This paper investigates the complete periodic synchronization of memristor-based neural networks with time-varying delays. Firstly, under the framework of Filippov solutions, by using
Memristor, as the fourth fundamental passive circuit, was firstly postulated by Chua [
Different from the previous works, in this paper, we will study complete periodic synchronization of memristor-based neural networks described by the following differential equation:
The rest of this paper is organized as follows. In Section
In this section, we give some definitions and properties, which are needed later.
Suppose
For the system
The initial value associated with system (
Suppose that
From the theoretical point of view, the above parameters
We say that real matrix
Let there exists a vector there exists a vector
Suppose that there exists a bounded open set satisfies each solution satisfies then differential inclusion (
To proceed with our analysis, we need the following assumptions for system ( For
In this section, we will give a sufficient condition which ensures the existence of periodic solution of memristor-based neural network (
Under assumptions
Set
Based on the conditions of Lemma
If
Therefore, we have
Up to now, we have proved that
Notice that a constant function can be regarded as a special periodic function with arbitrary period or zero amplitude. Hence, we can obtain the following result.
Suppose assumption
By employing the method based on the
In this paper, we consider model (
Suppose that all the conditions of Theorem
Consider the following Lyapunov functional:
The master system (
In the literature, some results on stability analysis of periodic solution (or equilibrium point) or synchronization (or antisynchronization) control of memristor-based neural network were obtained [
As far as we know, there is no work on the periodic synchronization of memristor-based neural network via adaptive control. Thus, our outcomes are brand new and original compared to the existing results ([
In this section, one example is offered to illustrate the effectiveness of the results obtained in this paper.
Consider the second-order memristor-based neural network (
We take
For numerical simulations, we choose the external input
In order to demonstrate the adaptive controller (
Time-domain behavior of the state variables
Time-domain behavior of the state variables
Phase plane behavior of the master system (
The synchronization errors
Trajectories of control parameters
Trajectories of control parameters
In this paper, complete periodic synchronization of a class of memristor-based neural networks has been investigated. The master system synchronizes with the slave system by using adaptive control. The obtained results are novel since there are few works about complete periodic synchronization issue of memristor-based neural networks via adaptive control. In addition, the easily testable condition which ensures the existence of periodic solution of a class of memristor-based recurrent neural network is also much different from the existing work. The obtained results are also applicable to the continuous systems without switching jumps. Finally, a numerical example has been given to illustrate the validity of the present results.
This work was supported by the Natural Science Foundation of Hebei Province of China (A2011203103) and the Hebei Province Education Foundation of China (2009157).