Exponential Synchronization for Fractional-order Time-delayed Memristive Neural Networks

Dawei, Ding; Yaqin, Zhang; Nian, Wang

doi:10.21307/ijanmc-2019-001

Full Article

I.

INTRODUCTION

Chua already supposed the existence of memristor in 1971 [1], however, the practical device of memristor in electronics is obtained in [2] until 2008. In addition to the existing three kinds of circuit elements, memristor is regarded as the fourth basic circuit element and is defined by a nonlinear charge-flux characteristic. As everyone knows, resistors can be used to work as connection weights so that it can emulate the synapses in artificial neural networks. However, in the neural networks of biological individual, long-term memories is essential in the synapses among neurons, but for the general resistors, it is impossible to have the function of memory. Recently, due to the memory characteristics of memristor, memristor can replace the resistor to develop a new neural networks that is memristor-based neural networks (MNN) [3-6].

In recent years, more and more attentions have been put on the dynamical analysis of memristor-based neural networks, such as the investigation of stability [7-10], periodicity [11-13], system synchronization [14-22], passivity analysis [23], dissipativity [24-25] and attractivity [26]. Particularly, the stability and synchronization of MNN has been widely studied in [27-30]. In fact, synchronization means the dynamics of nodes share the common time-spatial property. Therefore we can understand an unknown dynamical system by achieving the synchronization with the well-known dynamical systems [18]. Moreover, in the transmission of digital signals, communication will become security, reliable and secrecy by achieving synchronization between the various systems. Therefore, the synchronization of MNN is still worth further research.

Moreover, the fractional-order models can better describe the memory and genetic properties of various materials and process, so the fractional-order models have received a lot of research attentions than integer-order models. In recent years, with the improvement of fractional-order differential calculus and fractional-order differential equations, it is easy to model and analyze practical problems [31, 32]. Therefore, there have been a lot of researches about the dynamical analysis and synchronization of fractional-order memristor-based neural networks (FMNN) [34-39]. Finite-time synchronization, hybrid projective synchronization and adaptive synchronization of FMNN have all been researched [34-36]. However, there are only a very few research results on exponential synchronization of FMNN. In fact, the exponential synchronization of neural networks has been widely used in the theoretical research and practical application of many scientific fields, for example, associative memory, ecological system, combinatorial optimization, military field, artificial intelligence system and so on [40-43]. So the exponential synchronization of FMNN is still worth further studying as it is a significant academic problem.

On the other hand, the stability and synchronization of FMNN without time delay have been deeply studied such as in [33]. However, in hardware implementation of neural networks, time delay is unavoidable owing to the finite switching speeds of the amplifiers. And it will cause instability, oscillation and chaos phenomena of systems. So the investigation for stability and synchronization of FMNN cannot be independent on the time delay.

Motivated by the above discussion, this paper studies the exponential synchronization of FMNN with time-varying delays. The main contributions of this paper can be listed as follow. (1) This is the first attempt to achieve exponential synchronization of FMNN with time-varying delays by employing a simple linear error feedback controller. (2) The sufficient condition for exponential synchronization of FMNN with time delays is obtained based on comparison principle instead of the traditional Lyapunov theory. (3) Some previous research results of exponential synchronization for integer-order memristor-based system are the special cases of our results. Furthermore, some numerical examples are given to demonstrate the effectiveness and correctness of the main results.

The rest of this paper is organized as follows. Preliminaries including the introduction of Caputo fractional-order derivative, model description, assumptions, definitions and lemmas are presented in Section 2. Section 3 introduces the sufficient condition for exponential synchronization of the FMNN. In Section4, the numerical simulations are presented. Section5 gives the conclusion of this paper.

II.

PRELIMINARIES

Compared to the integer-order derivatives, we know the distinct advantage of Caputo derivative is that it only requires initial conditions from the Laplace transform of fractional derivative, and it can represent well-understood features of physical situations and making it more applicable to real world problems [36]. So in the rest of this paper, we apply the Caputo fractional-order derivative for the fractional-order memristor-based neural networks (FMNN) and investigate the exponential synchronization of FMNN.

A.

The Caputo fractional-order derivative

Definition1 [32] The Caputo fractional-order derivative is defined as follows: (1) $D_{t}^{q} f (t) = \frac{1}{Γ (m - q)} \int_{t_{0}}^{t} \frac{f^{(m)} (τ)}{{(t - τ)}^{q - m + 1}} d τ,$ where q is the order of fractional derivative, m is the first integer larger than q, m − 1 ≤ q < m,Γ(⋅) is the Gamma function, (2) $T (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t .$

Particularly, when 0<q<1, (3) $D_{t}^{q} f (t) = \frac{1}{Γ (1 - q)} \int_{t_{0}}^{t} \frac{f' (τ)}{{(t - t_{0})}^{q}} d τ .$

B.

