A new hyperchaotic system and its synchronization

In this paper, a four-dimensional (4D) continuous autonomous hyperchaotic system is introduced and analyzed. This hyperchaotic system is constructed by adding a linear controller to the 3D autonomous chaotic system with a reverse butterfly-shape attractor. Some of its basic dynamical properties, such as Lyapunov exponents, Poincare section, bifurcation diagram and the periodic orbits evolving into chaotic, hyperchaotic dynamical behavior by varying parameter d are studied. Furthermore, the full state hybrid projective synchronization (FSHPS) of new hyperchaotic system with unknown parameters including the unknown coefficients of nonlinear terms is studied by using adaptive control. Numerical simulations are presented to show the effective of the proposed chaos synchronization scheme.     

Hyperchaos in fractional order nonlinear systems

We numerically investigate hyperchaotic behavior in an autonomous nonlinear system of fractional order. It is demonstrated that hyperchaotic behavior of the integer order nonlinear system is preserved when the order becomes fractional. The system under study has been reported in the literature [Murali K, Tamasevicius A, Mykolaitis G, Namajunas A, Lindberg E. Hyperchaotic system with unstable oscillators. Nonlinear Phenom Complex Syst 3(1);2000:7–10], and consists of two nonlinearly coupled unstable oscillators, each consisting of an amplifier and an LC resonance loop. The fractional order model of this system is obtained by replacing one or both of its capacitors by fractional order capacitors. Hyperchaos is then assessed by studying the Lyapunov spectrum. The presence of multiple positive Lyapunov exponents in the spectrum is indicative of hyperchaos. Using the appropriate system control parameters, it is demonstrated that hyperchaotic attractors are obtained for a system order less than 4. Consequently, we present a conjecture that fourth-order hyperchaotic nonlinear systems can still produce hyperchaotic behavior with a total system order of 3 + ε, where 1 > ε > 0.     

Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters

This paper brings attention to hyperchaos anti-synchronization between two identical and different hyperchaotic systems by using adaptive control. The sufficient conditions for achieving the anti-synchronization of two hyperchaotic systems are derived based on Lyapunov stability theory. An adaptive control law and a parameter update rule for unknown parameters are introduced such that the hyperchaotic Chen system is controlled to be the hyperchaotic Lü system. Theoretical analysis and numerical simulations are shown to verify the results.     

Hyperchaotic secure communication via generalized function projective synchronization

This paper presents two different hyperchaotic secure communication schemes by using generalized function projective synchronization (GFPS), where the drive and response systems could be synchronized up to a desired scaling function matrix. The unpredictability of the scaling functions can additionally enhance the security of communication. First, a hyperchaotic secure communication scheme applying GFPS of the uncertain Chen hyperchaotic system is proposed. The transmitted information signal is modulated into the parameter of the Chen hyperchaotic system in the transmitter and it is assumed that the parameter of the receiver system is unknown. Based on the Lyapunov stability theory and the adaptive control technique, the controllers are designed to make two identical Chen hyperchaotic systems with unknown parameter asymptotically synchronized; thus, the uncertain parameter of the receiver system is identified. The information signal can be recovered accurately by the estimated parameter. Secondly, another secure communication scheme by the coupled GFPS of the Chen hyperchaotic system is introduced. The information signal transmitted can be extracted exactly through simple operation in the receiver. The corresponding theoretical proofs and numerical simulations demonstrate the validity and feasibility of the proposed hyperchaotic secure communication schemes.     

Solution and dynamics analysis of a fractional‐order hyperchaotic system

Numerical solution and chaotic behaviors of the fractional‐order simplified Lorenz hyperchaotic system are investigated in this paper. The solution of the fractional‐order hyperchaotic system is obtained by employing Adomian decomposition method. Lyapunov characteristic exponents algorithm for the fractional‐order chaotic system is designed. Dynamics of the fractional‐order hyperchaotic system are analyzed by means of bifurcation diagrams, Lyapunov characteristic exponents, C₀ complexity, and chaos diagram. It shows that this system has rich dynamical behaviors, and it is more complex when the fractional order q is small. It lays a foundation for the practical application of the fractional‐order hyperchaotic systems. Copyright © 2015 John Wiley & Sons, Ltd.     

