Weak, modified and function projective synchronization of chaotic memristive neural networks with time delays

doi:10.1016/j.neucom.2014.08.008

Neurocomputing

Volume 149, Part B, 3 February 2015, Pages 667-676

https://doi.org/10.1016/j.neucom.2014.08.008 Get rights and content

Highlights

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A state-feedback controller of weak projective synchronization is designed and unknown control gain is determined.
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An adaptive controller is proposed and unknown control parameters are found to achieve adaptive modified and function projective synchronization..
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In the proof of all theorems, the assumption used in the existing literature is abandoned, and the state switching jumps are concluded in all theorems.

Abstract

This paper deals with the weak, modified and function projective synchronization issues for chaotic memristive neural networks with time delays. By applying the generalized Halanay inequality, a state-feedback controller is designed, and a novel condition is proposed to ensure the network be weak synchronization with certain error level. By means of Lyapunov–Krasovskii functional approach, the adaptive state-feedback controllers are designed and unknown control parameters are determined by adaptive updated laws to achieve function and modified projective synchronization. Several weak, modified and function projective synchronization conditions are addressed to ensure the synchronization goal. Finally, an illustrative example is given to demonstrate the effectiveness of the theoretical results.

Introduction

Memristor, the fourth basic circuital element along with resistor, inductor and capacitor, was first theoretically predicted by Chua in 1971 [1] (Fig. 1 [2]). In late 2008, an actual physical memristor was realized as a ${TiO}_{2}$ nanocomponent in Hewlett–Packard Laboratory [2]. This new passive electronic device has attracted unprecedented worldwide attention because of its potential applications in various fields, from new high-speed low-power processors [3], filters [4] to new biological models for associative memory [5], learning models of simple organisms [6], and in particular chaotic circuit [7], [8], [9], to which memristor has been introduced as new technology process nodes.

Recently, memristive neural networks (MNNs) have been designed based on primitive network by replacing resistors with memristors. Pershin and Di Ventra constructed a simple MNN composed of three electronic neurons connected by two memristor emulator synapses [5], it demonstrated the formation of the associative memory in this network. In [10], Itoh and Chua designed a memristic cellular automaton and discrete-time cellular neural network, which acts in quite a few applications, such as logical operations, image processing operations, complex behaviors, higher brain functions, and RSA algorithm. More recently, the nonlinear dynamics of MNNs with time delays have also been investigated in [11], [12], [13], [14], [15], [16], [17], [18], [19].

We note that a detailed analytical study of synchronization or anti-synchronization control of the neural network is necessary. Synchronization, which means the dynamical behaviors of coupled systems achieve spatial state at the same time, has been applied in secure communication, biology systems, linguistic networks, and information processing. As we know, synchronization exists in various types, such as complete synchronization [20], anti-synchronization [21], lag synchronization [22], generalized synchronization [23], projective synchronization [24], [25], [26], [27], [28], phase synchronization [29] and so on. In [31], Wen et al. considered the adaptive synchronization of memristor-based Chua׳s circuits and designed an adaptive controller to achieve the synchronization goal. Wu et al. investigated the exponential synchronization of memristor-based recurrent neural networks with time delays in [30], [32]. It should be pointed out that (1) in [30], [31], [32], the following assumption $co {{\underset{̲}{a}}_{ij}, {\overset{̄̄}{a}}_{ij}} f_{j} (x_{j} (t)) - co {{\underset{̲}{a}}_{ij}, {\overset{̄̄}{a}}_{ij}} f_{j} (y_{j} (t)) \subseteq co {{\underset{̲}{a}}_{ij}, {\overset{̄̄}{a}}_{ij}} (f_{j} (x_{j} (t)) - f_{j} (y_{j} (t)))$ was used in the proof of the main results. It can be easily checked that this assumption holds only when $f_{j} (x_{j} (t))$ and $f_{j} (y_{j} (t))$ have different signs, or $f_{j} (x_{j} (t)) = 0$ or $f_{j} (y_{j} (t)) = 0$ , where co(Q) denotes the closure of the convex hull of Q. (2) The results in [30], [31], [32] are independent of the neuron state switching jumps $T_{i}, i = 1, 2, \dots, n$ . From the view of mathematics, the results obtained in these works are meaningless to the theory research and application.

In this paper, we treat the weak, modified and function projective synchronization issue of chaotic MNNs with time delays. The main novelty of our contribution lies in three aspects: (1) a novel state-feedback controller is designed, and a condition is presented to achieve weak projective synchronization with an error level; (2) an adaptive controller is proposed to achieve function projective synchronization. As a special case, an adaptive modified projective synchronization condition is also developed; (3) the assumption (1) is abandoned in this paper. More precisely, all the conclusions are related to the state switching jumps $T_{i}, i = 1, 2, \dots, n$ .