Model description

In this paper, referring to some relevant works on FMNN [35,36], we consider a class of FMNN with time-varying delays described by the following equation, (4) $\begin{matrix} D^{q} x_{i} (t) = - c_{i} x_{i} (t) + \sum_{j = 1}^{n} a_{i j} (x_{j} (t)) f_{j} (x_{j} (t)) + \sum_{j = 1}^{n} b_{i j} (x_{j} (t - τ_{j} (t))) g_{j} (x_{j} (t - τ_{j} (t))) + I_{i} \\ t \geq 0, i \in N . \\ a_{i j} (x_{j} (t)) = \frac{M_{i j}}{C_{i}} \times δ_{i j,} b_{i j} (x_{j} (t - τ_{j} (t))) = \frac{W_{i j}}{C_{i}} \times δ_{i j}, δ_{i j} = {\begin{array}{l} 1, & i \neq j, \\ - 1, & i = j, \end{array} \end{matrix}$ where x_i(t) is the state variable of the ith neuron (the voltage of capacitor C_i), q is the order of fractional derivative, c_i > 0 is the self-regulating parameters of the neurons, 0 ≤ τ_j ≤ τ and (τ is a constant ) represents the transmission time-varying delay. f_j,g_j :R → R are feedback functions without and with time-varying delay. a_ij(x_j(t)) and b_ij(x_j(t − τ_j(t))) are memristive connective weights, which denote the neuron interconnection matrix and the delayed neuron interconnection matrix, respectively. W_ij and M_ij denote the memductances of memristors R_ij and F_ij respectively. And R_ij represents the memristor between the feedback function f_i(x_i(t)) and x_i(t), F_ij represents the memristor between the feedback function g_i(x_i(t − τ_i(t))) and x_i(t). I_i represents the external input. According to the feature of memristor, we denote (5) $a_{i j} (x_{j} (t)) = {\begin{matrix} {\overset{⌢}{a}}_{i j}, & x_{j} (t) \leq 0, \\ {\overset{⌣}{a}}_{i j}, & x_{j} (t) > 0, \end{matrix} b_{i j} (x_{j} (t - τ_{j} (t))) = {\begin{matrix} {\overset{⌢}{b}}_{i j}, & x_{j} (t - τ_{j} (t)) \leq 0. \\ {\overset{⌣}{b}}_{i j}, & x_{j} (t - τ_{j} (t)) > 0. \end{matrix}$

C.

Assumptions, Definitions and Lemmas

In the rest of paper, we first make following assumption for system (4).

Assumption1: For j ∈ N, ∀s₁, s₂ ∈ R, the neuron activation functions f_j, g_j bounded, f_i(0) = g_j(0) = 0 and satisfy (6) $\begin{matrix} 0 \leq \frac{f_{j} (s_{1}) - f_{j} (s_{2})}{s_{1} - s_{2}} \leq σ_{j}, \\ 0 \leq \frac{g_{j} (s_{1}) - g_{j} (s_{2})}{s_{1} - s_{2}} \leq ρ_{j}, \end{matrix}$ where s₁ ≠ s₂ and σ_j, ρ_j are nonnegative constants.

We consider system (4) as drive system and corresponding response system is given as follows:3 (7) $\begin{matrix} D^{q} y_{i} (t) = - c_{i} y_{i} (t) + \sum_{j = 1}^{n} a_{i j} (y_{j} (t)) f_{j} (y_{j} (t)) + \sum_{j = 1}^{n} b_{i j} (y_{j} (t - τ_{j} (t))) g_{j} (y_{j} (t - τ_{j} (t))) + I_{i} + u_{i}, \\ t \geq 0, i \in N, \end{matrix}$

Where (8) $a_{i j} (y_{j} (t)) = {\begin{matrix} {\overset{⌢}{a}}_{i j}, & y_{j} (t) \leq 0, \\ {\overset{⌣}{a}}_{i j}, & y_{j} (t) > 0, \end{matrix} b_{i j} (y_{j} (t - τ_{j} (t))) = {\begin{matrix} {\overset{⌢}{b}}_{i j}, & y_{j} (t - τ_{j} (t)) \leq 0. \\ {\overset{⌣}{b}}_{i j}, & y_{j} (t - τ_{j} (t)) > 0. \end{matrix}$ and u_i(^t) is a liner error feedback control function which defined by u_i(^t) = ω_i(y_i(t) − x_i(t)), where ω_i, i ∈ N are constants, which denotes the control gain. Next, we define the synchronization error e(t) as e(t) = (e₁(t),e₂(t),….,e_n(t))^T, where e_i(t) = y_i(t) − x_i(t). According to the system (4) and system (7), the synchronization error system can be described as follows: (9) $\begin{matrix} D^{q} e_{i} (t) = - c_{i} e_{i} (t) + [\sum_{j = 1}^{n} a_{i j} (y_{j} (t)) f_{j} (y_{j} (t)) - \sum_{j = 1}^{n} a_{i j} (x_{j} (t)) f_{j} (x_{j} (t))] \\ + [\sum_{j = 1}^{n} b_{i j} (y_{j} (t - τ_{j} (t))) g_{j} (y_{j} (t - τ_{j} (t))) - \sum_{j = 1}^{n} b_{i j} (x_{j} (t - τ_{j} (t))) g_{j} (x_{j} (t - τ_{j} (t)))] \\ + u_{i} (t), t \geq 0, i \in N \end{matrix}$ where a_ij(y_j(t)), b_ij(y_j(t − τ_j(t))), a_ij(x_j(t)), b_ij(x_j(t − τ_j(t))) are the same as those defined above, u_i(^t) = ω_i(y_i(t) − x_i(t)) = ω_ie_i(t), where ω_i, i ∈ N are constants, which denotes the control again.

