Almost periodic solutions for a memristor-based neural networks with leakage, time-varying and distributed delays

doi:10.1016/j.neunet.2015.04.005

Neural Networks

Volume 68, August 2015, Pages 34-45

https://doi.org/10.1016/j.neunet.2015.04.005 Get rights and content

Abstract

In this paper, we study the existence and global exponential stability of almost periodic solution for memristor-based neural networks with leakage, time-varying and distributed delays. Using a new Lyapunov function method, we prove that this delayed neural network has a unique almost periodic solution, which is globally exponentially stable. Moreover, the obtained conclusion on the almost periodic solution is applied to prove the existence and stability of periodic solution (or equilibrium point) for this delayed neural network with periodic coefficients (or constant coefficients).

Introduction

During the last few years, a new circuit element, memristor have received increasing attention from its great potential application value in physics, computer science, electrical and electronic engineering communities. For more details, see the literature (Chua, 1971, Corinto et al., 2004, Di Ventra et al., 2009, Itoh and Chua, 2008b, Merrikh-Bayat and Shouraki, 2011a, Merrikh-Bayat and Shouraki, 2011b, Petras, 2010, Strukov et al., 2008, Tour and He, 2008) and references therein. At the same time, an interesting issue is to investigate the memristor-based neural networks, which is an ideal model for the case where the memristor-based circuit network process of more efficient learning with a realization of the famous Hebbian rule stating, in a simplified form, that “neurons that fire together, wire together”. Moreover, the analysis of dynamical behaviors for the memristor-based neural networks has been found useful to address a number of interesting engineering tasks and therefore have received a great deal of attention in the literature (see, for instance, Chen, Zeng, & Jiang, 2014a, Chen, Zeng, & Jiang, 2014b, Chen, Zeng, & Jiang, 2014c, Hu & Wang, 2010, Itoh & Chua, 2008a, Jiang, Wang, Mei, & Shen, 2015, Wu, Wen, & Zeng, 2012, Wu & Zeng, 2012, Zhang, Shen, & Sun, 2013, Zhang, Shen, & Wang, 2013 and Zhang, Shen, & Wang, 2015).

As well known, the delays are actually encountered in practical implementation, due to the finite switching speed of the neuron amplifiers and the finite signal propagation speed. It is important to study delayed memristor-based neural networks (DMNN). In particular, it is worth pointing out that, in a real nervous system, a typical time delay in the negative feedback terms which is known as leakage (or forgetting) delays have a tendency to destabilize the system (see Gopalsamy, 2007) and have great impact on the dynamical behavior of neural networks (see Li and Cao, 2010, Long et al., 2012, Peng and Wang, 2013, Tong, Zhou & Wang, 2014, Tong, Zhu, Zhou, Xu & Fang, 2013, Zhou, Tong, Gao, Ji & Su, 2012). Since leakage delays have a destabilizing influence on the dynamical behaviors of neural networks, it is necessary and important to consider the leakage delays’ effects on the study of state estimation of neural networks. More recently, there are more works on neural networks with leakage delays in the literature (see Balasubramanian, Kalpana, & Rakkiyappan, 2011a, Balasubramanian, Kalpana, & Rakkiyappan, 2011b, Gao, Wang, & Zhang, 2014, Lakshmanan, Park, Jung, & Balasubramaniam, 2012, Liu, 2013, Li & Huang, 2009, Li, Rakkiyappan, & Balasubramanian, 2011, Park, Kwon, Park, Lee, & Cha, 2012, Peng, 2010 and Zhang & Shao, 2013). However to the best of our knowledge, relative works on DMNN with leakage delay has not been investigated until now.

