A new 5D hyperchaotic system based on modified generalized Lorenz system

This paper reports a new five-dimensional (5D) hyperchaotic system with three positive Lyapunov exponents, which is generated by adding a linear controller to the second equation of a 4D system that is obtained by coupling of a 1D linear system and a 3D modified generalized Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest are the observations that the hyperchaotic system has a hyperchaotic attractor with three positive Lyapunov exponents under a unique equilibrium, three or infinite equilibria, and there are three types of coexisting attractors of this new 5D hyperchaotic system. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, Poincaré projections and power spectrum verifies the existence of the hyperchaotic and chaotic attractors. Moreover, stability of hyperbolic or non-hyperbolic equilibria and two complete mathematical characterization for 5D Hopf bifurcation are rigorously studied. Finally, some electronic circuits are designed to implement the 5D hyperchaotic system.     

Dynamics of a hyperchaotic Lorenz-type system

This paper discusses the complex dynamics of a new four-dimensional continuous-time autonomous hyperchaotic Lorenz-type system. The local dynamics, such as the stability, pitchfork bifurcation, and Hopf bifurcation at equilibria of this hyperchaotic system are analyzed by using the parameter-dependent center manifold theory and the normal form theory. The existence of homoclinic and heteroclinic orbits of this hyperchaotic system is further rigorously studied. More exactly, under some special parameter conditions, the fact that this hyperchaotic system has no homoclinic orbit but has two and only two heteroclinic orbits are proved.     

On the hyperchaotic complex Lü system

The aim of this paper is to introduce the new hyperchaotic complex Lü system. This system has complex nonlinear behavior which is studied and investigated in this work. Numerically the range of parameter values of the system at which hyperchaotic attractors exist is calculated. This new system has a whole circle of equilibria and three isolated fixed points, while the real counterpart has only three isolated ones. The stability analysis of the trivial fixed point is studied. Its dynamics is more rich in the sense that our system exhibits both chaotic and hyperchaotic attractors, as well as periodic and quasi-periodic solutions and solutions that approach fixed points. The nonlinear control method based on Lyapunov function is used to synchronize the hyperchaotic attractors. The control of these attractors is studied. Different forms of hyperchaotic complex Lü systems are constructed using the state feedback controller and complex periodic forcing.     

Hyperchaos in constrained Hamiltonian system and its control

This paper first formulates a Hamiltonian system with hyperchaotic phenomena and investigates the equilibrium point and double Hopf bifurcation of the system. We obtain the result that the Hamiltonian system has hyperchaotic behaviors when any system parameter varies. The influences of holonomic constraint and nonholonomic constraint on the equilibrium points, invariance and the hyperchaotic state of the Hamiltonian system are then studied. Finally, we achieve the hyperchaotic control of the Hamiltonian system by introducing the constraint method. The studies indicate that the constraint can not only change the Hamiltonian system from hyperchaotic state to periodic state or chaotic state, but also make the Hamiltonian system become globally asymptotically stable. Numerical simulations, including Lyapunov exponents, bifurcation diagrams, Poincaré maps and phase portraits for systems, exhibit the complex dynamical behaviors.     

A new fractional order hyperchaotic Rabinovich system and its dynamical behaviors

In the paper, the dynamical behaviors of a new fractional order hyperchaotic Rabinovich system are investigated, which include its local stability, hyperchaos, chaotic control and synchronization. Firstly, a new fractional order hyperchaotic Rabinovich system with Caputo derivative is proposed. Then, the hyperchaotic attractors of the commensurate and incommensurate fractional order hyperchaotic Rabinovich system are found. After that, four linear feedback controllers are designed to stabilize this fractional order system Finally, by using the active control method the synchronization is studied between the fractional order hyperchaotic and chaos controlled Rabinovich system In addition, the theoretical predictions are confirmed by numerical simulations.     

Projective synchronization of new hyperchaotic system with fully unknown parameters

Projective synchronization of new hyperchaotic Newton–Leipnik system with fully unknown parameters is investigated in this paper. Based on Lyapunov stability theory, a new adaptive controller with parameter update law is designed to projective synchronize between two hyperchaotic systems asymptotically and globally. Basic bifurcation analysis of the new system is investigated by means of Lyapunov exponent spectrum and bifurcation diagrams. It is found that the new hyperchaotic system possesses two positive Lyapunov exponents within a wide range of parameters. Numerical simulations on the hyperchaotic Newton–Leipnik system are used to verify the theoretical results.     

A time-varying hyperchaotic system and its realization in circuit

A hyperchaotic system is often used to generate secure keys or carrier wave for secure communication and the realistic hyperchaotic circuit often is made of capacitor, nonlinear resistor unit and induction coil. Parameters are often fixed in these hyperchaotic circuits and the hyperchaotic property of the system can be estimated by using a scheme of synchronization and time series analysis. In this paper, a time-varying hyperchaotic system is proposed by introducing changeable electric power source into the circuit; the changeable electric power source is combined with induction coil or capacitor in series to generate changeable output signals to excite the system. The diagrams of improved circuit are illustrated and critical parameters in experimental circuits are presented; the Lyapunov exponent spectrum vs. external applied electric power source is calculated. It is confirmed that the improved circuit always holds two positive Lyapunov exponents when the external electric power source works, and the chaotic attractors are much too different from the original one; thus, a more changeable hyperchaotic system is constructed in experiment.     