According to the theories of differential inclusions and set valued maps [40], if x_i(t) and y_i(t) are solutions of (4) and (7) respectively, system (4) and system (7) can be written as follow: (10) $\begin{matrix} D^{q} x_{i} (t) \in - c_{i} x_{i} (t) + \sum_{j = 1}^{n} c o [a_{i j} (x_{j} (t))] f_{j} (x_{j} (t)) \\ + \sum_{j = 1}^{n} c o [b_{i j} (x_{j} (t - τ_{j} (t)))] g_{j} (x_{j} (t - τ_{j} (t))) + I_{i}, t \geq 0, i \in N \end{matrix}$

And

(11)

\begin{matrix} D^{q} y_{i} (t) \in - c_{i} y_{i} (t) + \sum_{j = 1}^{n} c o [a_{i j} (y_{j} (t))] f_{j} (y_{j} (t)) \\ + \sum_{j = 1}^{n} c o [b_{i j} (y_{j} (t - τ_{j} (t)))] g_{j} (y_{j} (t - τ_{j} (t))) + I_{i} + u_{i}, t \geq 0, i \in N, \end{matrix}

Where

(12)

c o [a_{i j} (x_{j} (t))] = {\begin{array}{l} {\overset{⌢}{a}}_{i j}, x_{j} (t) < 0, \\ c o {{\overset{⌢}{a}}_{i j}, {\overset{⌣}{a}}_{i j}}, x_{j} (t) = 0 \\ {\overset{⌣}{a}}_{i j}, x_{j} (t) > 0, \end{array}, c o [a_{i j} (y_{j} (t))] = {\begin{array}{l} {\overset{⌢}{a}}_{i j}, y_{j} (t) < 0, \\ c o {{\overset{⌢}{a}}_{i j}, {\overset{⌣}{a}}_{i j}}, y_{j} (t) = 0 \\ {\overset{⌣}{a}}_{i j}, y_{j} (t) > 0, \end{array}

And

(13)

\begin{matrix} c o [b_{i j} (x_{j} (t - τ (j)))] = {\begin{array}{l} {\overset{⌢}{b}}_{i j}, x_{j} (t - τ (j)) < 0, \\ c o {{\overset{⌢}{b}}_{i j}, {\overset{⌣}{b}}_{i j}}, x_{j} (t - τ (j)) = 0, \\ {\overset{⌣}{b}}_{i j}, x_{j} (t - τ (j)) > 0, \end{array} \\ c o [b_{i j} (y_{j} (t - τ (j)))] = {\begin{array}{l} {\overset{⌢}{b}}_{i j}, y_{j} (t - τ (j)) < 0, \\ c o {{\overset{⌢}{b}}_{i j}, {\overset{⌣}{b}}_{i j}}, y_{j} (t - τ (j)) = 0, \\ {\overset{⌣}{b}}_{i j}, y_{j} (t - τ (j)) > 0, \end{array} \end{matrix}

where co{u, v} denotes the closure of convex hull generated by real numbers u and v or real matrices u and v. Then the synchronization error system can be described as follows: (14) $\begin{matrix} D^{q} e_{i} (t) \in - c_{i} e_{i} (t) + {\sum_{j = 1}^{n} c o [a_{i j} (y_{j} (t))] f_{j} (y_{j} (t)) - \sum_{j = 1}^{n} c o [a_{i j} (x_{j} (t))] f_{j} (x_{j} (t))} \\ + {\sum_{j = 1}^{n} c o [b_{i j} (y_{j} (t - τ_{j} (t)))] g_{j} (y_{j} (t - τ_{j} (t))) \\ - \sum_{j = 1}^{n} c o [b_{i j} (x_{j} (t - τ_{j} (t)))] g_{j} (x_{j} (t - τ_{j} (t)))} + ω_{i} e_{i}, t \geq 0, i \in N . \end{matrix}$

Definition2 [8] For ∀t ≥ 0, the exponential synchronization of system (4) and system (7) can be transformed to the exponential stability of the error system (9) (error approaches to zero). The error system (9) is said to be exponentially stable, if there exist constant Q_i > 0, P_i > 0, such that the solution e(t) = (e₁(t),e₂(t),…,e_n(t))^T of error system (9) with initial condition e(s) = ϕ(s) ∈ ([t₀ − τ, t₀],Rⁿ) satisfies $| e_{i} (t) | \leq Q_{i} \max_{1 \leq i \leq n} {\sup_{t_{0} - τ \leq s \leq t_{0}} | ϕ_{i} (s) |} \exp {- P_{i} (t - t_{0})}, t \geq t_{0} > 0,$ i = 1,2,…,n, where P_i is called the estimated rate of exponential convergence.

Lemma1 [14] Under the assumption1, the following estimation can be obtained:

(i) co[a_ij(y_j(t))]f_j(y_j(t)) − co[a_ij(x_j(t))]f_j(x_j(t)) ≤ A_ijF_j(e_j(t)),

(ii) co[b_ij(y_j(t − τ_j(t)))]g_j(y_j(t − τ_j(t))) − co[b_ij(x_j(t − τ_j(t)))]g_j(x_j(t − τ_j(t))) ≤ B_ijG_j(e_j(t − τ_j(t))),

where $A_{i j} = \max {| {\overset{⌢}{a}}_{i j} |, | {\overset{⌣}{a}}_{i j} |}, B_{i j} = \max {| {\overset{⌢}{b}}_{i j} |, | {\overset{⌣}{b}}_{i j} |}, i, j \in N,$ $F_{j} (e_{j} (t)) = f_{j} (y_{j} (t)) - f_{j} (x_{j} (t)), G_{j} (e_{j} (t - τ_{j} (t))) = g_{j} (y_{j} (t - τ_{j} (t))) - g_{j} (x_{j} (t - τ_{j} (t))), j \in N .$

