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A memristor-based five-dimensional (5D) hyperchaotic Chua’s circuit is proposed. Based on the Lyapunov stability theorem, the controllers are designed to realize the synchronization and lag synchronization between the hyperchaotic memristor-based Chua’s circuits under different initial values, respectively. Numerical simulations are also presented to show the effectiveness and feasibility of the theoretical results.

The fourth fundamental circuit element included along with the resistor, capacitor, and inductor, called the memristor, was first postulated by Chua in 1971 [

Itoh and Muthuswamy presented a fourth-order memristor based on Chua’s oscillator by replacing Chua’s diode with an active two-terminal circuit consisting of a conductance and a flux-controlled memristor and observed rich nonlinear dynamic behavior in such system [

Many works have been done about the stabilization and synchronization of memristor-based systems [

The rest of the paper is organized as follows. In Section

Referring to [

The hyperchaotic circuit based on memristor [

And the equations for the circuit are described by

Letting

When

The chaotic attractor of the memristor oscillator.

Let system (

Letting

We now state our main results.

Suppose that there exist positive constants

Choose the Lyapunov function as follows:

Then the differentiation of

According to Lyapunov theory, the inequality

In simulation, we select the parameters of memristor-based Chua’s system as initial values of drive and response systems are

Synchronization error of memristor-based hyperchaotic systems.

In this section, we study the lag synchronization of memristor-based Chua’s circuit. The system (

As for vector function

The above condition is considered as the uniform Lipschitz condition, and

We construct the response system as follows:

We now state our main results.

Suppose that there exist positive constants

Then, the synchronization error system (

Choose the Lyapunov function as follows:

Then the differentiation of

According to Lyapunov theory, the inequality

Let

If there exist positive constants

In simulation, we select the parameters of memristor-based Chua’s system as initial values of drive and response systems are

Based on the bound of the hyperchaotic attractor, we can choose

Lag synchronization error of memristor-based hyperchaotic systems.

The norm of lag synchronization error of memristor-based hyperchaotic systems.

This paper has studied the synchronization and lag synchronization of memristor-based 5D hyperchaotic circuits. The feedback controllers have been designed to stabilize the synchronization error system and lag synchronization error system. Simulation results were given to verify the effectiveness and feasibility of method.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This publication was made possible by NPRP Grant no. NPRP-4-1162-181 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. This work was also supported by Natural Science Foundation of China (Grant no. 61374078, Grant no. 61403050, and Grant no. 61302180).