Mittag-Leffler stabilization of anti-periodic solutions for fractional-order neural networks with time-varying delays

1.   Introduction

The stabilization and existence of anti-periodic solutions have major significance in dynamic behavior on nonlinear differential equations, which plays a key role in various physical phenomena, such as anti-periodic characteristics in vibration equations and so on ^[1,2,3,4,5]. As a special case of periodic solutions, many scholars have studied the existence and stabilization of anti-periodic solutions of several kinds of neural networks in recent years. The authors ^[6] studied the existence and stabilization of anti-periodic solutions for BAM Cohen-Grossberg neural networks. In ^[7] authors investigated the existence and global exponential stabilization of anti-periodic solutions for quaternion numerical cellular neural networks with impulse effect. The existence and exponential stabilization of anti-periodic solutions for BAM neural networks is studied in ^[8,9]. The authors ^[10] studied the global exponential stabilization of anti-periodic solutions for Cohen-Grossberg neural networks. All studies in ^[6,7,8,9,10] are integer-order models, however, the research on fractional-order neural networks has attracted attention and obtained important research results in recent years.

The existence and stabilization of anti-periodic solutions are of great significance in the dynamic behavior of nonlinear differential equations, such as ^[1,2,3,4,5]. From previous data, there are only discussions on the asymptotic $\omega$-periodic solution, almost periodic solutions and $s$-asymptotic $\omega$-periodic solutions for fractional-order neural networks (e.g., ^{[11,12,13,14,15]}), we haven't found the existence and stabilization of anti-periodic solutions yet. We focus on the problem of the existence of anti-periodic solutions and Mittag-Leffler stabilization for a class of fractional order neural networks in this paper, this is a new research topic, our characteristics mainly include three points:

1) Deriving the relationship between fractional-order integrals of state functions with and without time delay through the division of time interval and the properties of fractional-order calculus;

2) Constructing function sequence solution, and it uniformly converges to a continuous function with Arzela-Asoli theorem, then giving a sufficiency for the existence of anti-periodic solutions and Mittag-Leffler stabilization of the system, the results are new;

3) Verifying the correctness of the theorems by numerical simulation instances. It provides a new criterion for dynamic system research.

We consider fractional-order neural networks with time-varying delays:

Where $t \geqslant 0$, $D_t^\alpha$ is Riemann-Liouville derivative with $\alpha$-order, $0 < \alpha < 1$; ${x_i}(t)$ is the state of the $i$th neuron at time $t$; ${\beta _i} > 0$; ${a_{ij}}, {b_{ij}}$ are connection weights of neurons; ${f_j}(\cdot)$ is an excitation function of the $j$th neuron; ${I_i}(t)$ is an external input function of the $i$th neuron at time $t$; ${\tau _{ij}}(t)$ is a signal transmission delay between the $i$th neuron and the $j$th neuron, and ${\tau _{ij}}(t) > 0$.

Given the initial conditions of the system (1):

Here $\tau {\text{ = }}\mathop {\sup }\limits_{1 \leqslant i, j \leqslant n{\text{, }}t > 0} {\text{\{ }}{\tau _{ij}}{\text{(}}t{\text{)\} }}$, ${\varphi _i}(s), {\psi _i}(s)$ are bounded continuous functions.

The structure of this article is as follow. First a few preliminaries are given in Section 2. In Section 3, by the properties of fractional-order calculus, constructing function sequence solution, and the Arzela-Asoli theorem, a sufficient case is derived for the existence of anti-periodic solutions and Mittag-Leffler stabilization of the system. An illustrative example to show the effectiveness of the proposed theory in Section 4.

2.   Preliminaries

Definition 1. ^[16] Define the $q$-order fractional-order integral of $f(t)$ (Riemann-Liouville integral) as

where $t \geqslant {t_0} \geqslant 0$, $q$ is a positive real number, $\Gamma (\cdot)$ is a Gamma function, and $\Gamma (r) = \int_0^{ + \infty } {{t^{r - 1}}{e^{ - t}}{\text{d}}t}, r > 0.$

Definition 2. ^[16] Define the $q$-order fractional-order derivative of $f(t)$ (Riemann-Liourille derivative) as

where $t \geqslant {t_0} \geqslant 0$, $n - 1 \leqslant q < n, n \in {Z^ + }$, $\Gamma (\cdot)$ is a Gamma function.