Proof: If y_i(t) = 0, x_i(t) = 0, i ∈ N we can easily have part(i) hold. From (9) and (10), we can get

(1)
For y_i(t) < 0, x_i(t) < 0, then $\begin{array}{l} c o [a_{i j} (y_{j} (t))] f_{j} (y_{j} (t)) - c o [a_{i j} (x_{j} (t))] f_{j} (x_{j} (t)) = {\overset{⌢}{a}}_{i j} f_{j} (y_{j} (t)) - {\overset{⌣}{a}}_{i j} f_{j} (x_{j} (t) \\ = {\overset{⌢}{a}}_{i j} F_{j} (e_{j} (t)) \leq A_{i j} F_{j} (e_{j} (t)) . \end{array}$
(2)
For y_i(t) > 0, x_i(t) > 0, then $\begin{matrix} c o [a_{i j} (y_{j} (t))] f_{j} (y_{j} (t)) - c o [a_{i j} (x_{j} (t))] f_{j} (x_{j} (t)) = {\overset{⌣}{a}}_{i j} f_{j} (y_{j} (t)) - {\overset{⌢}{a}}_{i j} f_{j} (x_{j} (t) \\ = {\overset{⌣}{a}}_{i j} F_{j} (e_{j} (t)) \leq A_{i j} F_{j} (e_{j} (t)) . \end{matrix}$
(3)
For x_i(t) < 0 < y_i(t) or y_i(t) < 0 < x_i(t), then $\begin{array}{l} c o [a_{i j} (y_{j} (t))] f_{j} (y_{j} (t)) - c o [a_{i j} (x_{j} (t))] f_{j} (x_{j} (t)) \\ = {\overset{⌣}{a}}_{i j} f_{j} (y_{j} (t)) - f (0)) + {\overset{⌢}{a}}_{i j} (f (0) - f_{j} (x_{j} (t))) \\ \leq A_{i j} (f_{j} (y_{j} (t)) - f (0)) + A_{i j} (f (0) - f_{j} (x_{j} (t))) = A_{i j} (f_{j} (y_{j} (t)) - f_{j} (x_{j} (t))) = A_{i j} F_{j} (e_{j} (t)) . \end{array}$

Then complete the proof of part (i). In the similar way, part(ii) can be easily hold.

III.

MAIN RESULTS

We present the exponential stability results for the synchronization error system of FMNN when the error system (9) is exponentially stable, the system (4) and system (7) will achieve the exponential synchronization.

t \geq t_{0} > 0, i \in {1, 2, \dots, n}

Theorem1 If there exist positive constant ε,η₁,η₂,…,η_n such that for any (15) $\begin{matrix} (- c_{i} + ω_{i} + ε) η_{i} + \sum_{j = 1}^{n} A_{i j} σ_{j} η_{j} + \sum_{j = 1}^{n} B_{i j} ρ_{j} η_{j} \exp {ε τ_{j} (t)} < 0, \end{matrix}$ then the error system (9) is globally exponentially stable.

Proof: Consider W_i(t) = |e_i(t)|/η_i,i = 1,2,…,n, according to the error system (9) or (14) and lemma1, we can get the following inequality (16) $\begin{matrix} D^{q} e_{i} (t) \leq - c_{i} e_{i} (t) + \sum_{j = 1}^{n} A_{i j} F_{j} (e_{j} (t)) + \sum_{j = 1}^{n} B_{i j} G_{j} (e_{j} (t - τ_{j} (t))) + ω_{i} e_{i} (t) \\ = (- c_{i} + ω_{i}) e_{i} (t) + \sum_{j = 1}^{n} A_{i j} F_{j} (e_{j} (t)) + \sum_{j = 1}^{n} B_{i j} G_{j} (e_{j} (t - τ_{j} (t))) . \end{matrix}$

Evaluating the fractional order derivative of W_i(t) along the trajectory of error system, then (17) $\begin{matrix} D^{q} W_{i} (t) \leq (- c_{i} + ω_{i}) | e_{i} (t) | / η_{i} + 1 / η_{i} [\sum_{j = 1}^{n} A_{i j} σ_{j} | e_{j} (t) | + \sum_{j = 1}^{n} B_{i j} ρ_{j} | e_{j} (t - τ_{j} (t)) |] \\ = (- c_{i} + ω_{i}) W_{i} (t) + 1 / η_{i} [\sum_{j = 1}^{n} A_{i j} σ_{j} η_{j} W_{j} (t) + \sum_{j = 1}^{n} B_{i j} ρ_{j} η_{j} W_{j} (t - τ_{j} (t)))] . \end{matrix}$

Define ${\tilde{W}}_{i} (t) = W_{i} (t) - \bar{W} (t_{0}) \exp {- ε (t - t_{0})}, t \geq t_{0} > 0, i = 1, 2, \dots, n$ , where $\bar{W} (t_{0}) = \max_{1 \leq i \leq n} {\sup_{t_{0} - τ \leq s \leq t_{0}} | e_{i} (s) | / η_{i}} .$