The framework of this paper is organized as follows: In Section 2, the mathematics models of memristor and chaotic MNNs are described, and some preliminaries are introduced. In Section 3, a general scheme of weak projective synchronization for chaotic MNNs is presented. The adaptive modified and function projective synchronization are discussed in Section 4. In Section 5, a numerical example is presented to demonstrate the validity of the proposed results. Section 6 is the conclusions of this paper.

Notation: $R$ denotes the set of real numbers, $R^{n}$ denotes the n-dimensional Euclidean space, $R^{m \times n}$ denotes the set of all $m \times n$ real matrices. For $r > 0$ , $C ([- r, 0]; R^{n})$ denotes the family of continuous function φ from $[- r, 0]$ to $R^{n}$ with the norm $∥ φ ∥ = \sup_{- r \leq s \leq 0} | φ (s) |$ . $λ_{\max} (P), λ_{\min} (P)$ denote the maximum eigenvalue and minimum eigenvalue of $P \in R^{n \times n}$ , respectively. $P^{T}, P^{- 1}$ represent the transpose and inverse of $P \in R^{n \times n}$ , respectively. The Euclidian norm of the square matrix P is denoted by $∥ P ∥$ , where $∥ P ∥ = \sqrt{λ_{\max} ({PP}^{T})}$ .

Section snippets

Mathematics model of memristor

In the TiO₂ memristor, as shown in Fig. 2, a thin undoped titanium dioxide (TiO₂) layer and a thin oxygen-deficient doped titanium dioxide (TiO_2−x) layer are sandwiched between two platinum electrodes. When a voltage (or current) is applied to the device, the width of the TiO₂ and TiO_2−x layer changes as a function of the applied voltage (or current). As a result, the resistance between the two electrodes is altered.

Let D and ω denote the thickness of the sandwiched area and the doped area

Weak projective synchronization

Set $\overset{´}{D} = diag ({\overset{´}{d}}_{1}, {\overset{´}{d}}_{2}, \dots, {\overset{´}{d}}_{n})$ , $\overset{´}{A} = {({\overset{´}{a}}_{ij})}_{n \times n}, \overset{´}{B} = {({\overset{´}{b}}_{ij})}_{n \times n}$ , $\overset{`}{D} = diag ({\overset{`}{d}}_{1}, {\overset{`}{d}}_{2}, \dots, {\overset{`}{d}}_{n})$ , $\overset{`}{A} = {({\overset{`}{a}}_{ij})}_{n \times n}$ , $\overset{`}{B} = {({\overset{`}{b}}_{ij})}_{n \times n}$ , $Γ = diag (ρ_{1}, ρ_{2}, \dots, ρ_{n})$ , $Λ = diag (ϱ_{1}, ϱ_{2}, \dots, ϱ_{n})$ .

Define the weak projective synchronization error signal $e (t) = {(e_{1} (t), e_{2} (t), \dots, e_{n} (t))}^{T}$ as $e_{i} (t) = y_{i} (t) - α_{i} x_{i} (t), i = 1, 2, \dots, n,$ then the initial value associated with (6) is $e_{i} (t) = ϕ_{i} (t) - α_{i} ψ_{i} (t), t \in [- τ, 0], i = 1, 2, \dots, n$ .

Theorem 3.1

For the given scalars $α_{i}, i = 1, \dots, n$ , if there exist scalars $s_{i} > 0$ , $l_{i} > 0$ , $i = 1, \dots, 6$ , and the appropriate gain parameter matrix $K = diag (k_{1}, k_{2}, \dots, k_{n})$ , such that $β >$

Function projective synchronization

Define the function projective synchronization error signal e_i(t) as $e_{i} (t) = y_{i} (t) - α_{i} (t) x_{i} (t), i = 1, 2, \dots, n,$ where $α_{i} (t)$ is differentiable and bounded, i.e., there exist scalars $ξ_{i} > 0$ , such that $| α_{i} (t) | \leq ξ_{i}$ , $i = 1, 2, \dots, n$ .

Theorem 4.1

For the given synchronization scalar functions $α_{i} (t), i = 1, 2, \dots, n$ , the MNN system (2), (5) can achieve function projective synchronization under the following adaptive controller: $u_{i} (t) = - \sum_{j = 1}^{n} {\overset{`}{a}}_{ij} f_{j} (α_{j} (t) x_{j} (t)) + α_{i} (t) \sum_{j = 1}^{n} {\overset{`}{a}}_{ij} f_{j} (x_{j} (t)) - \sum_{j = 1}^{n} {\overset{`}{b}}_{ij} g_{j} (α_{j} (t - τ) x_{j} (t - τ)) + α_{i} (t) \sum_{j = 1}^{n} {\overset{'}{b}}_{ij} g_{j} (x_{j} (t - τ)) + {\dot{α}}_{i} (t)$