Hopf bifurcation and intermittent transition to hyperchaos in a novel strong four-dimensional hyperchaotic system

This paper presents a new four-dimensional autonomous system having complex hyperchaotic dynamics. Basic properties of this new system are analyzed, and the complex dynamical behaviors are investigated by dynamical analysis approaches, such as time series, Lyapunov exponents’ spectra, bifurcation diagram, phase portraits. Moreover, when this new system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of hyperchaotic systems reported before, which implies the new system has strong hyperchaotic dynamics in itself. The Kaplan–Yorke dimension, Poincaré sections and the frequency spectra are also utilized to demonstrate the complexity of the hyperchaotic attractor. It is also observed that the system undergoes an intermittent transition from period directly to hyperchaos. The statistical analysis of the intermittency transition process reveals that the mean lifetime of laminar state between bursts obeys the power-law distribution. It is shown that in such four-dimensional continuous system, the occurrence of intermittency may indicate a transition from period to hyperchaos not only to chaos, which provides a possible route to hyperchaos. Besides, the local bifurcation in this system is analyzed and then a Hopf bifurcation is proved to occur when the appropriate bifurcation parameter passes the critical value. All the conditions of Hopf bifurcation are derived by applying center manifold theorem and Poincaré–Andronov–Hopf bifurcation theorem. Numerical simulation results show consistency with our theoretical analysis.     

Dynamic analysis and control of a new hyperchaotic finance system

In this paper, a new hyperchaotic finance system which is constructed based on a chaotic finance system by adding an additional state variable is presented. The basic dynamical behaviors of this hyperchaotic finance system are investigated, such as the equilibrium, stability, hyperchaotic attractor, Lyapunov exponents, and bifurcation analysis. Furthermore, effective speed feedback controllers and linear feedback controllers are designed for stabilizing hyperchaos to unstable equilibrium points. Numerical simulations are given to illustrate and verify the results.     

Generalized Darboux transformation and parameter-dependent rogue wave solutions to a nonlinear Schrödinger system

Objectives of the paper are (1) to design two new real and complex no equilibrium point hyperchaotic systems, (2) to design synchronisation technique for the new systems using the contraction theory and (3) to validate the results by using circuit realisation. First a new no equilibrium point hyperchaotic system is developed using a 3-D generalised Lorenz system; then using the new system a new complex no equilibrium point hyperchaotic system is reported. Both the new systems have hidden chaotic attractors. Various dynamical behaviours are observed in the new systems like chaotic, periodic, quasi-periodic and hyperchaotic. Both the systems have inverse crisis route to chaos with the variation of parameter a and crisis route to chaos with the variation of parameters \(b,\ c\) and d. These phenomena along with hidden attractors in a complex hyperchaotic system are not seen in the literature. Synchronisation between the identical new hyperchaotic systems is achieved using the contraction theory. Further the synchronisation between the identical new complex hyperchaotic systems is achieved using adaptive contraction theory. The proposed synchronisation strategies are validated using the MATLAB simulation and circuit implementation results. Further, an application of the proposed system is shown by transmitting and receiving an audio signal.     

The existence of homoclinic orbits in a 4D Lorenz-type hyperchaotic system

Little seems to be known about the homoclinic orbits of hyperchaotic system. Through the deep researches of a 4D Lorenz-type hyperchaotic system, with the help of Fishing Principle, we obtain the existence conditions of homoclinic orbits of this hyperchaotic system. In order to justify the theoretical analysis, by using the numerical methods, a set of approximate bifurcation parameters and its corresponding homoclinic orbits are obtained.     

Generalized projective synchronization of the fractional-order Chen hyperchaotic system

In this paper, we numerically investigate the hyperchaotic behaviors in the fractional-order Chen hyperchaotic systems. By utilizing the fractional calculus techniques, we find that hyperchaos exists in the fractional-order Chen hyperchaotic system with the order less than 4. We found that the lowest order for hyperchaos to have in this system is 3.72. Our results are validated by the existence of two positive Lyapunov exponents. The generalized projective synchronization method is also presented for synchronizing the fractional-order Chen hyperchaotic systems. The present technique is based on the Laplace transform theory. This simple and theoretically rigorous synchronization approach enables synchronization of fractional-order hyperchaotic systems to be achieved and does not require the computation of the conditional Lyapunov exponents. Numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme.     