Definition 3. ^[17] A Mittag-Leffler function with parameter $q$ is defined

where $Re(q) > 0$ is the real part of $q$, $z$ is plural, $\Gamma (\cdot)$ is a Gamma function.

Definition 4. Let ${X^{\text{T}}}(t)$ and ${\bar X^{\text{T}}}(t)$ are the solutions of ${x_i}(s) = {\varphi _i}(s), D_t^\alpha {x_i}(s) = {\psi _i}(s)$ and ${\bar x_i}(s) = {\bar \varphi _i}(s)$, $D_t^\alpha {\bar x_i}(s) = {\bar \psi _i}(s), {\text{ }} - \tau \leqslant s \leqslant 0$. If there exist ${\rho _1} > 0$, ${\rho _2} > 0$, ${\bar X^{\text{T}}}(t)$ and ${X^{\text{T}}}(t)$ satisfy

then the system (1) is Mittag-Leffler stabilization, where $X(t) = {({x_1}(t), {x_2}(t), \cdots {x_n}(t))^{\text{T}}}, {\text{ }}\bar X(t) = {({\bar x_1}(t), {\bar x_2}(t), \cdots {\bar x_n}(t))^{\text{T}}}, {\text{ }}\varphi (t) = {({\varphi _1}(t), {\varphi _2}(t), \cdots {\varphi _n}(t))^{\text{T}}}$, $\bar \varphi (t) = {({\bar \varphi _1}(t), {\bar \varphi _2}(t), \cdots {\bar \varphi _n}(t))^{\text{T}}}, {\text{ }}M(\varphi - \bar \varphi) \geqslant 0, M(0) = 0.$ ${E_q}(\cdot)$ is a Mittag-Leffler function with a parameter $q$.

Lemma 1. ^[18] $x(t)$ is a continuously differentiable function on $[0, \delta](\delta > 0)$, then $D_{^t}^{ - p}D_{^t}^qx(t) = D_{^t}^{ - p + q}x(t), 0 < q < 1, n - 1 \leqslant p < n, n \in {Z^ + }.$

Lemma 2. ^[19] $u(t)$ is a continuous function on $[0, + \infty)$, there exists ${d_1} > 0$ and ${d_2} > 0$, such that $u(t) \leqslant - {d_1}D_t^{ - q}u(t) + {d_2}, {\text{ }}t \geqslant 0,$ then $u(t) \leqslant {d_2}{E_q}(- {d_1}{t^q})$, where $0 < q < 1$, ${E_q}(\cdot)$ is a Mittag-Leffler function with a parameter $q$.

Lemma 3. ^[18] If $r(t)$ is differentiable and $r'(t)$ is continuous, thus $\frac{1}{2}D_t^q{r^2}(t) \leqslant r(t)D_t^qr(t), {\text{ 0 < }}q \leqslant 1.$

Definition 5. ^[20] For $u(t) \in C(R)$, if $u(t + \omega) = - u(t)$ for $t \in R$, thus $u(t)$ is an anti-periodic function, where $\omega$ is a normal number.

Assumptions used in this article:

${{\text{H}}_{\text{1}}}:$${f_i}(t)$ is bounded continuous excitation function and satisfies Lipschitz conditions, there exist

${l_i} > 0$, ${\bar f_i} > 0$ such $\left| {{f_i}({\xi _1}) - {f_i}({\xi _2})} \right| \leqslant {l_i}\left| {{\xi _1} - {\xi _2}} \right|, {\text{ }}\left| {{f_i}(t)} \right| \leqslant {\bar f_i}, {\text{ }}{\xi _1}, {\xi _2} \in R, {\text{ }}i = 1, 2, \cdots n.$

${{\text{H}}_2}:$ Excitation function ${f_i}(t)$ satisfies ${f_i}(u) = - {f_i}(- u), {\text{ }}u \in R, {\text{ }}i = 1, 2, \cdots, n.$

${{\text{H}}_3}:$ Input function ${I_i}(t)$ satisfies ${I_i}(t + \omega) = - {I_i}(t), {\text{ }}\left| {{I_i}(t)} \right| \leqslant {\bar I_i}$, where $\omega > 0, {\text{ }}{\bar I_i} \geqslant 0, {\text{ }}i = 1, 2, \cdots, n.$

${{\text{H}}_4}:$ Time-varying delays function ${\tau _{ij}}(t)$ is bounded, and differentiable, and satisfy $0 \leqslant {\dot \tau _{ij}}(t) \leqslant {\tau ^*} < 1, {\text{ }}t > 0$, ${\text{ }}i = 1, 2, \cdots, n$.

