This paper deals with a class of neutral-type complex-valued neural networks with delays. By means of Mawhin’s continuation theorem, some criteria on existence of periodic solutions are established for the neutral-type complex-valued neural networks. By constructing an appropriate Lyapunov-Krasovskii functional, some sufficient conditions are derived for the global exponential stability of periodic solutions to the neutral-type complex-valued neural networks. Finally, numerical examples are given to show the effectiveness and merits of the present results.

As it is well known, in a large amount of applications, complex signals often occur and the complex-valued neural networks (CVNNs) are preferable. Therefore, there have been increasing research interests in the dynamical behaviors of complex-valued recurrent neural networks; see [

On the other hand, neutral-type CNNs can be used for describing these complicated dynamic properties of neural cells. Generally, it can be described as

To our knowledge, no studies have been reported on the properties of neutral-type CVNNs with time-varying delays until now. This motivated us to carry out a study in this paper. In this study, we discuss the existence and exponential stability of periodic solution for the following neutral-type CVNNs with time-varying delays:

System (

Throughout the paper, we give some notations:

The neural network model (

When

In general, when

We also make the following assumptions:

There exist nonnegative constants

The distinctive contributions of this paper are outlined as follows: (1) the neutral-type neural network model (

By separating the state, the connection weight, the activation function, and the external input into its real and imaginary part, then system (

In this section, we state some useful definitions and lemmas.

Define operator

Suppose that

where

The periodic solution of (

In order to investigate the periodicity of system (

Assume that conditions (

From (

Obviously,

Since the condition

In this paper, we use Mawhin’s continuous theorem and some mathematical analysis technique for obtaining existence results. We can also use other methods (e.g., some fixed points theorem) to discuss the existence of

In this section, we establish some results for the uniqueness and exponential stability of the

Under conditions of Theorem

there exist

where

Using conditions of Theorem

According to condition (

It is well known that Lyapunov method has been widely used for studying stability problems. In this paper we construct a novel Lyapunov functional for studying the stability of periodic solutions, which is different from traditional Lyapunov functional method. And the proposed analysis method is also easy to extend to the case of other type neural networks. In the future, we will further study the synchronization problem and/or the Markovian jumping problem of complex-valued neural networks.

In studying the stability problems of time-delay systems, various methods have been developed. The important methods are based on the LKF methods [

In order to verify the feasibility of our results, consider the following neutral-type CVNNs:

The numerical simulations of (

Transient states of the real and imaginary parts of system (

State trajectories of the real and imaginary parts of system (

We greatly want to provide the circuit diagram of system (

In this paper, we have investigated stability problems of periodic solutions for a class of neutral-type complex-valued neural networks with time-varying delays. By utilizing novel Lyapunov-Krasovskii functionals, the sufficient conditions are derived to guarantee global exponential stability for the involved systems. A simulation example has been provided to show the usefulness of the proposed global exponential stability conditions.

We mention here that some finer approaches to deal with time delays would be the delay-slope-dependent method [

The authors declare to have no competing interests in this paper.

The authors acknowledge the funding of NNSF (no. 11571136) of China.