Adaptive modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system with uncertain parameters

This Letter investigates modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system using adaptive method. By Lyapunov stability theory, the adaptive control law and the parameter update law are derived to make the state of two hyperchaotic systems modified function projective synchronized. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.     

General methods for modified projective synchronization of hyperchaotic systems with known or unknown parameters

This work is concerned with the general methods for modified projective synchronization of hyperchaotic systems. A systematic method of active control is developed to synchronize two hyperchaotic systems with known parameters. Moreover, by combining the adaptive control and linear feedback methods, general sufficient conditions for the modified projective synchronization of identical or different chaotic systems with fully unknown or partially unknown parameters are presented. Meanwhile, the speed of parameters identification can be regulated by adjusting adaptive gain matrix. Numerical simulations verify the effectiveness of the proposed methods.     

A Hyperchaotic Attractor with Multiple Positive Lyapunov Exponents

There are many hyperchaotic systems, but few systems can generate hyperchaotic attractors with more than three PLEs （positive Lyapunov exponents）. A new hyperchaotic system, constructed by adding an approximate time-delay state feedback to a five-dimensional hyperchaotic system, is presented. With the increasing number of phase-shift units used in this system, the number of PLEs also steadily increases. Hyperchaotic attractors with 25 PLEs can be generated by this system with 32 phase-shift units. The sum of the PLEs will reach the maximum value when 23 phase-shift units are used. A simple electronic circuit, consisting of 16 operational amplifiers and two analogy multipliers, is presented for confirming hyperchaos of order 5, i.e., with 5 PLEs.     

Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system

The stability conditions in fractional order hyperchaotic systems are derived. These conditions are applied to a novel fractional order hyperchaotic system. The proposed system is also shown to exhibit hyperchaos for orders less than 4. Based on the Routh-Hurwitz conditions, the conditions for controlling hyperchaos via feedback control are also obtained. A specific condition for controlling only fractional order hyperchaotic systems is achieved. Numerical simulations are used to verify the theoretical analysis.     

A Hyperchaotic Attractor Coined from Chaotic Lü System

We report a new hyperchaotic attractor coined from the chaotic Lü system by using a state feedback controller. Theoretical analyses and simulation experiments are conducted to investigate the dynamical behaviour of the proposed hyperchaotic system     

Comparison of feedback control methods for a hyperchaotic Lorenz system

More and more attention has been payed to the hyperchaotic system for the huge potential applications of hyperchaotic system such as secure communication and more complex structure than chaotic system. So at present the controlling of the hyperchaotic system simply and effectively is a frontier topic of nonlinear science. In this Letter, for the latest hyperchaotic Lorenz system, four feedback control methods were studied with analytic solution and necessary numerical simulations. It is found that the enhancing feedback control approach is the best choice of the given four feedback control methods for its relatively simple external inputs and relatively small necessary feedback coefficient after comparison. The conclusion is a helpful for the choice of control methods of any other chaotic and hyperchaotic systems.     

Unstable manifolds for the hyperchaotic Rössler system

We analyse stability of the generalized four-variable Rössler oscillating system depending on selected control parameters, by using analytic and Hurwitz-Routh methods. In contrast to the usual three-dimensional Rössler and Lorenz systems, we show that always there exists at least one unstable direction, and the number of positive local Lyapunov exponents may be different for both fixed points. We have found two new types of Hopf bifurcation, in which the dimension of the unstable manifold can be increased or reduced by two. Hence there are many possibilities for hyperchaotic unstable manifolds of various dimensions. We have also calculated various ranges of the control parameters for which different unstable manifolds can be obtained. This allows a better characterization of stability of the attractors in the hyperchaotic regime.     

Adaptive Functional Projective Lag Synchronization of a Hyperchaotic Rössler System

We explain the functional projective lag synchronization of a hyperchaotic Rössler system with four unknown parameters, where the output of the master system lags behind the output of the slave system proportionally. Based on Lyapunov stability theory, an active control method and adaptive control law are employed to make the states of two hyperchaotic Rössler systems asymptotically synchronized. Finally, some numerical examples are provided to show the effectiveness of our results.     