Numerical example

Example 1

Consider the two-dimensional MNNs ${\begin{matrix} {\dot{x}}_{1} (t) & = & - d_{1} (x_{1} (t)) x_{1} (t) + a_{11} (x_{1} (t)) f (x_{1} (t)) \\ + a_{12} (x_{1} (t)) f (x_{2} (t)) + b_{11} (x_{1} (t)) g (x_{1} (t - 1)) \\ + b_{12} (x_{1} (t)) g (x_{2} (t - 1)), \\ {\dot{x}}_{2} (t) & = & - d_{2} (x_{2} (t)) x_{1} (t) + a_{21} (x_{2} (t)) f (x_{1} (t)) \\ + a_{22} (x_{2} (t)) f (x_{2} (t)) + b_{21} (x_{2} (t)) g (x_{1} (t - 1)) \\ + b_{22} (x_{2} (t)) g (x_{2} (t - 1)), \end{matrix}$ where $d_{1} (x_{1} (t)) = {\begin{matrix} 1, & | x_{1} (t) | \leq \frac{1}{4}, \\ 1.01, & | x_{1} (t) | > \frac{1}{4}, \end{matrix} d_{2} (x_{2} (t)) = {\begin{matrix} 1.01, & | x_{2} (t) | \leq \frac{1}{4}, \\ 1, & | x_{2} (t) | > \frac{1}{4}, \end{matrix} a_{11} (x_{1} (t)) = {\begin{matrix} 2, & | x_{1} (t) | \leq \frac{1}{4}, \\ 1.7, & | x_{1} (t) | > \frac{1}{4}, \end{matrix} a_{12} (x_{1} (t)) = {\begin{matrix} - 0.1, & | x_{1} (t) | \leq \frac{1}{4}, \\ - 0.06, & | x_{1} (t) | > \frac{1}{4}, \end{matrix} a_{21} (x_{2} (t)) = {\begin{matrix} - 4.7, & | x_{2} (t) | \leq \frac{1}{4}, \\ - 5, & | x_{2} (t) | > \frac{1}{4}, \end{matrix} a_{22} (x_{2} (t)) = {\begin{matrix} 2.4, & | x_{2} (t) | \leq \frac{1}{4}, \\ 2.9, & | x_{2} (t) | > \frac{1}{4}, \end{matrix} b_{11} (x_{1}$

Conclusion

In this paper, the weak, modified and function projective synchronization for chaotic MNNs with time delays have been investigated. By applying the generalized Halanay inequality, a weak projective synchronization scheme has been proposed to ensure that the chaotic MNNs can achieve weak projective synchronization with an error level under the designed state-feedback controller. The new adaptive controllers have been developed, and unknown control parameters have also been determined by adaptive

Acknowledgment

The authors are very grateful to the Associate Editor and Reviewers for their valuable comments and constructive suggestions, which help us enriching the content and improving the presentation of this paper.

Huaiqin Wu is a professor of mathematics and applied mathematics at the Yanshan University, Hebei, China. Wu received the B.S. and Ph.D. degrees in mathematics from Harbin Institute of Technology, Heilongjiang, China, in 1987 and 2008, respectively. His main scientific interests are in the field of differential inclusion and neural network theory. He is an author of more than 50 scientific papers.

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Ruoxia Li is a master candidate in the Department of Science, Yanshan University, Qinhuangdao, China. She received the B.S. degree from Huanghuai University, Zhumadian, China, in 2012. Her current research interests include neural networks and memristors.

Rong Yao is a master candidate in the Department of Science, Yanshan University, Qinhuangdao, China. She received the B.S. degree in mathematics from TaiYuan Normal University, Shanxi, China in 2012. Her main research interest is the neural network for solving the optimization problems.

Xiaowei Zhang is a master candidate in the Department of Science, Yanshan University, Qinhuangdao, China. Zhang received the B.S. degree in mathematics from Hanhan University, Handan, China, in 2012. His main research interest is the dynamic behavior of the discontinuous neural networks.

^☆: This work was supported by the Natural Science Foundation of Hebei Province of China (A2011203103).

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Weak, modified and function projective synchronization of chaotic memristive neural networks with time delays☆

Highlights

Abstract

Introduction

Section snippets

Mathematics model of memristor

Weak projective synchronization

Function projective synchronization

Numerical example

Conclusion

Acknowledgment

Neural Netw.

Neural Netw.

Neurocomputing

Neurocomputing

Phys. Lett. A

Inf. Sci.

Nonlinear Anal. Real World Appl.

Chaos Solitons Fractals

Appl. Math. Comput.

Chaos Solitons Fractals

Inf. Sci.

Phys. Lett. A

Neurocomputing

Memristor—the missing circuit element

IEEE Trans. Circuit Theory

The missing memristor found

Nature

The fourth element

Nature

Memristive adaptive filters

Appl. Phys. Lett.

Memristive model of amoeba learning

Phys. Rev. E