We will prove that ${\tilde{W}}_{i} (t) \leq 0, i = 1, 2, \dots, n$ , for any t≥t₀>0. Otherwise, since ${\tilde{W}}_{i} (t) \leq 0, i = 1, 2, \dots, n$ for t ∈ [t₀ − τ, t₀], there must exist t₁ ≥ t₀ and some ς such that $D^{q} {\tilde{W}}_{ς} (t_{1}) \geq 0$ and ${\tilde{W}}_{ς} (t_{1}) = 0$ . Then (18) $\begin{matrix} D^{q} {\tilde{W}}_{ς} (t_{1}) \leq (- c_{ς} + ω_{ς}) W_{ς} (t_{1}) + 1 / η_{ς} [\sum_{j = 1}^{n} A_{ς j} σ_{j} η_{j} W_{j} (t_{1}) + \sum_{j = 1}^{n} B_{ς j} ρ_{j} η_{j} W_{j} (t - τ_{j} (t_{1}))] \\ + ε \bar{W} (t_{0}) \exp {- ε (t_{1} - t_{0})} \\ = (- c_{ς} + ω_{ς}) \bar{W} (t_{0}) \exp {- ε (t_{1} - t_{0})} + 1 / η_{ς} [\sum_{j = 1}^{n} A_{ς j} σ_{j} η_{j} \bar{W} (t_{0}) \exp {- ε (t_{1} - t_{0})} \\ + \sum_{j = 1}^{n} B_{ς j} ρ_{j} η_{j} \bar{W} (t_{0}) \exp {- ε (t_{1} - τ_{j} (t_{1}) - t_{0})}] + ε \bar{W} (t_{0}) \exp {- ε (t_{1} - t_{0})} \\ = (- c_{ς} + ω_{ς} + ε) \bar{W} (t_{0}) \exp {- ε (t_{1} - t_{0})} + 1 / η_{ς} [\sum_{j = 1}^{n} A_{ς j} σ_{j} η_{j} \bar{W} (t_{0}) \exp {- ε (t_{1} - t_{0})} \\ + \sum_{j = 1}^{n} B_{ς j} ρ_{j} η_{j} \bar{W} (t_{0}) \exp {- ε (t_{1} - τ_{j} (t_{1}) - t_{0})}] \\ = (- c_{ς} + ω_{ς} + ε) \bar{W} (t_{0}) \exp {- ε (t_{1} - t_{0})} \\ + 1 / η_{ς} [\sum_{j = 1}^{n} A_{ς j} σ_{j} η_{j} \bar{W} (t_{0}) + \sum_{j = 1}^{n} B_{ς j} ρ_{j} η_{j} \bar{W} (t_{0}) \exp {ε τ_{j} (t_{1})}] \exp {- ε (t_{1} - t_{0})} . \end{matrix}$

Moreover, from inequality(15), we have $(- c_{ς} + ω_{ς} + ε) + 1 / η_{i} [\sum_{j = 1}^{n} A_{i j} σ_{j} η_{j} + \sum_{j = 1}^{n} B_{i j} σ_{j} η_{j} \exp {ε τ_{j} (t)}] < 0, t \geq t_{0} > 0, i = 1, 2, \dots, n,$

Therefore (19) $\begin{array}{l} (- c_{i} + ω_{i} + ε) \bar{W} (t_{0}) \exp {- ε (t_{1} - t_{0})} \\ + 1 / η_{i} [\sum_{j = 1}^{n} A_{i j} σ_{j} η_{j} + \sum_{j = 1}^{n} B_{i j} ρ_{j} η_{j} \exp {ε τ_{j} (t)}] \bar{W} (t_{0}) \exp {- ε (t_{1} - t_{0})} < 0, \\ t \geq t_{0} > 0, i = 1, 2, \dots, n, \end{array}$ so it is easy to find that $D^{q} {\tilde{W}}_{ς} (t_{1}) < 0$ , which contradicts $D^{q} {\tilde{W}}_{ς} (t_{1}) \geq 0$ . That shows ${\tilde{W}}_{i} (t) = W_{i} (t) - \bar{W} (t_{0}) \exp {- ε (t - t_{0})} \leq 0, t \geq t_{0} > 0, i = 1, 2, \dots, n$ . Thus $| e_{i} (t) | / η_{i} \leq \max_{1 \leq i \leq n} {\sup_{t_{0} - τ \leq s \leq t_{0}} | e_{i} (s) | / η_{i}} \exp {- ε (t - t_{0})}, t \geq t_{0} > 0, i = 1, 2, \dots, n .$

It shows (20) $| e_{i} (t) | \leq η_{i} \max_{1 \leq i \leq n} {\sup_{t_{0} - τ \leq s \leq t_{0}} | e_{i} (s) | / η_{i}} \exp {- ε (t - t_{0})}, t \geq t_{0} > 0, i = 1, 2, \dots, n .$

This completes the proof.

IV.

NUMERICAL RESULTS

In this section, we will give two numerical examples to demonstrate our analysis on exponential synchronization of FMNN.