Synchronization in coupled nonidentical incommensurate fractional-order systems

Synchronization of fractional-order nonlinear systems has received considerable attention for many research activities in recent years. In this Letter, we consider the synchronization between two nonidentical fractional-order systems. Based on the open-plus-closed-loop control method, a general coupling applied to the response system is proposed for synchronizing two nonidentical incommensurate fractional-order systems. We also derive a local stability criterion for such synchronization behavior by utilizing the stability theory of linear incommensurate fractional-order differential equations. Feasibility of the proposed coupling scheme is illustrated through numerical simulations of a limit cycle system, a chaotic system and a hyperchaotic system.     

A New High Performance Realization of Hyperchaotic Modified Canonical Chua Circuit Using JFETs

A new implementation of hyperchaotic modified canonical Chua circuit using junction field-effect transistors （JFETs） is proposed. The design is based on a source coupled JFET circuit to approximate a smooth cubic nonlinearity and a two-terminal negative resistance element containing a p-n-p silicon transistor and an n-channel JFET. The realization is supported by Orcad Pspice simulation and numerical MATLAB results. The hyperchaotic nature is confirmed by two positive Lyapunov exponents associated with the attractor which is a fractal with a Lyapunov dimension between 3 and 4.     

Symmetric bursting of focus-focus type in the controlled Lorenz system with two time scales

By employing a special feedback controlling scheme, a hyperchaotic Lorenz system with the structure of two time scales is constructed. Two kinds of bursting phenomena, symmetric fold/fold bursting and symmetric sub-Hopf/sub-Hopf bursting, can be observed in this system. Their respective dynamical behaviors are investigated by means of slow-fast analysis. In particular, symmetric fold/fold bursting is of focus-focus type, namely, both the up-state and the down-state are stable focus, which is different from the usual fold/fold bursting; Symmetric sub-Hopf/sub-Hopf bursting is also of focus-focus type, which has not been reported in previous work. Furthermore, phase plane analysis has been introduced to explore the evolution details of the fast subsystem for symmetric sub-Hopf/sub-Hopf bursting. With the variation of the parameter, symmetric sub-Hopf/sub-Hopf bursting can evolve to symmetric chaotic bursting or even hyperchaos.     

Reconstruction of systems with delayed feedback: II. Application

We apply a recently proposed method for the analysis of time series from systems with delayed feedback to experimental data generated by a laser. The method allows estimating the delay time with an error of the order of the sampling interval, while an approach based on the peaks of either the autocorrelation function, or the time delayed mutual information would yield systematically larger values. We reconstruct rather accurately the equations of motion and, in turn, estimate the Lyapunov spectrum even for high dimensional attractors. By comparing models constructed for different “embedding dimensions” with the original data, we are able to find the minimal faithful model. For short delays, the results of our procedure have been cross-checked using a conventional Takens time-delay embedding. For large delays, the standard analysis is inapplicable since the dynamics becomes hyperchaotic. In such a regime we provide the first experimental evidence that the Lyapunov spectrum, rescaled according to the delay time, is independent of the delay time itself. This is in full analogy with the independence of the system size found in spatially extended systems. Received 17 December 1999     

A novel hyperchaos in the quantum Zakharov system for plasmas

The nonlinear interaction of quantum Langmuir waves (QLWs) and quantum ion-acoustic waves (QIAWs) described by the one-dimensional quantum Zakharov equations (QZEs) is reinvestigated. A Galerkin type approximation is used to reduce the QZS to a simplified system (SS) of nonlinear ordinary differential equations which governs the temporal behaviors of the slowly varying envelope of the high-frequency electric field and the low frequency density fluctuation. This SS is then shown to establish the coexistence of novel hyperchaotic attractors, whose appearance is explained by means of the analysis of Lyapunov exponent spectra as well as the Kaplan-Yorke dimension. The system has an equilibrium point which depends parametrically on the nondimensional quantum parameter (H) proportional to quantum diffraction, the plasmon number (N) and the wave number of perturbation (α), and which can evolve into periodic, quasi-periodic, chaotic and hyperchaotic states in both semiclassical and quantum cases.     

