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The purpose of this note is to study impulsive control and synchronization of memristor based chaotic circuits shown by Muthuswamy. We first establish a less conservative sufficient condition for the stability of memristor based chaotic circuits. After that, we discuss the effect of errors on stability. Meanwhile, we also discuss impulsive synchronization of two memristor based chaotic systems. Our results are more general and more applicable than the ones shown by Yang, Li, and Huang. Finally, several numerical examples are given to show the effectiveness of our methods.

The memristor was postulated as the fourth circuit element by Chua [

In practical applications, impulsive control has some advantages: for example, impulsive control provides an efficient way in dealing with systems especially which cannot endure continuous control inputs. During the last several decades, impulsive control theory has attracted considerable attention because impulsive control method can be employed in many fields, such as the stabilization and synchronization of chaotic systems [

Recently, complex dynamical systems are receiving much attention, and there is no exception for chaotic systems. Muthuswamy [

In this note, we shall also consider the asymptotic stabilization and synchronization of memristor based chaotic circuits, as in [

The equations for the memristor based chaotic circuit presented in [

The state trajectory of the memristor based chaotic circuit shown in (

In the sequel, we mainly adopt the notation and terminology in [

In this section, we design impulsive control for the memristor based chaotic circuit shown by Muthuswamy.

Let

Let us construct the following Lyapunov function:

From the fact that

Let

In many practical applications, we cannot guarantee the impulses without any error due to the limit of equipment and technology. So we should take into account the influence of impulsive control gain errors on the systems. Motivated by the above discussions, we will study the stabilization of system (

Let

Let us construct the following Lyapunov function:

In many practical applications, the parameters

Let

Let us construct the following Lyapunov function:

In this section, we investigate impulsive synchronization of two memristor based chaotic circuits. Equation (

Let

Let us construct the following Lyapunov function:

In this section, some numerical examples are given to illustrate the effectiveness of our results. The initial condition of the system (

It is easy to see that

The state trajectory of the controlled memristor based chaotic circuit.

In this example, the coefficient matrix

The state trajectory of the controlled memristor based chaotic circuit.

In this example, the matrices

The state trajectory of the controlled memristor based chaotic circuit.

In this example, the matrix

Simulation results of synchronization errors.

In this note, we discuss impulsive control and synchronization of memristor based chaotic circuits shown by Muthuswamy [

The authors declare that they have no conflicts of interest.

All authors contributed equally to the writing of this paper. All authors read and approved the final version of this paper.

This work is funded by Chongqing Research Program of Basic Research and Frontier Technology (no. cstc2017jcyjAX0032), the National Natural Science Foundation of China under Grant no 11601047, Key Laboratory of Chongqing Municipal Institutions of Higher Education (Grant no.