3.   Main results

Theorem 1. The solution of system (1) is bounded on $[0, T](0 \leqslant T < + \infty)$ when ${{\text{H}}_{\text{1}}}$ and ${{\text{H}}_3}$ hold.

Proof. There is $D_t^\alpha \left| {g(x)} \right| \leqslant {sgn} (g(x))D_t^\alpha g(x)$ for a continuous function $g(x)$ and Definition 2. We get from (1):

Combined with Lemma 1, it can be deduced from (3):

From Lemma 2:

That is the solution $x(t) = {({x_1}(t), {x_2}(t), \cdots, {x_n}(t))^{\text{T}}}$ is bounded on $0 \leqslant t \leqslant T < + \infty$, where ${E_\alpha }(\cdot)$ is a

Mittag-Leffler function with a parameter $\alpha$.

Theorem 2. The solution of system (1) is Mittag-Leffler stabilization on $[0, T](T < + \infty)$, if

when ${{\text{H}}_{\text{1}}}$ and ${{\text{H}}_3}$ hold.

Proof. Suppose ${x^*}(t) = {(x_1^*(t), x_2^*(t), \cdots, x_n^*(t))^{\text{T}}}$ and $x(t) = {({x_1}(t), {x_2}(t), \cdots, {x_n}(t))^{\text{T}}}$ are the solutions of $x_i^*(s) = \varphi _i^*(s), {\text{ }}D_t^\alpha x_i^*(s) = \psi _i^*(s)$ and ${x_i}(s) = {\varphi _i}(s), {\text{ }}D_t^\alpha {x_i}(s) = {\psi _i}(s)$. Let ${y_i}(t) = {x_i}(t) - x_i^*(t)$, combined formula (1):

We get $D_t^\alpha y_i^2(t) \leqslant 2{y_i}(t)D_t^\alpha {y_i}(t)$ from Lemma 3, and from (4):

From (5):

$t - {\tau _{ij}}(t) \in [- {\tau _{ij}}(t), 0]$ when $t \in [0, {\tau _{ij}}(t)]$. Let $u = s - {\tau _{ij}}(s)$, then

where $\varphi _i^* = \mathop {\sup }\limits_{ - \tau \leqslant s \leqslant 0} \{ {(\varphi _i^*(s) - {\varphi _i}(s))^2}\}, {\text{ }}i = 1, 2, \cdots, n.$

$t - {\tau _{ij}}(t) \in [0, + \infty)$ when $t \in [{\tau _{ij}}(t), + \infty)$. Let $u = s - {\tau _{ij}}(s)$, then

where $\varphi _i^* = \mathop {\sup }\limits_{ - \tau \leqslant s \leqslant 0} \{ {(\varphi _i^*(s) - {\varphi _i}(s))^2}\}, {\text{ }}i = 1, 2, \cdots, n.$

We obtain from (7) and (8):

Substitute the result of (9) into (6):

Combined with Lemma 2, it can be deduced from (10):

where $M(\varphi - {\varphi ^*}) = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {\left| {{b_{ji}}} \right|} } {l_i}\frac{{\varphi _i^*{T^\alpha }}}{{(1 - {\tau ^*})\Gamma (\alpha + 1)}}$, $\eta = \mathop {\min }\limits_{1 \leqslant i \leqslant n} \{ 2{\beta _i} - \sum\limits_{j = 1}^n {(\left| {{a_{ij}}} \right| + \left| {{b_{ij}}} \right|){l_j} - \sum\limits_{j = 1}^n {(\left| {{a_{ji}}} \right| + \frac{{\left| {{b_{ji}}} \right|}}{{1 - \tau *}}){l_i}} } \} > 0.$ Obviously $M(\varphi - \varphi *) \geqslant 0, {\text{ }}$ and $M(0) = 0$, thus the solution of system (1) is Mittag-Leffler stabilization from Definition 4.

Theorem 3. System (1) has an anti-periodic solution when the Theorem 2 and ${{\text{H}}_2}$ hold, and the solution is Mittag-Leffler stabilization.