Example1 Consider two-dimension fractional-order memristor-based neural networks (21) ${\begin{array}{l} D^{q} x_{1} (t) = - c_{1} x_{1} (t) + a_{11} (x_{1} (t)) f_{1} (x_{1} (t)) + a_{12} (x_{2} (t)) f_{2} (x_{2} (t)) \\ + b_{11} (x_{1} (t - τ_{1} (t))) g_{1} (x_{1} (t - τ_{1} (t))) + b_{12} (x_{2} (t - τ_{2} (t))) g_{2} (x_{2} (t - τ_{2} (t))) + I_{1} \\ D^{q} x_{2} (t) = - c_{2} x_{2} (t) + a_{21} (x_{1} (t)) f_{1} (x_{1} (t)) + a_{22} (x_{2} (t)) f_{2} (x_{2} (t)) \\ + b_{21} (x_{1} (t - τ_{1} (t))) g_{1} (x_{1} (t - τ_{1} (t))) + b_{22} (x_{2} (t - τ_{2} (t))) g_{2} (x_{2} (t - τ_{2} (t))) + I_{2} \end{array}$ where c₁ = c₂ = 1, a₁₁(x₁(t)) = 1, a₂₂(x₂(t)) = 1.8, $a_{12} (x_{2} (t)) = {\begin{array}{l} 12, & x_{2} (t) \leq 0, \\ 14, & x_{2} (t) > 0, \end{array} a_{21} (x_{1} (t)) = {\begin{array}{l} 0.1, & x_{1} (t) \leq 0, \\ 0.05, & x_{1} (t) > 0, \end{array}$ $\begin{array}{l} b_{11} (x_{1} (t - τ_{1} (t))) = {\begin{array}{l} - 1.2, & x_{1} (t - τ_{1} (t)) \leq 0, \\ - 1.5, & x_{1} (t - τ_{1} (t)) > 0, \end{array} b_{12} (x_{2} (t - τ_{2} (t))) = {\begin{array}{l} 0.8, & x_{2} (t - τ_{2} (t)) \leq 0, \\ 1.0, & x_{2} (t - τ_{2} (t)) > 0, \end{array} \\ b_{21} (x_{1} (t - τ_{1} (t))) = {\begin{array}{l} 0.05, & x_{1} (t - τ_{1} (t)) \leq 0, \\ 0.1, & x_{1} (t - τ_{1} (t)) > 0, \end{array} b_{22} (x_{2} (t - τ_{2} (t))) = {\begin{array}{l} - 1.6, & x_{2} (t - τ_{2} (t)) \leq 0, \\ - 1.4, & x_{2} (t - τ_{2} (t)) > 0, \end{array} \end{array}$ where τ_j(t) = e^t/1 + e^t, I = (I₁,I₂)^T = (0,0)^T, q = 0.92 and take the activation function as f_i(x_i) = sin(x_i), g_i(x_i) = 0.5(|x_i + 1| − |x_i − 1|), i, j = 1,2. The model (21) has chaotic attractors with initial values x(0) = (0.45,0.65)^T which can be seen in Figure1.

We consider system (21) as the drive system and corresponding response system is defined as Eq.(7). And for the controller u_i(t) = ω_i(y_i(t) − x_i(t)), the parameter ω_i is chosen as ω₁ = −9.5, ω₂ = −10.5. From Theorem1, when we take ε = 0.7, τ_j(t) = 1, η₁ = η₂ = ρ₁ = ρ₂ = σ₁= σ₂ = 0.1, we can easily know $(- c_{i} + ω_{i} + ε) η_{i} + \sum_{j = 1}^{n} A_{i j} σ_{j} η_{j} + \sum_{j = 1}^{n} B_{i j} ρ_{j} η_{j} \exp {ε τ_{j} (t)} < 0$ ω₁ = −9.5, ω₂ = −10.5, we can get is true when ω₁ < −1.703, ω₂ < −0.232. So when $\begin{array}{l} (- c_{1} + ω_{1} + ε) η_{1} + A_{11} σ_{1} η_{1} + A_{12} σ_{2} η_{2} + B_{11} ρ_{1} η_{1} \exp {ε τ_{1} (t)} + B_{12} ρ_{2} η_{2} \exp {ε τ_{2} (t)} = - 0.798 < 0, \\ (- c_{2} + ω_{2} + ε) η_{2} + A_{21} σ_{1} η_{1} + A_{22} σ_{2} η_{2} + B_{21} ρ_{1} η_{1} \exp {ε τ_{1} (t)} + B_{22} ρ_{2} η_{2} \exp {ε τ_{2} (t)} = - 1.027 < 0. \end{array}$

It satisfies the condition of Theorem 1, then the exponential synchronization of drive-response system is achieved.

When the response system with this controller, we get state trajectories of variable x₁(t), y₁(t) and x₂(t), y₂(t) are depicted in Figure2a and 2b. Moreover, Figure3a and 3b depict the synchronization error curves e₁(t), e₂(t) between the drive system and response system. These numerical simulations show the state trajectories of variable x₁(t), y₁(t) and x₂(t), y₂(t) are synchronous and synchronization error e₁(t), e₂(t) are converge to zero. These prove the correctness of the Theorem1.

