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The modified function projective lag synchronization of the memristor-based five-order chaotic circuit system with unknown bounded disturbances is investigated. Based on the LMI approach and Lyapunov stability theorem, an adaptive control law is established to make the states of two different memristor-based five-order chaotic circuit systems asymptotically synchronized up to a desired scaling function matrix, while the parameter controlling strength update law is designed to estimate the parameters well. Finally, the simulation is put forward to demonstrate the correctness and effectiveness of the proposed methods. The control method involved is simple and practical.

The memristor, an abbreviation for memory resistor studied by Chua in 1971 [

As is known to all, the synchronization of chaotic systems has been a subject of active research field due to its potential applications for secure communications and control. Up to now, many types of synchronization methods have been put forward in dynamical systems, such as complete synchronization (CS) [

Recently a more general form of FPS called modified function projective synchronization (MFPS) [

To the best of our knowledge, the MFPLS of memristor-based five-order chaotic circuit system with unknown disturbances has not been reported yet. Motivated by the above discussion, we will give a comprehensive study on this topic in this article. Based on the parameter modulation, the adaptive control technique, and Lyapunov stability theorem, the adaptive control laws are designed to make the states of two different memristor-based five-order chaotic circuit systems asymptotically synchronized up to a desired scaling function matrix.

By replacing Chua’s diode with an active flux-controlled memristor circuit, Bao derived a memristor-based five-order chaotic circuit from four-order Chua’s oscillator. This new chaotic circuit can be shown in Figure

Memristor-based five-order chaotic circuit.

Denote

If we set

Attractor of the memristor-based five-order chaotic circuit (a).

Attractor of the memristor-based five-order chaotic circuit (b).

Attractor of the memristor-based five-order chaotic circuit (c).

Attractor of the memristor-based five-order chaotic circuit (d).

Taking into account the external disturbances, for the sake of convenience, we reexpress system (

Taking system (

The unknown external time-varying disturbances

For the drive system (

It is clear that (

When

Denote

It is obvious that

Noted that

There exist three nonnegative constants

The main purpose of this paper is to design an appropriate controller

Let us define the MFPLS error vector

Combining systems (

Furthermore, we can obtain

It followed by designing an adaptive controller to achieve MFPLS of systems (

We start with a simple case in which the bounds

If there exists symmetric positive definite matrix

then systems (

Substituting the control law (

Calculating the time derivative of

Utilizing Lyapunov stability theorem, we get

We now consider the general case in which the bounds

If there exist a positive constant

then the two systems (

Substituting the controller (

Applying Lyapunov stability theorem, we can obtain

More generally, if all the bounds

For this case, the adaptive control law and parameter update rule is chosen by

If there exist positive constants

then systems (

Substituting the controller (

The Lyapunov function is designed as

The time derivative of

Substituting (

According to Lyapunov stability theorem, we can get

In this section, two different memristor-based five-order chaotic circuit systems with unknown bounded disturbances are considered as the master system and the slave system, respectively, which can be described by

The drive system is initialized with

Using the control method proposed in Theorem

Time response of MFPLS error

Time response of the input controller

Time response of the estimated value of

In this paper, the problem of MFPLS of memristor-based five-order chaotic circuit systems with unknown bounded disturbances has been addressed. Combining the LMI approach with Lyapunov stability theory, an adaptive control law is designed to make the states of two different memristor-based five-order chaotic circuit systems asymptotically synchronized up to a desired scaling function matrix and the unknown parameters can be estimated accurately. At the end of the paper, the corresponding numerical simulations have been given to verify the effectiveness of the proposed control techniques. The proposed method is also suitable for the MFPLS of other chaotic systems and has broad application in secure communication, image processing, and other fields.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This paper is supported by the National Natural Science Foundation of China (61373174 and 11301409), and thanks are due to all the references authors.

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