A unified approach to fuzzy modelling and robust synchronization of different hyperchaotic systems

In this paper, a Takagi Sugeno （T-S） fuzzy model-based method is proposed to deal with the problem of synchronization of two identical or different hyperchaotic systems. The T S fuzzy models with a small number of fuzzy IF-THEN rules are employed to represent many typical hyperchaotic systems exactly. The benefit of employing the T-S fuzzy models lies in mathematical simplicity of analysis. Based on the T-S fuzzy hyperchaotic models, two fuzzy controllers arc designed via parallel distributed compensation （PDC） and exact linearization （EL） techniques to synchronize two identical hyperchaotic systems with uncertain parameters and two different hyperchaotic systems, respectively. The sufficient conditions for the robust synchronization of two identical hyperchaotic systems with uncertain parameters and the asymptotic synchronization of two different hyperchaotic systems are derived by applying the Lyapunov stability theory. This method is a universal one of synchronizing two identical or different hyperchaotic systems. Numerical examples are given to demonstrate the validity of the proposed fuzzy model and hyperchaotic synchronization scheme.     

Function projective synchronization of different chaotic systems with uncertain parameters

This Letter investigates the function projective synchronization of different chaotic systems with unknown parameters. By Lyapunov stability theory, the adaptive control law and the parameter update law are derived to make the states of two different chaotic systems asymptotically synchronized up to a desired scaling function. Numerical simulations on Lorenz system and Newton-Leipnik system are presented to verify the effectiveness of the proposed scheme.     

Chaos synchronization between two different 4D hyperchaotic Chen systems

This paper presents chaos synchronization between two different four-dimensional (4D) hyperchaotic Chen systems by nonlinear feedback control laws. A modified 4D hyperchaotic Chen system is obtained by changing the nonlinear function of the 4D hyperchaotic Chen system, furthermore, an electronic circuit to realize two different 4D hyperchaotic Chen systems is designed. With nonlinear feedback control method, chaos synchronization between two different 4D hyperchaotic Chen systems is achieved. Based on the stability theory, the functions of the nonlinear feedback control for synchronization of two different 4D hyperchaotic Chen systems is derived, the range of feedback gains is determined. Numerical simulations are shown to verify the theoretical results.     

Different Types of Synchronization in Time-Delayed Systems

We investigate different types of synchronization between two unidirectionally nonlinearly coupled identical delay- differential systems related to optical bistable or hybrid optical bistable devices. This system can represent some kinds of delay-differential models, i.e. Ikeda model, Vall~e model, sine-square model, Mackey Glass model, and so on. We find existence and sufficient stability conditions by theoretical analysis and test the correctness by＂ numerical simulations. Lag, complete and anticipating synchronization are observed, respectively. It is found that the time-delay system can be divided into two parts~ one is the instant term and the other is the delay term. Synchronization between two identical chaotic systems can be derived by adding a coupled term to the delay term in the driven system.     

Anti-synchronization Between Lorenz and Liu Hyperchaotic Systems

Anti-synchronization between different hyperchaotic systems is presented using Lorenz and Liu systems. When the parameters of two systems are known, one can use active synchronization. When the parameters are unknown or uncertain, the adaptive synchronization is applied. The simulation results verify the effectiveness of the proposed two schemes for anti-synchronization between different hyperchaotic systems.     

Projective synchronization of a new hyperchaotic Lorenz system

This Letter mainly concerns projective synchronization (PS) of a new hyperchaotic Lorenz system. PS with both identical and different scaling factors between two hyperchaotic Lorenz systems are realized. A general sufficient condition for PS in a certain class of chaotic (hyperchaotic) system with uncertainties is obtained by using adaptive control. Numerical simulations are performed to verify and illustrate the analytical results.     

Anti-synchronization of chaotic systems with uncertain parameters via adaptive control

In this Letter, an adaptive control scheme is developed to study the anti-synchronization behavior between two identical and different chaotic systems with unknown parameters. This adaptive anti-synchronization controller is designed based on Lyapunov stability theory and an analytic expression of the controller with its adaptive laws of parameters is shown. The adaptive anti-synchronization between two identical systems (Chen system) and different systems (Genesio and Lü systems) are taken as two illustrative examples to show the effectiveness of the proposed method. Theoretical analysis and numerical simulations are shown to verify the results.