In many applications, knowing the property of periodic oscillatory solutions is very interesting and valuable. For example, the human brain is often in periodic oscillatory or chaos state, hence it is of prime importance to study periodic oscillatory and chaos phenomenon of neural networks for understanding the function of the human brain. The studies of the periodic oscillation of various neural networks such as the Hopfield network, cellular neural networks, and bidirectional associative memories are all reported in the literature (see, for instance, Chen et al., 2014c, Peng, 2010, Zhang, Shen, & Sun, 2013 and Chen & Wang, 2004, Chen & Wang, 2005, Jiang, Zhang, & Teng, 2005 and Tan & Tan, 2009 and references therein). Meanwhile, in practice, almost periodic phenomenon is more common than periodic phenomenon (see, for instance, Levitan & Zhikov, 1982). For example, upon considering long-term dynamical behaviors, the periodic parameters of the neural networks often turn out to experience certain perturbations, that is, parameters become periodic up to a small error. So, almost periodic oscillatory behavior is considered to be more accordant with reality. There have been a great number of results on almost periodic oscillation of the neural networks with or without delay (see, for instance, Allegretto, Papini, & Forti, 2010, Gao et al., 2014, Huang & Cao, 2003, Jiang et al., 2005, Levitan & Zhikov, 1982, Qin, Xue, & Wang, 2013, Wang, Lu, & Chen, 2009, Wang, 2010, Xiang & Cao, 2009, and Zhang & Shao, 2013,). In Chen et al. (2014a), we have studied the existence and global exponential stability of almost periodic solution for the class of a memristor-based neural networks with time-varying.

This paper is a continuation of the paper (Chen et al., 2014a), the main purpose is to give the conditions for the existence and exponential stability of the almost periodic solutions for a DMNN with leakage, time-varying and distributed delays. The main advantages are highlighted as follows:

•
The studies of dynamical behaviors on DMNN with time delays in the leakage term is firstly put forward.
•
By applying a new Lyapunov function techniques, we derive some new sufficient conditions ensuring the existence, uniqueness and exponential stability of the almost periodic DMNN with leakage, time-varying and distributed delays.
•
It is worth pointing that our model involves more general memristor-based weights, time-varying delays, distributed delays and leakage delays. Therefore, many relative works are included in this paper.
•
It is noted that our result is more general, which include the existence, and global exponential stability of almost periodic solution, periodic solution and equilibrium point and it is valid for the usual neural networks model.

The rest of the paper is organized as follows. A DMNN model with leakage, time-varying and distributed delays is introduced and some necessary definitions are given in Section 2. A sufficient criterion is obtained to ensure the global existence and boundedness of some solutions, the existence and exponential stability of an almost periodic solution of the networks in Section 3. We show the applications of our main results, that are the existence and global exponential stability of periodic solution and equilibrium point in Section 4. Examples and simulations are obtained in Section 5. Finally, the paper is concluded in Section 6.

Section snippets

Model description and preliminaries

According to the physical characteristics of a memristor, the addressed memristor-based neural networks with time-varying delays, distributed delays and leakage delays are described by $\frac{d x_{i} (t)}{d t} = - d_{i} (t) x_{i} (t - η_{i} (t)) + \sum_{j = 1}^{n} a_{i j} (t, x_{j} (t)) f_{j} (x_{j} (t)) + \sum_{j = 1}^{n} b_{i j} (t, x_{j} (t - τ_{i j} (t))) g_{j} (x_{j} (t - τ_{i j} (t))) + \sum_{j = 1}^{n} \int_{0}^{\infty} p_{i j} (s) c_{i j} (t, x_{j} (t - s)) h_{j} (x_{j} (t - s)) d s + I_{i} (t),$ for $i = 1, 2, \dots, n$ , where $n$ corresponds to the number of units in a neural network; $x_{i} (t)$ denotes the state variable associated with the $i$ th neuron; $f_{j}$ , $g_{j}$ and $h_{j}$ are

Boundedness and almost periodicity

In order to prove existence of bounded solution for DMNN (2.1), we show that the solutions of DMNN (2.1) with initial condition (2.5) and $| φ_{i} (t) - \int_{t - η_{i} (t)}^{t} d_{i} (s) φ_{i} (s) d s | < μ_{i} \frac{γ}{λ}$ for $i = 1, \dots, n$ and $t \leq 0$ , are bounded where $γ = max_{1 \leq i \leq n} {\sum_{j = 1}^{n} μ_{j} [{\bar{a}}_{i j}^{u} | f_{j} (0) | + {\bar{b}}_{i j}^{u} | g_{j} (0) | + {\bar{c}}_{i j}^{u} | h_{j} (0) | \int_{0}^{\infty} p_{i j} (s) d s] + {\bar{I}}_{i}} .$

