Discrete-time chaotic circuit implementations of a tent map and a Bernoulli map using switched-current （SI） techniques are presented. The two circuits can be constructed with 16 MOSFET＇s and 2 capacitors. The simul...Discrete-time chaotic circuit implementations of a tent map and a Bernoulli map using switched-current （SI） techniques are presented. The two circuits can be constructed with 16 MOSFET＇s and 2 capacitors. The simulations and experiments built with commercially available IC＇s for the circuits have demonstrated the validity of the circuit designs. The experiment results also indicate that the proposed circuits are integrable by a standard CMOS technology. The implementations are useful for studies and applications of chaos.展开更多
A two-SBT-memristor-based chaotic circuit was proposed. The stability of the equilibrium point was studied by theoretical analysis. The close dependence of the circuit dynamic characteristics on its initial conditions...A two-SBT-memristor-based chaotic circuit was proposed. The stability of the equilibrium point was studied by theoretical analysis. The close dependence of the circuit dynamic characteristics on its initial conditions and circuit parameters was investigated by utilizing Lyapunov exponents spectra, bifurcation diagrams, phase diagrams, and Poincaré maps. The analysis showed that the circuit system had complex dynamic behaviors, such as stable points, period, chaos, limit cycles,and so on. In particular, the chaotic circuit produced the multistability phenomenon, such as coexisting attractors and coexisting periods.展开更多
We evaluate the influence of temperature on the behavior of a three-phase clock-driven metal–oxide–semiconductor （MOS） chaotic circuit. The chaotic circuit consists of two nonlinear functions, a level shifter, and...We evaluate the influence of temperature on the behavior of a three-phase clock-driven metal–oxide–semiconductor （MOS） chaotic circuit. The chaotic circuit consists of two nonlinear functions, a level shifter, and three sample and hold blocks. It is necessary to analyze a CMOS-based chaotic circuit with respect to variation in temperature for stability because the circuit is sensitive to the behavior of the circuit design parameters. The temperature dependence of the proposed chaotic circuit is investigated via the simulation program with integrated circuit emphasis （SPICE） using 0.6-μm CMOS process technology with a 5-V power supply and a 20-kHz clock frequency. The simulation results demonstrate the effects of temperature on the chaotic dynamics of the proposed chaotic circuit. The time series, frequency spectra, bifurcation phenomena, and Lyapunov exponent results are provided.展开更多
Based on Chua’s chaotic oscillation circuit, a fifth-order chaotic circuit with two memristors is designed and its corresponding dimensionless mathematic model is established. By using conventional dynamical analysis...Based on Chua’s chaotic oscillation circuit, a fifth-order chaotic circuit with two memristors is designed and its corresponding dimensionless mathematic model is established. By using conventional dynamical analysis methods, stability analysis of the equilibrium set of the circuit is performed, the distribution of stable and unstable regions corresponding to the memristor initial states is achieved, and the complex dynamical behaviors of the circuit depending on the circuit parameters and the memristor initial states are investigated. The theoretical analysis and numerical simulation results demonstrate that the proposed chaotic circuit with two memristors has an equilibrium set located on the plane constituted by the inner state variables of two memristors. The stability of the equilibrium set depends on both the circuit parameters and the initial states of the two memristors. Rich nonlinear dynamical phenomena, such as state transitions, transient hyperchaos and so on, are expected.展开更多
In the paper, the Liu system with a feedback controller is discussed. The influence of the feedback coefficient of the controlled system is studied through Lyapunov exponents spectrum and bifurcation diagram. Various ...In the paper, the Liu system with a feedback controller is discussed. The influence of the feedback coefficient of the controlled system is studied through Lyapunov exponents spectrum and bifurcation diagram. Various attractors are demonstrated not only by numerical simulations but also by circuit experiments. Only one feedback channel is used in our study, which is useful in communication. The circuit experiments show that our study has significance in practical applications.展开更多
Nonlinear oscillators and circuits can be coupled to reach synchronization and consensus. The occurrence of complete synchronization means that all oscillators can maintain the same amplitude and phase, and it is ofte...Nonlinear oscillators and circuits can be coupled to reach synchronization and consensus. The occurrence of complete synchronization means that all oscillators can maintain the same amplitude and phase, and it is often detected between identical oscillators. However, phase synchronization means that the coupled oscillators just keep pace in oscillation even though the amplitude of each node could be different. For dimensionless dynamical systems and oscillators, the synchronization approach depends a great deal on the selection of coupling variable and type. For nonlinear circuits, a resistor is often used to bridge the connection between two or more circuits, so voltage coupling can be activated to generate feedback on the coupled circuits. In this paper, capacitor coupling is applied between two Pikovsk-Rabinovich(PR) circuits, and electric field coupling explains the potential mechanism for differential coupling. Then symmetric coupling and cross coupling are activated to detect synchronization stability, separately. It is found that resistor-based voltage coupling via a single variable can stabilize the synchronization, and the energy flow of the controller is decreased when synchronization is realized. Furthermore, by applying appropriate intensity for the coupling capacitor, synchronization is also reached and the energy flow across the coupling capacitor is helpful in regulating the dynamical behaviors of coupled circuits, which are supported by a continuous energy exchange between capacitors and the inductor. It is also confirmed that the realization of synchronization is dependent on the selection of a coupling channel. The approach and stability of complete synchronization depend on symmetric coupling, which is activated between the same variables. Cross coupling between different variables just triggers phase synchronization. The capacitor coupling can avoid energy consumption for the case with resistor coupling, and it can also enhance the energy exchange between two coupled circuits.展开更多