An image encryption scheme based on time-delay and hyperchaotic system

In this paper, a novel image encryption scheme based on time-delay and hyperchaotic system is suggested. The time-delay phenomenon is commonly observed in daily life and is incorporated in the generation of pseudo-random chaotic sequences. To further increase the degree of randomness, the output of the hyperchaotic system is processed before appending to the generated sequence. A novel permutation function for shuffling the position index, together with the double diffusion operations in both forward and reverse directions, is employed to enhance the encryption performance. Experimental results and security analyses show that the proposed scheme has a large key space and can resist known-plaintext and chosen-plaintext attacks. Moreover, the encryption scheme can be easily modified to adopt other hyperchaotic systems under the same structure.     

A novel fractional-order hyperchaotic system stabilization via fractional sliding-mode control

In this paper we numerically investigate the fractional-order sliding-mode control for a novel fractional-order hyperchaotic system. Firstly, the dynamic analysis approaches of the hyperchaotic system involving phase portraits, Lyapunov exponents, bifurcation diagram, Lyapunov dimension, and Poincaré maps are investigated. Then the fractional-order generalizations of the chaotic and hyperchaotic systems are studied briefly. The minimum orders we found for chaos and hyperchaos to exist in such systems are 2.89 and 3.66, respectively. Finally, the fractional-order sliding-mode controller is designed to control the fractional-order hyperchaotic system. Numerical experimental examples are shown to verify the theoretical results.     

Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form

In this paper, a new simple 4D smooth autonomous system is proposed, which illustrates two interesting rare phenomena: first, this system can generate a four-wing hyperchaotic and a four-wing chaotic attractor and second, this generation occurs under condition that the system has only one equilibrium point at the origin. The dynamic analysis approach in the paper involves time series, phase portraits, Lyapunov exponents, bifurcation diagram, and Poincaré maps, to investigate some basic dynamical behaviors of the proposed 4D system. The physical existence of the four-wing hyperchaotic attractor is verified by an electronic circuit. Finally, it is shown that the fractional-order form of the system can also generate a chaotic four-wing attractor.     

Hyperchaotic memristive system with hidden attractors and its adaptive control scheme

In this research work a novel 4-D memristive system is presented. The proposed system belongs to the category of dynamical systems with hidden attractors as it displays a line of equilibrium points. Also, it has an hyperchaotic dynamical behavior in a particular range of its parameters space. System’s behavior is investigated through numerical simulations, by using well-known tools of nonlinear theory, such as phase portrait, bifurcation diagram, Lyapunov exponents and Poincaré map. Next, the case of chaos control of the system with unknown parameters using adaptive control method is investigated. Finally, an electronic circuit realization of the novel hyperchaotic system using Spice is presented in detail to confirm the feasibility of the theoretical model.     

Generalized projective lag synchronization between?different hyperchaotic systems with uncertain parameters

Generalized projective lag synchronization (GPLS) is characterized by the output of the drive system proportionally lagging behind the output of the response system. In this paper, GPLS between different hyperchaotic systems with uncertain parameters, i.e., GPLS between Lorenz and Lü hyperchaotic systems, and between Lorenz?CStenflo and Lorenz hyperchaotic systems, is studied by applying an adaptive control method. Based on Lyapunov stability theory, the adaptive controllers and corresponding parameter update rules are constructed to make the states of two diverse hyperchaotic systems asymptotically synchronize up to the desired scaling matrix and to estimate the uncertain parameters. Some numerical simulations are provided to show the effectiveness of our results.     

Tracking control and synchronization of the new hyperchaotic Liu system via backstepping techniques

In this paper, recursive and active backstepping nonlinear techniques are employed to design control functions for the respective, control, and synchronization of the new hyperchaotic Liu system. The designed recursive backstepping nonlinear controllers are capable of stabilizing the hyperchaotic Liu system at any position as well as controlling it to track any trajectory that is a smooth function of time. The designed active backstepping nonlinear controllers are effective in globally synchronizing two identical hyperchaotic Liu systems evolving from different initial conditions. The results are all validated by numerical simulations.     

Circuit implementation and finite-time synchronization of the 4D Rabinovich hyperchaotic system

This paper studies the problem of the circuit implementation and the finite-time synchronization for the 4D (four-dimensional) Rabinovich hyperchaotic system. The electronic circuit of 4D hyperchaotic system is designed. It is rigorously proven that global finite-time synchronization can be achieved for hyperchaotic systems which have uncertain parameters.     

Dynamics,circuit realization,control and synchronization of a hyperchaotic hyperjerk system with coexisting attractors

Although different hyperjerk systems have been discovered, a few hyperjerk systems can exhibit hyperchaotic behavior. In this work, we introduce a new hyperjerk system with hyperchaotic attractors. By investigating dynamics of the system, we have observed the different coexisting attractors such as coexistence of period-2 attractors, or coexistence of period-2 attractor and quasiperiodic attractor. It is worth noting that this striking phenomenon is rarely reported in a hyperjerk system. The proposed system has been realized with electronic components. The agreement between the simulation and experimental results indicates the feasibility of the hyperjerk system. Moreover, chaos control and synchronization of such hyperjerk system have been also reported.