Adaptive projective synchronization of unified chaotic systems and its application to secure communication

In this paper, a simple adaptive linear feedback control method is proposed for controlling the scaling factor between two coupled unified chaotic systems to a desired value, based on the invarianee principle of differential equations. Under this control strategy, one can arbitrarily select the scaling factor. Numerical simulations are given to support the effectiveness of the proposed method and show the robustness against noise. Furthermore, a secure communication scheme based on the adaptive projective synchronization of unified chaotic systems is presented and numerical simulation shows its feasibility.

Synchronization of Unified Chaotic System Using Occasional Driving

The synchronization of the unified chaotic systems using occasional driving technique is studied. The relation among interim period T , sampling interval 2 ε , feedback gain r and the parameter α of the system is thoroughly investigated. Numerical results show that smaller interim period T and properly larger sampling interval 2 ε can accelerate the synchronizing pace. Furthermore, for a unified chaotic system in which α is given, we can achieve satisfying synchronizing results as long as T,ε and r are appropriately chosen. As we adopt the occasional driving method, we greatly reduce the control cost. Therefore with this method we can obtain the expecting goals with little control cost.

The feedback control of fractional order unified chaotic system

This paper studies the stability of the fractional order unified chaotic system. On the unstable equilibrium points, the equivalent passivity＇＇ method is used to design the nonlinear controller. With the definition of fractional derivatives and integrals, the Lyapunov function is constructed by which it is proved that the controlled fractional order system is stable. With Laplace transform theory, the equivalent integer order state equation from the fractional order nonlinear system is obtained, and the system output can be solved. The simulation results validate the effectiveness of the theory.

A simple asymptotic trajectory control of full states of a unified chaotic system

A simple full-state asymptotic trajectory control （FSATC） scheme is proposed to asymptotically drive full states of a unified chaotic system （UCS） to arbitrary desired trajectories. The FSATC uses only information, i.e. one state of the UCS. A sinusoidal wave and two chaotic variables are taken as illustrative tracking trajectories to verify that using the proposed FSATC can make full UCS states track desired trajectories with high tracking accuracy in a finite time.

Estimating the globally attractive set and positively invariant set of a unified chaotic system

By constructing two suitable generalized Lyapunov functions,we derived a generalized ellipsoidal estimate of the globally attractive set and positively invariant set of the unified chaotic system with the parameters α=1/29 and 1/29<α<2/29,respectively,which extends some related results of Li,et al. [Li DM,Lu JA,Wu XQ,Chen GR,Estimating the global basin of attraction and positively invariant set for the Lorenz system and a unified chaotic system,Journal of Mathematical Analysis and Applications,2006,323(2): 844-853]. The theoretical results obtained in this paper will find wide application in chaos control and synchronization.

Comparison between two different sliding mode controllers for a fractional-order unified chaotic system

Two different sliding mode controllers for a fractional order unified chaotic system are presented. The controller for an integer-order unified chaotic system is substituted directly into the fractional-order counterpart system, and the fractional-order system can be made asymptotically stable by this controller. By proving the existence of a sliding manifold containing fractional integral, the controller for a fractional-order system is obtained, which can stabilize it. A comparison between these different methods shows that the performance of a sliding mode controller with a fractional integral is more robust than the other for controlling a fractional order unified chaotic system.

Circuit realization of the fractional-order unified chaotic system

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.

The stability control of fractional order unified chaotic system with sliding mode control theory

This paper studies the stability of the fractional order unified chaotic system with sliding mode control theory. The sliding manifold is constructed by the definition of fractional order derivative and integral for the fractional order unified chaotic system. By the existing proof of sliding manifold, the sliding mode controller is designed. To improve the convergence rate, the equivalent controller includes two parts： the continuous part and switching part. With Gronwall＇s inequality and the boundness of chaotic attractor, the finite stabilization of the fractional order unified chaotic system is proved, and the controlling parameters can be obtained. Simulation results are made to verify the effectiveness of this method.

Cascade adaptive control of uncertain unified chaotic systems

The chaos control of uncertain unified chaotic systems is considered. Cascade adaptive control approach with only one control input is presented to stabilize states of the uncertain unified chaotic system at the zero equilibrium point. Since an adaptive controller based on dynamic compensation mechanism is employed, the exact model of the unified chaotic system is not necessarily required. By choosing appropriate controller parameters, chaotic phenomenon can be suppressed and the response speed is tunable. Sufficient condition for the asymptotic stability of the approach is derived. Numerical simulation results confirm that the cascade adaptive control approach with only one control signal is valid in chaos control of uncertain unified chaotic systems.

Constructing a one-way hash function based on the unified chaotic system

A new one-way hash function based on the unified chaotic system is constructed. With different values of a key parameter, the unified chaotic system represents different chaotic systems, based on which the one-way hash function algorithm is constructed with three round operations and an initial vector on an input message. In each round operation, the parameters are processed by three different chaotic systems generated from the unified chaotic system. Feed-forwards are used at the end of each round operation and at the end of each element of the message processing. Meanwhile, in each round operation, parameter-exchanging operations are implemented. Then, the hash value of length 160 bits is obtained from the last six parameters. Simulation and analysis both demonstrate that the algorithm has great flexibility, satisfactory hash performance, weak collision property, and high security.

Robust finite-time stabilization of unified chaotic complex systems with certain and uncertain parameters

This paper deals with the finite-time stabilization of unified chaotic complex systems with known and unknown parameters. Based on the finite-time stability theory, nonlinear control laws are presented to achieve finite-time chaos control of the determined and uncertain unified chaotic complex systems, respectively. The two controllers are simple, and one of the uncertain unified chaotic complex systems is robust. For the design of a finite-time controller on uncertain unified chaotic complex systems, only some of the unknown parameters need to be bounded. Simulation results for the chaotic complex Lorenz, Lu¨ and Chen systems are presented to validate the design and analysis.

An observer based asymptotic trajectory control using a scalar state for chaotic systems

A state-observer based full-state asymptotic trajectory control （OFSTC） method requiring a scalar state is presented to asymptotically drive all the states of chaotic systems to arbitrary desired trajectories. It is no surprise that OFSTC can obtain good tracking performance as desired due to using a state-observer. Significantly OFSTC requires only a scalar state of chaotic systems. A sinusoidal wave and two chaotic variables were taken as illustrative tracking trajectories to validate that using OFSTC can make all the states of a unified chaotic system track the desired trajectories with high tracking accuracy and in a finite time. It is noted that this is the first time that the state-observer of chaotic systems is designed on the basis of Kharitonov＇s Theorem.

A new kind of nonlinear fractional-order chaotic phenomenon in coupled systems： coexistence of anti-phase and complete synchronization

In this paper, we have found a kind of interesting nonlinear phenomenon hybrid synchronization in linearly coupled fractional-order chaotic systems. This new synchronization mechanism, i.e., part of state variables are anti- phase synchronized and part completely synchronized, can be achieved using a single linear controller with only one drive variable. Based on the stability theory of the fractional-order system, we investigated the possible existence of this new synchronization mechanism. Moreover, a helpful theorem, serving as a determinant for the gain of the controller, is also presented. Solutions of coupled systems are obtained numerically by an improved Adams Bashforth-Moulton algorithm. To support our theoretical analysis, simulation results are given.