This paper studies the chaotic behaviours of the fractional-order unified chaotic system. Based on the approximation method in frequency domain, it proposes an electronic circuit model of tree shape to realize the fra...This paper studies the chaotic behaviours of the fractional-order unified chaotic system. Based on the approximation method in frequency domain, it proposes an electronic circuit model of tree shape to realize the fractional-order operator. According to the tree shape model, an electronic circuit is designed to realize the 2.7-order unified chaotic system. Numerical simulations and circuit experiments have verified the existence of chaos in the fraction-order unified system.展开更多
To develop real world memristor application circuits, an equivalent circuit model which imitates memductance （mem- ory conductance） of the HP memristor is presented. The equivalent circuit can be used for breadboard...To develop real world memristor application circuits, an equivalent circuit model which imitates memductance （mem- ory conductance） of the HP memristor is presented. The equivalent circuit can be used for breadboard experiments for various application circuit designs of memristor. Based on memductance of the realistic HP memristor and Chua＇s circuit a new chaotic oscillator is designed. Some basic dynamical behaviors of the oscillator, including equilibrium set, Lyapunov exponent spectrum, and bifurcations with various circuit parameters are investigated theoretically and numerically. To con- firm the correction of the proposed oscillator an analog circuit is designed using the proposed equivalent circuit model of an HP memristor, and the circuit simulations and the experimental results are given.展开更多
This paper reports a new three-dimensional autonomous chaotic system. It contains six control parameters and three nonlinear terms. Two cross-product terms are respectively in two equations. And one square term is in ...This paper reports a new three-dimensional autonomous chaotic system. It contains six control parameters and three nonlinear terms. Two cross-product terms are respectively in two equations. And one square term is in the third equation. Basic dynamic properties of the new system are investigated by means of theoretical analysis, numerical simulation, sensitivity to initial, power spectrum, Lyapunov exponent, and Poincar~ diagrams. The dynamic properties affected by variable parameters are also analysed. Finally, the chaotic system is simulated by circuit. The results verify the existence and implementation of the system.展开更多
In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter va...In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter varies. The system has rich and complex dynamical behaviors, and it is investigated in terms of Lyapunov exponents, bifurcation diagrams, Poincare maps, frequency spectrum, and numerical simulations. In addition, the theoretical analysis shows that the system undergoes a Hopf bifurcation as one parameter varies, which is illustrated by the numerical simulation. Finally, an analog circuit is designed to implement this hyper-chaotic system.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
Modularized circuit designs for chaotic systems are introduced in this paper.Especially,a typical improved modularized design strategy is proposed and applied to a new hyper-chaotic system circuit implementation.In th...Modularized circuit designs for chaotic systems are introduced in this paper.Especially,a typical improved modularized design strategy is proposed and applied to a new hyper-chaotic system circuit implementation.In this paper,the detailed design procedures are described.Multisim simulations and physical experiments are conducted,and the simulation results are compared with Matlab simulation results for different system parameter pairs.These results are consistent with each other and they verify the existence of the hyper-chaotic attractor for this new hyper-chaotic system.展开更多
This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms,and the system can generate a single four-wing chaotic attractor with wide...This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms,and the system can generate a single four-wing chaotic attractor with wide parameter ranges. Through theoretical analysis,the Hopf bifurcation processes are proved to arise at certain equilibrium points.Numerical bifurcation analysis shows that the system has many interesting complex dynamical behaviours;the system trajectory can evolve to a chaotic attractor from a periodic orbit or a fixed point as the proper parameter varies. Finally,an analog electronic circuit is designed to physically realize the chaotic system;the existence of four-wing chaotic attractor is verified by the analog circuit realization.展开更多
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
基金Supported by the National Natural Science Foundation of China (No.60372004) and Natural Science Foundation of Guangdong Province (No.20820)
文摘Discrete-time chaotic circuit implementations of a tent map and a Bernoulli map using switched-current （SI） techniques are presented. The two circuits can be constructed with 16 MOSFET＇s and 2 capacitors. The simulations and experiments built with commercially available IC＇s for the circuits have demonstrated the validity of the circuit designs. The experiment results also indicate that the proposed circuits are integrable by a standard CMOS technology. The implementations are useful for studies and applications of chaos.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 61703247 and 61703246)the Qingdao Science and Technology Plan Project, China (Grant No. 19-6-2-2-cg)the Elite Project of Shandong University of Science and Technology, and the Taishan Scholar Project of Shandong Province of China
文摘A two-SBT-memristor-based chaotic circuit was proposed. The stability of the equilibrium point was studied by theoretical analysis. The close dependence of the circuit dynamic characteristics on its initial conditions and circuit parameters was investigated by utilizing Lyapunov exponents spectra, bifurcation diagrams, phase diagrams, and Poincaré maps. The analysis showed that the circuit system had complex dynamic behaviors, such as stable points, period, chaos, limit cycles,and so on. In particular, the chaotic circuit produced the multistability phenomenon, such as coexisting attractors and coexisting periods.
