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In this paper, a new projective lag synchronization is proposed, where a driven chaotic system synchronizes the past state of the driver up to a scaling factor α . An active control method is employed to design a controller to achieve the global synchronization of two identical chaotic systems. Based on Lyapunov stability theorem, a sufficient condition is then given for the asymptotical stability of the null solution of an error dynamics. The effectiveness of the proposed schemes is verified via numerical simulations.

A novel inverse lag synchronization is proposed, where a driven chaotic system anti-synchronizes to the past state of the driver. From observer theory, the slave system is designed. The method is tested on the famous hyperchaotic generalized Henon map and the numerical simulations fully support the analytical approach.

In this Letter, generalized synchronization with a kind of function relationship between the states of drive and response chaotic systems is achieved. From matrix measure theory, some sufficient conditions for generalized synchronization are derived through suitable separation by decomposing the system as the linear part and the nonlinear one. Simulation results are provided for illustration and verification of the proposed method.

In this paper, an adaptive backstepping design is proposed to synchronize and control the Colpitts oscillator. The proposed control approach enables stabilization of chaotic motion to a steady state as well as synchronization by recursively interlacing the choice of a Lyapunov function with the design of feedback control in a systematical way. Numerical simulations verify the effectiveness of the approach.

A modified projective synchronization is proposed to acquire a general kind of proportional relationships between the drive and response systems. From rigorously control theory, a sufficient condition is attained for the stability of the error dynamics, and is applied to guiding the design of the controllers. Finally, we take Lorenz system as an example for illustration and verification.

On the basis of active backstepping design, this paper presents the generalized projective synchronization between two different chaotic systems: Lorenz system and Chen’s system. The proposed method combines backstepping methods and active control without having to calculate the Lyapunov exponents and the eigenvalues of the Jacobian matrix, which makes it simple and convenient. Numerical simulations show that this method works very well.

Based on the active control theory, anti-synchronization between two different chaotic systems is investigated. Through rigorous mathematical theory, the sufficient condition is drawn for the stability of the error dynamics, where the controllers are designed by using the sum of the relevant variables in chaotic systems. Numerical simulations are performed for the Lorenz–Lu system and Lorenz–Chen system to demonstrate the effectiveness of the proposed control strategy.

A new approach for constructing a projective-synchronized chaotic slave system is proposed in this paper. This method is based on suitable separation by decomposing the system as the linear part and the nonlinear one. From matrix measure theory, some simple but efficient criteria are derived for projective synchronization of chaotic system. Numerical simulations for the Lorenz system show that this control method works very well.

In this paper, an active control method is proposed to projective-synchronize two chaotic systems by constructing the response system no matter whether they are identical or not. The proposed technique is applied to achieve generalized projective synchronization for the Lorenz and Chen’s systems, where all state variables are in a proportional way. This property allows us to arbitrarily direct the scaling factor onto a desired value. Feasibility of the proposed control scheme is illustrated through the numerical examples.

An effective backstepping design is applied to projective synchronization in a general class of the so-called strict-feedback chaotic systems. Only one controller is required via backstepping design technique that recursively interlaces the choice of a Lyapunov function with the design of feedback control. Moreover, dead-beat synchronization in finite time can be achieved. This control method also allows us to arbitrarily amplify or reduce the scale of the dynamics of the slave system through a control. The chaotic Henon system is taken as an example to illustrate the effectiveness of the proposed approach.

Anti-synchronization is characterized by the phenomenon that the state vectors of synchronized systems have the same absolute values but opposite signs. By means of techniques from the classic state observer, an anti-phase synchronization of a class of chaotic systems is investigated in this Letter. Compared with other strategies, the technique presented here is rather simple and the convergence rate can be determined by adjusting the eigenvalues through the pole placement technique. Numerical simulations are provided to verify the effectiveness of the proposed methods.

Chaos control with driving parameter scheme in uncoupled identical chaotic oscillators is presented. By driving the parameter of chaotic systems using external chaotic signal, synchronization and anti-synchronization can be implemented. Numerical simulations show that either synchronization or anti-synchronization can appear depending significantly on initial condition and on driving strength. The proposed method is particularly suited for a variety of chaotic systems, which cannot couple with each other in engineering.