua's circuit state variables,and exhibit the chaotic attractors under the autonomous working mode.
基金Project supported by the National Natural Science Foundation of China (Grant No. 50877007)
文摘The eigenvalue space of the canonical four-dimensional Chua's circuit which can realize every eigenvalue for fourdimensional system is studied in this paper. First, the analytical relations between the circuit parameters and the eigenvalues of the system are established, and therefore all the circuit parameters can be determined explicitly by any given set of eigenvalues. Then, the eigenvalue space of the circuit is investigated in two cases by the nonlinear elements used. According to the types of the eigenvalues, some novel hyperchaotic attractors are presented. Further, the dynamic behaviours of the circuit are studied by the bifurcation diagrams and the Lyapunov spectra of the eigenvalues.
文摘This paper proposes a new simple autonomous chaotic system which can generate multi-scroll chaotic attractors. The characteristic of this new multi-scroll chaotic system is that the 4n + 2rn + 4-scroll chaotic attractors are generated easily with n and m vaxying under n ≤ m. Various number of scroll chaotic attractors are illustrated not only by computer simulation but also by the realization of an electronic circuit experiment on EWB (Electronics Workbench).
文摘A new method is presented to generate two-directional (2D) grid multi-scroll chaotic attractors via a specific form of the sine function and sign function series, which are applied to increase saddle points of index 2. The scroll number in the x-direction is modified easily through changing the thresholds of the specific form of the sine function, while the scroll number in the y-direction is controlled by the sign function series. Some basic dynamical properties, such as equilibrium points, bifurcation diagram, phase portraits, and Lyapunov exponents spectrum are studied. Furthermore, the electronic circuit of the system is designed and its simulation results are given by Multisim 10.