基金Project supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education(Grant No.2011-0011698)
文摘We evaluate the influence of temperature on the behavior of a three-phase clock-driven metal–oxide–semiconductor （MOS） chaotic circuit. The chaotic circuit consists of two nonlinear functions, a level shifter, and three sample and hold blocks. It is necessary to analyze a CMOS-based chaotic circuit with respect to variation in temperature for stability because the circuit is sensitive to the behavior of the circuit design parameters. The temperature dependence of the proposed chaotic circuit is investigated via the simulation program with integrated circuit emphasis （SPICE） using 0.6-μm CMOS process technology with a 5-V power supply and a 20-kHz clock frequency. The simulation results demonstrate the effects of temperature on the chaotic dynamics of the proposed chaotic circuit. The time series, frequency spectra, bifurcation phenomena, and Lyapunov exponent results are provided.
基金supported by the National Natural Science Foundation of China (Grant No. 60971090)the Natural Science Foundations of Jiangsu Province, China (Grant No. BK2009105)
文摘Based on Chua’s chaotic oscillation circuit, a fifth-order chaotic circuit with two memristors is designed and its corresponding dimensionless mathematic model is established. By using conventional dynamical analysis methods, stability analysis of the equilibrium set of the circuit is performed, the distribution of stable and unstable regions corresponding to the memristor initial states is achieved, and the complex dynamical behaviors of the circuit depending on the circuit parameters and the memristor initial states are investigated. The theoretical analysis and numerical simulation results demonstrate that the proposed chaotic circuit with two memristors has an equilibrium set located on the plane constituted by the inner state variables of two memristors. The stability of the equilibrium set depends on both the circuit parameters and the initial states of the two memristors. Rich nonlinear dynamical phenomena, such as state transitions, transient hyperchaos and so on, are expected.
文摘In the paper, the Liu system with a feedback controller is discussed. The influence of the feedback coefficient of the controlled system is studied through Lyapunov exponents spectrum and bifurcation diagram. Various attractors are demonstrated not only by numerical simulations but also by circuit experiments. Only one feedback channel is used in our study, which is useful in communication. The circuit experiments show that our study has significance in practical applications.
基金Project supported by the National Natural Science Foundation of China(No.11672122)
文摘Nonlinear oscillators and circuits can be coupled to reach synchronization and consensus. The occurrence of complete synchronization means that all oscillators can maintain the same amplitude and phase, and it is often detected between identical oscillators. However, phase synchronization means that the coupled oscillators just keep pace in oscillation even though the amplitude of each node could be different. For dimensionless dynamical systems and oscillators, the synchronization approach depends a great deal on the selection of coupling variable and type. For nonlinear circuits, a resistor is often used to bridge the connection between two or more circuits, so voltage coupling can be activated to generate feedback on the coupled circuits. In this paper, capacitor coupling is applied between two Pikovsk-Rabinovich(PR) circuits, and electric field coupling explains the potential mechanism for differential coupling. Then symmetric coupling and cross coupling are activated to detect synchronization stability, separately. It is found that resistor-based voltage coupling via a single variable can stabilize the synchronization, and the energy flow of the controller is decreased when synchronization is realized. Furthermore, by applying appropriate intensity for the coupling capacitor, synchronization is also reached and the energy flow across the coupling capacitor is helpful in regulating the dynamical behaviors of coupled circuits, which are supported by a continuous energy exchange between capacitors and the inductor. It is also confirmed that the realization of synchronization is dependent on the selection of a coupling channel. The approach and stability of complete synchronization depend on symmetric coupling, which is activated between the same variables. Cross coupling between different variables just triggers phase synchronization. The capacitor coupling can avoid energy consumption for the case with resistor coupling, and it can also enhance the energy exchange between two coupled circuits.