Hyperchaos in fractional order nonlinear systems

We numerically investigate hyperchaotic behavior in an autonomous nonlinear system of fractional order. It is demonstrated that hyperchaotic behavior of the integer order nonlinear system is preserved when the order becomes fractional. The system under study has been reported in the literature [Murali K, Tamasevicius A, Mykolaitis G, Namajunas A, Lindberg E. Hyperchaotic system with unstable oscillators. Nonlinear Phenom Complex Syst 3(1);2000:7–10], and consists of two nonlinearly coupled unstable oscillators, each consisting of an amplifier and an LC resonance loop. The fractional order model of this system is obtained by replacing one or both of its capacitors by fractional order capacitors. Hyperchaos is then assessed by studying the Lyapunov spectrum. The presence of multiple positive Lyapunov exponents in the spectrum is indicative of hyperchaos. Using the appropriate system control parameters, it is demonstrated that hyperchaotic attractors are obtained for a system order less than 4. Consequently, we present a conjecture that fourth-order hyperchaotic nonlinear systems can still produce hyperchaotic behavior with a total system order of 3 + ε, where 1 > ε > 0.     

Achieving synchronization between the fractional-order hyperchaotic Novel and Chen systems via a new nonlinear control technique

In this work, we discuss the stability conditions for a nonlinear fractional-order hyperchaotic system. The fractional-order hyperchaotic Novel and Chen systems are introduced. The existence and uniqueness of solutions for two classes of fractional-order hyperchaotic Novel and Chen systems are investigated. On the basis of the stability conditions for nonlinear fractional-order hyperchaotic systems, we study synchronization between the proposed systems by using a new nonlinear control technique. The states of the fractional-order hyperchaotic Novel system are used to control the states of the fractional-order hyperchaotic Chen system. Numerical simulations are used to show the effectiveness of the proposed synchronization scheme.     

A new hyperchaotic system from the Lü system and its control

This paper presents a 4D new hyperchaotic system which is constructed by a linear controller to a 3D Lü system. Some complex dynamical behaviors such as Hopf bifurcation, chaos and hyperchaos of the simple 4D autonomous system are investigated and analyzed. The corresponding hyperchaotic and chaotic attractor is first numerically verified through investigating phase trajectories, Lyapunove exponents, bifurcation path, analysis of power spectrum and Poincaré projections. Furthermore, the design is illustrated with both simulations and experiments. Finally, the control problem of a new hyperchaotic system is investigated using negative feedback control. Ordinary feedback control, dislocated feedback control and speed feedback control are used to suppress hyperchaos to an unstable equilibrium. Numerical simulations are presented to demonstrate the effectiveness of the proposed controllers.     

Adaptive synchronization between two different hyperchaotic systems

This work presents chaos synchronization between two different hyperchaotic systems using adaptive control. The sufficient conditions for achieving synchronization of two high dimensional chaotic systems are derived based on Lyapunov stability theory, and an adaptive control law and a parameter update rule for unknown parameters are given such that generalized Henon–Heiles system is controlled to be hyperchaotic Chen system. Theoretical analysis and numerical simulations are shown to verify the results.     

A new scheme of general hybrid projective complete dislocated synchronization

Based on the Lyapunov stability theorem, a new type of chaos synchronization, general hybrid projective complete dislocated synchronization (GHPCDS), is proposed under the framework of drive-response systems. The difference between the GHPCDS and complete synchronization is that every state variable of drive system does not equal the corresponding state variable, but equal other ones of response system while evolving in time. The GHPCDS includes complete dislocated synchronization, dislocated anti-synchronization and projective dislocated synchronization as its special item. As examples, the Lorenz chaotic system, Rössler chaotic system, hyperchaotic Chen system and hyperchaotic Lü system are discussed. Numerical simulations are given to show the effectiveness of these methods.     

A new four-dimensional energy resources system and its linear feedback control

This paper reports a new four-dimensional energy resources chaotic system. The system is obtained by adding a new variable to a three-dimensional energy resource demand–supply system established for two regions of China. The dynamics behavior of the system will be analyzed by means of Lyapunov exponents and bifurcation diagrams. Linear feedback control methods are used to suppress chaos to unstable equilibrium or unstable periodic orbits. Numerical simulations are presented to show these results.     

Generation of hyperchaos from the Chen–Lee system via sinusoidal perturbation

A system with more than one positive Lyapunov exponent can be classified as a hyperchaotic system. In this study, a sinusoidal perturbation was designed for generating hyperchaos from the Chen–Lee chaotic system. The hyperchaos was identified by the existence of two positive Lyapunov exponents and bifurcation diagrams. The system is hyperchaotic in several different regions of the parameters c, ε, and ω. It was found that this method not only can enhance or suppress chaotic behavior, but also induces chaos in non-chaotic parameter ranges. In addition, two interesting dynamical behaviors, Hopf bifurcation and intermittency, were also found in this study.     