The chaotic attractors of fractional-order memristor-based neural networks(18)

Exponential synchronization of state variable with cntroller (a: x₁(t),y₁(t),b : x₂(t),y₂(t))

Synchronization error between the drive and response system (a : e₁(t),b : e₂(t))

Example2 Consider three-dimension fractional-order memristor-based neural networks (22) ${\begin{array}{l} D^{q} x_{1} (t) = - c_{1} x_{1} (t) + a_{11} (x_{1} (t)) f_{1} (x_{1} (t)) + a_{12} (x_{2} (t)) f_{2} (x_{2} (t)) + a_{13} (x_{3} (t)) f_{3} (x_{3} (t)) \\ + b_{11} (x_{1} (t - τ_{1} (t))) g_{1} (x_{1} (t - τ_{1} (t))) + b_{12} (x_{2} (t - τ_{2} (t))) g_{2} (x_{2} (t - τ_{2} (t))) \\ + b_{13} (x_{3} (t - τ_{3} (t))) g_{3} (x_{3} (t - τ_{3} (t))) + I_{1} \\ D^{q} x_{2} (t) = - c_{2} x_{2} (t) + a_{21} (x_{1} (t)) f_{1} (x_{1} (t)) + a_{22} (x_{2} (t)) f_{2} (x_{2} (t)) + a_{23} (x_{3} (t)) f_{3} (x_{3} (t)) \\ + b_{21} (x_{1} (t - τ_{1} (t))) g_{1} (x_{1} (t - τ_{1} (t))) + b_{22} (x_{2} (t - τ_{2} (t))) g_{2} (x_{2} (t - τ_{2} (t))) \\ + b_{23} (x_{3} (t - τ_{3} (t))) g_{3} (x_{3} (t - τ_{3} (t))) + I_{2} \\ D^{q} x_{3} (t) = - c_{3} x_{3} (t) + a_{31} (x_{1} (t)) f_{1} (x_{1} (t)) + a_{32} (x_{2} (t)) f_{2} (x_{2} (t)) + a_{33} (x_{3} (t)) f_{3} (x_{3} (t)) \\ + b_{31} (x_{1} (t - τ_{1} (t))) g_{1} (x_{1} (t - τ_{1} (t))) + b_{32} (x_{2} (t - τ_{2} (t))) g_{2} (x_{2} (t - τ_{2} (t))) \\ + b_{33} (x_{3} (t - τ_{3} (t))) g_{3} (x_{3} (t - τ_{3} (t))) + I_{3} \end{array}$ where c₁ = c₂ = c₃ = 1, $\begin{array}{l} a_{11} (x_{1} (t)) = {\begin{array}{l} 1, & x_{1} (t) \leq 0, \\ - 1 & x_{1} (t) > 0, \end{array} a_{21} (x_{1} (t)) = {\begin{array}{l} 1, & x_{1} (t) \leq 0, \\ - 1 & x_{1} (t) > 0, \end{array} a_{31} (x_{1} (t)) = {\begin{array}{l} 1, & x_{1} (t) \leq 0, \\ - 1 & x_{1} (t) > 0, \end{array} \\ a_{12} (x_{2} (t)) = {\begin{array}{l} 1, & x_{2} (t) \leq 0, \\ - 1 & x_{2} (t) > 0, \end{array} a_{22} (x_{2} (t)) = {\begin{array}{l} 1, & x_{2} (t) \leq 0, \\ - 1 & x_{2} (t) > 0, \end{array} a_{32} (x_{2} (t)) = {\begin{array}{l} 1, & x_{2} (t) \leq 0, \\ - 1 & x_{2} (t) > 0, \end{array} \\ a_{13} (x_{3} (t)) = {\begin{array}{l} 1, & x_{3} (t) \leq 0, \\ - 1 & x_{3} (t) > 0, \end{array} a_{23} (x_{3} (t)) = {\begin{array}{l} 1, & x_{3} (t) \leq 0, \\ - 1 & x_{3} (t) > 0, \end{array} a_{33} (x_{3} (t)) = {\begin{array}{l} 1, & x_{3} (t) \leq 0, \\ - 1 & x_{3} (t) > 0, \end{array} \end{array}$ $\begin{array}{l} b_{11} (x_{1} (t - τ_{1} (t))) = {\begin{array}{l} 1, & x_{1} (t - τ_{1} (t)) \leq 0, \\ - 1 & x_{1} (t - τ_{1} (t)) > 0, \end{array} b_{12} (x_{2} (t - τ_{2} (t))) = {\begin{array}{l} 1, & x_{2} (t - τ_{2} (t)) \leq 0, \\ - 1 & x_{2} (t - τ_{2} (t)) > 0, \end{array} \\ b_{21} (x_{1} (t - τ_{1} (t))) = {\begin{array}{l} 1, & x_{1} (t - τ_{1} (t)) \leq 0, \\ - 1 & x_{1} (t - τ_{1} (t)) > 0, \end{array} b_{22} (x_{2} (t - τ_{2} (t))) = {\begin{array}{l} 1, & x_{2} (t - τ_{2} (t)) \leq 0, \\ - 1 & x_{2} (t - τ_{2} (t)) > 0, \end{array} \\ b_{31} (x_{2} (t - τ_{2} (t))) = {\begin{array}{l} 1, & x_{2} (t - τ_{2} (t)) \leq 0, \\ - 1 & x_{2} (t - τ_{2} (t)) > 0, \end{array} b_{32} (x_{2} (t - τ_{2} (t))) = {\begin{array}{l} 1, & x_{2} (t - τ_{2} (t)) \leq 0, \\ - 1 & x_{2} (t - τ_{2} (t)) > 0, \end{array} \\ b_{13} (x_{3} (t - τ_{3} (t))) = {\begin{array}{l} 1, & x_{3} (t - τ_{3} (t)) \leq 0, \\ - 1 & x_{3} (t - τ_{3} (t)) > 0, \end{array} \\ b_{23} (x_{3} (t - τ_{3} (t))) = {\begin{array}{l} 1, & x_{3} (t - τ_{3} (t)) \leq 0, \\ - 1 & x_{3} (t - τ_{3} (t)) > 0, \end{array} \\ b_{33} (x_{3} (t - τ_{3} (t))) = {\begin{array}{l} 1, & x_{3} (t - τ_{3} (t)) \leq 0, \\ - 1 & x_{3} (t - τ_{3} (t)) > 0. \end{array} \end{array}$

