Analysis, circuit implementation and synchronization control of a hyperchaotic system

In this paper, a novel four-dimensional memristive chaotic system is constructed by incorporating a memristor model into a three-dimensional chaotic system. Through the analysis of the Lyapunov exponent, bifurcation diagram, and Poincaré cross-section of the system, it has been observed that the system is capable of exhibiting a stable chaotic state, as well as complex dynamic behaviors, such as attractor coexistence, transient chaos, and offset boosting. To validate the actual existence of the system, a real circuit has been built based on Multisim simulation, and the numerical simulation results, along with the actual simulation results, are in agreement, thereby confirming the practical feasibility of the circuit. To achieve weak signal detection, a backstepping synchronization controller has been designed, which can detect the frequency and amplitude of unknown signals. It is obvious that this method does not require the determination of the critical threshold. Instead of, it relies on the system being in a chaotic state. The proposed detection method provides a new perspective for weak signal detection.

Based on the chaos anti-control theory, a new four-dimensional autonomous hyperchaotic system is formulated by adding a periodic driving signal rsin(cos(¿t)) to the second equation of the three-dimensional autonomous Chen chaotic system in this paper. The basic dynamical behavior of the new hyperchaotic system was not only demonstrated by numerical simulation, but also verified with Lyapunov exponents and bifurcation analysis. The new hyperchaotic system can be periodic, quasi-periodic, chaotic and hyperchaotic with the variation of the parameter ¿. It also has broad and stable hyperchaotic region which makes the new hyperchaotic system has a good application prospect.

This paper proposes a new five-dimensional memristive conservative chaotic system (5D MCCS) by introducing a flux-controlled memristor into a conservative chaotic system. Dynamic characteristics of the 5D MCCS are validated through spectral entropy(SE), bifurcation diagrams, Lyapunov exponents (LEs), and Poincaré mappings. Compared with other conservative chaotic systems, the 5D MCCS contains only three nonlinear terms yet can generate complex dynamic characteristics. Additionally, under fixed initial values, parameter variations lead the 5D MCCS to switch between chaotic states and quasi-periodic states. The 5D MCCS exhibits chaotic states over wide parameter and initial value range. Afterwards, the 5D MCCS is realized on Field Programmable Gate Array (FPGA) platform. Finally, the pseudorandom sequences generated by the 5D MCCS have been certified through the National Institute of Standards and Technology (NIST) and are utilized for image encryption.

Simple Memristive Chaotic Systems with Complex Dynamics

A novel memristive chaotic circuit is proposed by replacing the Chua’s diode in modified Chua’s circuit with a smooth active memristor, and the corresponding memristive model is analyzed and validated. The equilibrium point set, dissipativity and stability of this new chaotic circuit are developed theoretically. The dynamic characteristics for the new system are presented by means of phase diagrams, Lyapunov exponents, bifurcation diagrams and Poincare maps. The coexistence of the memristive system is carried out from the perspective of asymmetric coexistence and symmetry coexistence. In addition, the coexistence of multiple states is studied by a more direct method with initial value as the system variable to gain a more intuitive observation. The circuit model of the memristive chaotic system is designed through Multisim simulation software. Finally, the memristive chaotic sequence is used to encrypt the image, and the influence of multistability on the encryption is investigated by the histogram, correlation and key sensitivity. The results show that the proposed new memristive chaotic system has high security.

The paper presents a new four-dimensional autonomous hyper-chaotic system by introducing a state variable into the three-dimensional autonomous chaotic system based on studying and analyzing some hyper-chaos. Compared with other hyper-chaotic systems，the bigger positive Lyapunov exponent of the new system is bigger, and the range of hyper-chaos is also wider when the parameter varies. Numerical simulations and bifurcation analysis are done in this paper, and an analog electronic circuit is designed to verify the new system. The experiment results show that various attractors of the hyper-chaotic circuit, including periodic attractor, quasi-periodic attractor，chaotic attractor，hyper-chaotic attractor，are well agree with the simulation results.

This paper reports a new five-dimensional four-wing hyperchaotic system with hidden attractor. First, this paper discusses the dynamic properties of the new four-wing system with a detailed bifurcation analysis, coexistence of attractors and multistability, offset boosting, Lyapunov exponents, etc. It is shown that the new four-wing system has no rest point and thus it exhibits hidden attractor. The new four-wing system exhibits two positive Lyapunov characteristic exponents and a large value of Kaplan-Yorke dimension indicating high complexity of the system. We realise the dynamic equations of the new four-wing system with an electronic circuit and simulations via MultiSIM. As a control application, we derive new results for the complete synchronisation of the new four-wing system via integral sliding mode control. MATLAB simulations are adequately provided to illustrate modelling and applications of the new four-wing system with hyperchaotic four-wing attractor.

This paper reports a new five-dimensional four-wing hyperchaotic system with hidden attractor. First, this paper discusses the dynamic properties of the new four-wing system with a detailed bifurcation analysis, coexistence of attractors and multistability, offset boosting, Lyapunov exponents, etc. It is shown that the new four-wing system has no rest point and thus it exhibits hidden attractor. The new four-wing system exhibits two positive Lyapunov characteristic exponents and a large value of Kaplan-Yorke dimension indicating high complexity of the system. We realise the dynamic equations of the new four-wing system with an electronic circuit and simulations via MultiSIM. As a control application, we derive new results for the complete synchronisation of the new four-wing system via integral sliding mode control. MATLAB simulations are adequately provided to illustrate modelling and applications of the new four-wing system with hyperchaotic four-wing attractor.