A simple multi-scroll hyperchaotic system

We propose a simple autonomous hyperchaotic system that can generate multi-scroll attractors. The proposed system has a canonical structure, one control parameter, and a switching-type nonlinearity. If multiple breakpoints are added to the system nonlinearity, multi-scroll behavior can be obtained. We numerically demonstrate hyperchaotic behavior of the proposed system, under different nonlinearities, as its control parameter is changed. Furthermore, we study hyperchaos in the proposed system when it assumes a fractional order, and demonstrate that hyperchaotic behavior can be obtained in systems less than fourth order. Throughout the study, hyperchaos is verified by examining the Lyapunov spectrum, where the presence of multiple positive Lyapunov exponents in the spectrum is indicative of hyperchaos.     

A hyperchaotic system from the Rabinovich system

This paper presents a new 4D hyperchaotic system which is constructed by a linear controller to the 3D Rabinovich chaotic system. Some complex dynamical behaviors such as boundedness, chaos and hyperchaos of the 4D autonomous system are investigated and analyzed. A theoretical and numerical study indicates that chaos and hyperchaos are produced with the help of a Liénard-like oscillatory motion around a hypersaddle stationary point at the origin. The corresponding bounded hyperchaotic and chaotic attractors are first numerically verified through investigating phase trajectories, Lyapunov exponents, bifurcation path and Poincaré projections. Finally, two complete mathematical characterizations for 4D Hopf bifurcation are rigorously derived and studied.     

The evolution of a novel four-dimensional autonomous system: Among 3-torus, limit cycle, 2-torus, chaos and hyperchaos

In this paper, a novel four-dimensional autonomous system in which each equation contains a quadratic cross-product term is constructed. It exhibits extremely rich dynamical behaviors, including 3-tori (triple tori), 2-tori (quasi-periodic), limit cycles (periodic), chaotic and hyperchaotic attractors. In particular, we observe 3-torus phenomena, which have been rarely reported in four-dimensional autonomous systems in previous work. With the parameter r varying in quite a wide range, the evolution process of the system begins from 3-tori, and after going through a series of periodic, quasi-periodic and chaotic attractors in so many different shapes coming into being alternately, it evolves into hyperchaos, finally it degenerates to periodic attractor. Moreover, when the system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of the hyperchaotic systems already reported, especially the largest Lyapunov exponents. We also observe a chaotic attractor of a very special shape. The complex dynamical behaviors of the system are further investigated by means of Lyapunov exponents spectrum, bifurcation diagram and phase portraits.     

A new hyperchaotic system and its circuit implementation

In this paper, a new hyperchaotic system is presented by adding a nonlinear controller to the three-dimensional autonomous chaotic system. The generated hyperchaotic system undergoes hyperchaos, chaos, and some different periodic orbits with control parameters changed. The complex dynamic behaviors are verified by means of Lyapunov exponent spectrum, bifurcation analysis, phase portraits and circuit realization. The Multisim results of the hyperchaotic circuit were well agreed with the simulation results.     

Bifurcations and synchronization of the fractional-order simplified Lorenz hyperchaotic systems

In this paper, dynamics of the fractional-order simplied Lorenz hyperchaotic system is investigated. Modied Adams-Bashforth-Moulton method is applied for numerical simulation. Chaotic regions and periodic windows are identied. Dierent types of motions are shown along the routes to chaos by means of phase portraits, bifurcation diagrams, and the largest Lyapunov exponent. The lowest fractional order to generate chaos is 3.8584. Synchronization between two fractional-order simplied Lorenz hyperchaotic systems is achieved by using active control method. The synchronization performances are studied by changing the fractional order, eigenvalues and eigenvalue standard deviation of the error system.     

Secure communications based on the synchronization of the hyperchaotic Chen and the unified chaotic systems

This paper deals with the adaptive synchronization of two identical hyperchaotic master and slave systems. The master system and the slave system each consists of two subsystems: a hyperchaotic Chen subsystem and a unified chaotic subsystem. The asymptotic convergence of the errors between the states of the master system and the states of the slave system is proven using Lyapunov theory. Simulation results are presented to illustrate the ability of the control law to synchronize the master and slave systems. Moreover, the proposed control scheme is applied to encrypt and decrypt discrete signals such as digital images where computer simulation results are provided to show that the proposed control law works well.     

Adaptive control and synchronization of a new modified hyperchaotic Lü system with uncertain parameters

This paper addresses problems of control and synchronization for a new modified hyperchaotic Lü system with uncertain parameters. This new modified uncertain hyperchaotic Lü system is stabilized to its unique unstable equilibrium by using adaptive control. Furthermore, an adaptive control law and a parameter estimation update law are derived to synchronize two identical modified hyperchaotic Lü systems with uncertain parameters. Numerical examples are proposed to demonstrate and verify the theoretical analysis.