文摘This paper studies the chaotic behaviours of the fractional-order unified chaotic system. Based on the approximation method in frequency domain, it proposes an electronic circuit model of tree shape to realize the fractional-order operator. According to the tree shape model, an electronic circuit is designed to realize the 2.7-order unified chaotic system. Numerical simulations and circuit experiments have verified the existence of chaos in the fraction-order unified system.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61271064 and 60971046)the Natural Science Foundation of Zhejiang Province,China(Grant No.LZ12F01001)the Program for Zhejiang Leading Team of Science and Technology Innovation,China(Grant No.2010R50010-07)
文摘To develop real world memristor application circuits, an equivalent circuit model which imitates memductance （mem- ory conductance） of the HP memristor is presented. The equivalent circuit can be used for breadboard experiments for various application circuit designs of memristor. Based on memductance of the realistic HP memristor and Chua＇s circuit a new chaotic oscillator is designed. Some basic dynamical behaviors of the oscillator, including equilibrium set, Lyapunov exponent spectrum, and bifurcations with various circuit parameters are investigated theoretically and numerically. To con- firm the correction of the proposed oscillator an analog circuit is designed using the proposed equivalent circuit model of an HP memristor, and the circuit simulations and the experimental results are given.
文摘This paper reports a new three-dimensional autonomous chaotic system. It contains six control parameters and three nonlinear terms. Two cross-product terms are respectively in two equations. And one square term is in the third equation. Basic dynamic properties of the new system are investigated by means of theoretical analysis, numerical simulation, sensitivity to initial, power spectrum, Lyapunov exponent, and Poincar~ diagrams. The dynamic properties affected by variable parameters are also analysed. Finally, the chaotic system is simulated by circuit. The results verify the existence and implementation of the system.
基金supported by the National Natural Science Foundation of China(Grant Nos.10772135 and 60874028)the Young Scientists Fund of the National Natural Science Foundation of China(Grant No.11202148)+2 种基金the Incentive Funding of the National Research Foundation of South Africa(GrantNo.IFR2009090800049)the Eskom Tertiary Education Support Programme of South Africathe Research Foundation of Tianjin University of Science and Technology
文摘In the paper, a novel four-wing hyper-chaotic system is proposed and analyzed. A rare dynamic phenomenon is found that this new system with one equilibrium generates a four-wing-hyper-chaotic attractor as parameter varies. The system has rich and complex dynamical behaviors, and it is investigated in terms of Lyapunov exponents, bifurcation diagrams, Poincare maps, frequency spectrum, and numerical simulations. In addition, the theoretical analysis shows that the system undergoes a Hopf bifurcation as one parameter varies, which is illustrated by the numerical simulation. Finally, an analog circuit is designed to implement this hyper-chaotic system.
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supported by the Young Scientists Fund of the National Natural Science Foundation of China(Grant No.61403395)the Natural Science Foundation of Tianjin,China(Grant No.13JCYBJC39000)+3 种基金the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry of Chinathe Fund from the Tianjin Key Laboratory of Civil Aircraft Airworthiness and Maintenance in Civil Aviation of China(Grant No.104003020106)the National Basic Research Program of China(Grant No.2014CB744904)the Fund for the Scholars of Civil Aviation University of China(Grant No.2012QD21x)
文摘Modularized circuit designs for chaotic systems are introduced in this paper.Especially,a typical improved modularized design strategy is proposed and applied to a new hyper-chaotic system circuit implementation.In this paper,the detailed design procedures are described.Multisim simulations and physical experiments are conducted,and the simulation results are compared with Matlab simulation results for different system parameter pairs.These results are consistent with each other and they verify the existence of the hyper-chaotic attractor for this new hyper-chaotic system.
基金Project supported by the National Natural Science Foundation of China(Grant Nos 60774088 and 10772135)the Foundation of the Application Base and Frontier Technology Research Project of Tianjin,China (Grant Nos 07JCZDJC09600,08JCZDJC21900 and 08JCZDJC18600)the Tianjin Key Laboratory for Control Theory & Applications in Complicated Industry Systems of China
文摘This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms,and the system can generate a single four-wing chaotic attractor with wide parameter ranges. Through theoretical analysis,the Hopf bifurcation processes are proved to arise at certain equilibrium points.Numerical bifurcation analysis shows that the system has many interesting complex dynamical behaviours;the system trajectory can evolve to a chaotic attractor from a periodic orbit or a fixed point as the proper parameter varies. Finally,an analog electronic circuit is designed to physically realize the chaotic system;the existence of four-wing chaotic attractor is verified by the analog circuit realization.
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.