Theorem 3.1

Assume that the assumptions (A1), (A2) and (A3) hold. Let $x (t)$ be the solution of DMNN (2.1) with initial condition (2.5), (3.1). Then, for $t$ in the interval of existence and $i = 1, \dots, n$ , $| x_{i} (t) - \int_{t - η_{i} (t)}^{t} d_{i} ($

Periodicity and Stability

In this section, we consider the global exponential periodicity and stability of DRNN model (2.1).

Firstly since any periodic function can be regarded as an almost periodic function, Theorem 3.2 can be directly applied into the periodic case, i.e., we have the following result.

Theorem 4.1

Let $η_{i} (t)$ , $τ_{i j} (t)$ , ${\hat{a}}_{i j} (t)$ , ${\overset{̌}{a}}_{i j} (t)$ , ${\hat{b}}_{i j} (t)$ , ${\overset{̌}{b}}_{i j} (t)$ , ${\hat{c}}_{i j} (t)$ , ${\overset{̌}{c}}_{i j} (t)$ , $d_{i} (t)$ and $I_{i} (t)$ are all $ω$ -periodic functions. If the assumptions (A1), (A2), (A3) and (A4) hold, then there exists a unique $ω$ -periodic solution

Numerical examples

In this section, we will give an example to demonstrate the results.

Example

Consider the following DMNN with leakage, time-varying and distributed delays, ${\begin{matrix} \frac{d x_{1} (t)}{d t} = - 4 x_{1} (t - η (t)) + \frac{1}{8} a (t, x_{1} (t)) tanh (| x_{1} (t) | - 1) + \frac{1}{4} a (t, x_{2} (t)) tanh (| x_{2} (t) | - 1) + \frac{1}{4} b (t, x_{1} (t - τ (t))) tanh (| x_{1} (t - τ (t)) | - 1) + \frac{1}{8} b (t, x_{2} (t - {sin}^{2} t)) tanh (| x_{2} (t - τ (t)) | - 1) + \frac{1}{4} \int_{0}^{\infty} e^{- s} c (t, x_{1} (t - s)) tanh (| x_{1} (t - s) | - 1) d s + \frac{1}{8} \int_{0}^{\infty} e^{- s} c (t, x_{2} (t - s)) tanh (| x_{2} (t - s) | - 1) d s + sin t + sin \sqrt{3} t, \\ \frac{d x_{2} (t)}{d t} = - 4 x_{2} (t - η (t)) + \frac{1}{4} a (t, x_{1} (t)) tanh (| x_{1} (t) | - 1) + \frac{1}{4} a (t, x_{2} (t)) tanh (| x_{2} (t) | - 1) + \frac{1}{4} b (t, x_{1} (t - τ (t))) tanh (| x_{1} (t \end{matrix}$

Conclusion

In this paper, we use the concept of the Filippov solution and differential inclusion to study the dynamics of a memristor-based neural networks with leakage, time-varying and distributed delays. Under some conditions, we prove existence, uniqueness and global exponential stability of almost periodic solution of the neural network by using a new Lyapunov function technique. It should be pointed out that there are many works on the existence and the globally exponential stability of the almost

Acknowledgments

This work is supported by the Natural Science Foundation of China under Grant 61125303, National Basic Research Program of China (973 Program) under Grant 2011CB710606, the Program for Science and Technology in Wuhan of China under Grant 2014010101010004, the Program for Changjiang Scholars and Innovative Research Team in University of China under Grant IRT1245.

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Almost periodic solutions for a memristor-based neural networks with leakage, time-varying and distributed delays

Abstract

Introduction

Section snippets

Model description and preliminaries

Boundedness and almost periodicity

Periodicity and Stability

Numerical examples

Conclusion

Acknowledgments

Physics Letters A

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