Biological nervous system function is closely related to its dynamical behaviors, and some dynamical phenomena observed in biological systems can be detected in the simplified neural models. In this paper, the chaotic dynamics in a three-neuron-based Hopfield neural network (HNN) with stimulation of electromagnetic radiation is investigated. The neural network is modeled by utilizing a flux-controlled memristor to describe the effects of electromagnetic field on neurons. The simple neural model affected by electromagnetic radiation does not contain any equilibrium points, but can induce coexisting infinitely many hidden attractors, such as hyperchaos, transient hyperchaos, period, quasi-period, chaos as well as transient chaos with different chaotic times. In particular, the dynamics of hidden extreme multistability with hyperchaos and transient chaos in the neural network highly depends on the system parameters and state initial values. The coexistence of multiple hidden attractors is revealed via applying a host of numerical analysis methods including phase plots, time sequence waveforms, bifurcation diagrams, Lyapunov exponents and attraction basins. Besides, a HNN-based circuit consisting of commercially available electronic elements is designed to verify the theoretical analysis. Hardware measurement and MULTISIM simulation results are basically consistent with MATLAB numerical simulation results.

A new 4-D hyperchaotic two-scroll system with three quadratic nonlinearities and a cubic nonlinearity is proposed in this paper. The dynamical properties of the new hyperchaotic system are described in terms of phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, symmetry, dissipativity, etc. We also establish that the new hyperchaotic system has multistability with coexisting attractors. As a control application, we use integral sliding mode control for active self-synchronization of the new hyperchaotic systems as master-slave systems. As an engineering application, an electronic circuit design of the new hyperchaotic two-scroll system is developed in MultiSIM, which confirms the feasibility of the system.

We propose a new hyperchaotic complex system. The hyperchaotic complex system has a distinctive chaotic attractor, which is called butterfly-like attractor. We find that the state of the system has a complex evolution process with the change of parameters, and the system will produce at least two kinds of attractors, which we call parametric attractors. We systematically study the mechanics, symmetry, dissipation, Lyapunov exponents and Poincaré maps of the hyperchaotic complex system. We extend the hyperchaotic complex system to fractional order, and discuss the influence of order and parameters’ change on fractional order new hyperchaotic system, and compare it with integer order. What is interesting is that there exist parametric attractors in both integer order and fractional order of the new system. Based on the integer order and fractional order, we further analyze the new hyperchaotic complex system of hybrid order, propose the definition of complex chaotic family and carry out mechanical analysis.

We construct a three-dimensional chaotic system based on TANG chaotic system and analyze the system dynamics based on the equilibrium points, phase diagram, Lyapunov exponential spectrum, bifurcation diagram, and complexity. The more complexly dynamic behaviors indicate that our designed system has different topology, bifurcation anti-monotonic property and attractor coexistence phenomenon from TANG system. The system has the coexistence of chaotic attractor and point attractor and can select the initial value that is suitable for chaotic secure communication by analyzing the coexistence attractor. The actual digital circuit is implemented with field programmable gate array (FPGA). The system is extremely sensitive to the selections of initial values and system parameters and has good and rich dynamic behavior, which is suitable for the field of chaotic confidential communication. The linear feedback method is used to control the system synchronous and the simulation of synchronous circuit is realized by Multisim software. The simulation results are consistent with the numerical analysis, which lays a theoretical foundation for the application of the system in the field of synchronized confidential communication.

A new multi-scroll Chua’s circuit with composite hyperbolic tangent-cubic nonlinearity: Complex dynamics, Hardware implementation and Image encryption application

Hyperchaotic systems are defined as chaotic systems with more than one positive Lyapunov exponent. Combined with one null Lyapunov exponent along the flow and one negative Lyapunov exponent to ensure boundedness of the solution, the minimal dimension for a continuous hyperchaotic system is four. The hyperchaotic systems are known to have important applications in secure communications and cryptosystems. First, this work describes an eleven-term 4-D novel hyperchaotic system with four quadratic nonlinearities. The qualitative properties of the novel hyperchaotic system are described in detail. The Lyapunov exponents of the system are obtained as \( L_{1} = 0.7781,L_{2} = 0.2299,L_{3} = 0 \) and \( L_{4} = - 12.5062 \). The maximal Lyapunov exponent of the system (MLE) is \( L_{1} = 0.7781 \). The Lyapunov dimension of the novel hyperchaotic system is obtained as \( D_{L} = 3.0806 \). Next, the work describes an adaptive controller design for the global chaos control of the novel hyperchaotic system. The main result for the adaptive controller design has been proved using Lyapunov stability theory. MATLAB simulations are described in detail for all the main results derived in this work for the eleven-term 4-D novel hyperchaotic system with four quadratic nonlinearities.

This paper investigates the hybrid synchronization of identical hyperbolic Qi systems and hybrid synchronization of hyperchaotic Qi and Lu systems. The hyperchaotic Qi system (2008) and hyperchaotic Lu system (2006) are important models of hyperchaotic systems. Hybrid synchronization of the hyperchaotic systems is achieved through synchronization of two pairs of states and anti-synchronization of the other two pairs of states of the two hyperchaotic systems. Nonlinear control is the method used for the hybrid synchronization of identical hyperbolic Qi systems and hybrid synchronization of hyperchaotic Qi and Lu systems. Since the Lyapunov exponents are not required for these calculations, this method is effective and convenient to achieve hybrid synchronization of the two hyperchaotic systems. Numerical simulations are shown to verify the results.