And τ_j(t) = e^t/1 + e^t, I = (I₁,I₂,I₃)^T = (0,0,0)^Tq = 0.92 and take the activation function as f_i(x_i) = g_i(x_i) = tanh(x_i),i = 1,2,3. We consider system(22) as the drive system and the corresponding response system is defined in Eq.(7). And for the controller u_i(^t) = ω_i(y_i(t) − x_i(t)), ω_i is chosen as ω₁ = −9.5, ω₂ = −10.5, ω₃ = −11. From Theorem1, we take ε = 0.7, τ_j(t) = 1 and choose η₁ = η₂ = 0.1 ρ₁ = ρ₂ = σ₁ = σ₂ = 0.1. According to $A_{i j} = \max {| {\overset{⌢}{a}}_{i j} |, | {\overset{⌣}{a}}_{i j} |}, B_{i j} = \max {| {\overset{⌢}{b}}_{i j} |, | {\overset{⌣}{b}}_{i j} |}$ i,j = 1,2,3 A_ij = B_ij = 1, we can easily know $(- c_{i} + ω_{i} + ε) η_{i} + \sum_{j = 1}^{n} A_{i j} σ_{j} η_{j} + \sum_{j = 1}^{n} B_{i j} ρ_{j} η_{j} \exp {ε τ_{j} (t)} < 0$ is true when ω_i < −0.604. So when ω₁ = −9.5, ω₂ = −10.5, ω₃ = −11 we can get $\begin{array}{l} (- c_{1} + ω_{1} + ε) η_{1} + A_{11} σ_{1} η_{1} + A_{12} σ_{2} η_{2} + A_{13} σ_{3} η_{3} + (B_{11} ρ_{1} η_{1} + B_{12} ρ_{2} η_{2} + B_{13} ρ_{3} η_{3}) \exp {ε τ_{j} (t)} = - 0.89 < 0, \\ (- c_{2} + ω_{2} + ε) η_{2} + A_{21} σ_{1} η_{1} + A_{22} σ_{2} η_{2} + A_{23} σ_{3} η_{3} + (B_{21} ρ_{1} η_{1} + B_{22} ρ_{2} η_{2} + B_{23} ρ_{3} η_{3}) \exp {ε τ_{j} (t)} = - 0.99 < 0, \\ (- c_{3} + ω_{3} + ε) η_{1} + A_{31} σ_{1} η_{1} + A_{32} σ_{2} η_{2} + A_{33} σ_{3} η_{3} + (B_{31} ρ_{1} η_{1} + B_{32} ρ_{2} η_{2} + B_{33} ρ_{3} η_{3}) \exp {ε τ_{j} (t)} = - 1.04 < 0. \end{array}$

It suggests the condition of Theorem 1 is satisfied, then drive-response system achieves the synchronization.

When the response system with this controller, we get state trajectories of variable x₁(t), y₁(t) and x₂(t), y₂(t) and x₃(t), y₃(t) are depicted in Figure 4a,4b,4c. Moreover, Figure 5a,5b,5c depict the synchronization error curves e₁(t), e₂(t), e₃(t) between the drive system and response system. It’s easy to see that the state trajectories of variable x₁(t), y₁(t), x₂(t), y₂(t), and x₃(t), y₃(t) are synchronous and synchronization error e₁(t), e₂(t), e₃(t) are converge to zero. So the Theorem1 is proved to be correct.

In addition, we choose ω₁ = −9.5, ω₂ = −10.5, ω₃ = −11, according to the Theorem1, it needs the following inequalities to hold: ${\begin{cases} τ < \frac{1}{ε} \ln (\frac{117}{3} - \frac{10}{3} ε) \\ τ < \frac{1}{ε} \ln (\frac{112}{3} - \frac{10}{3} ε) \\ τ < \frac{1}{ε} \ln (\frac{102}{3} - \frac{10}{3} ε) \end{cases}$

So, we just need $τ < \frac{1}{ε} \ln (\frac{102}{3} - \frac{10}{3} ε)$ holds. We have the exponential convergence rate 0 < ε < 1, figure 6 depicts the relation of time-varying delay τ and exponential convergence rate ε.

Synchronization of state variable with controller (a:x₁(t),y₁(t),b:x₂(t),y₂(t),c:x₃(t),y₃(t))

Synchronization error between the drive and response system (a:e₁(t),b:e₂(t),c:e₃(t))

The relation of time-varying delay τ and exponential convergence rate ε.

V.

CONCLUSION

This paper achieves the exponential synchronization of a class of FMNN with time-varying delays by using linear error feedback controller. Based on comparison principle, the new theorem is derived to guarantee the exponential synchronization between the drive system and response system. The methods proposed for synchronization is effective and it is easy to achieve than other complex control methods. Moreover, it can be extended to investigate other dynamical behaviors of fractional-order memristive neural networks, such as realizing the lag synchronization or anti-synchronizaton of this system based on the suitable controller. These issues will be the topic of future research. Finally, numerical examples are given to illustrate the effectiveness of the proposed theory.

Exponential Synchronization for Fractional-order Time-delayed Memristive Neural Networks

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