Proof. For a positive integer $k$ and a normal number $\omega$ from ${{\text{H}}_2}$ and ${{\text{H}}_3}$, we obtain from (1):

So ${(- 1)^{k + 1}}{x_i}(t + (k + 1)\omega)$ is the solution of system (1) for a positive integer $k$. $x(t)$ is bounded from Theorem 1, then there exists a positive constant $N$ such that:

Because $0 \leqslant {E_\alpha }(- {(\lambda t)^\alpha }) \leqslant 1, \lambda > 0$, so the sequence $\{ {(- 1)^{k + 1}}{x_i}(t + k\omega)\}$ is equicontinuous and bounded uniformly. Reapplication Arzela-Ascoli theorem ${\{ {(- 1)^k}{x_i}(t + k\omega)\} _{k \in N}}$ converges to a continuous function $x_i^*(t)$ uniformly on any compact set in $[0, + \infty]$ by selecting a subsequence ${\{ k\omega \} _{k \in N}}$, that is $\mathop {\lim }\limits_{k \to + \infty } {(- 1)^k}{x_i}(t + k\omega) = x_i^*(t), {\text{ }}i = 1, 2, \cdots, n.$

On the other hand, owing to $x_i^*(t + \omega) = \mathop {\lim }\limits_{k \to + \infty } {(- 1)^k}{x_i}(t + \omega + k\omega) = - \mathop {\lim }\limits_{k \to + \infty } {(- 1)^{k + 1}}{x_i}(t + (k + 1)\omega) = - x_i^*(t), {\text{ }}i = 1, 2, \cdots, n.$

So ${x^*}(t)$ is $\omega$-anti-periodic function. Owing ${(- 1)^k}{x_i}(t + k\omega)$ is the solution of system (1) for any $k \in N$, we obtain from (1):

We can continue to get when ${f_i}(\cdot)$ is continuous, then

So ${x^*}(t)$ is an anti-periodic solution of system (1). For any $x(t)$, the inequality holds from (11):

so ${x^*}(t)$ is an anti-periodic solution and Mittag-Leffler stabilization.

Remark: The stabilization and existence of anti-periodic solutions of nonlinear differential equations are of great significance in dynamic behavior, which plays a key role in physical phenomena ^[1,2,3,4,5]. The model of integer- order neural network system is a nonlinear differential equation, and fractional-order neural network system is a generalization of integer-order neural network system, so fractional-order neural network system is also a model of nonlinear differential equation generalization. From previous data, there are only discussions on the boundedness and asymptotic stabilization of almost periodic solution and $\omega$-periodic solution for fractional-order neural networks (e.g., ^{[11,12,13,14,15]}), we have not seen the results of authors exploring the dynamic behavior of the anti-periodic solution of a system. In the article, we mainly give the sufficient conditions for the existence of anti-periodic solutions and Mittag-Leffler stabilization of fractional order neural network systems. The results are new. This provides a new basis to further explore the dynamic properties of a system in theoretical research and practical application.

4.   Numerical simulation

We consider fractional-order neural networks with time-varying delays:

We get $\alpha = 0.85$, ${\beta _1} = 1.85, {\text{ }}{\beta _2} = 1.9$, ${a_{11}} = \frac{1}{{16}},$ ${a_{12}} = \frac{1}{{32}},$ ${a_{21}} = - \frac{1}{{16}},$ ${a_{22}} = - \frac{1}{{32}},$ ${b_{11}} = - \frac{1}{{16}},$ ${b_{12}} = \frac{1}{{32}},$${b_{21}} = \frac{1}{{16}},$${b_{22}} = - \frac{1}{{32}},$ ${f_i}(x) = \frac{{\left| {{x_i} + 1} \right| - \left| {{x_i} - 1} \right|}}{{50}},$ ${I_i}(t) = \frac{{\cos (8t)}}{{150}},$ ${\tau _{ij}}(t) = \frac{{2 - {e^{ - t}}}}{3},$ therefore ${I_i}(t + \frac{\pi }{8}) = - {I_i}(t), {\text{ }}$${f_i}(- x) = - \frac{{\left| {{x_i} + 1} \right| - \left| {{x_i} - 1} \right|}}{{50}} = - {f_i}(x), {\text{ }}i = 1, 2.$

Let ${l_i} = \frac{1}{{25}},$${\bar f_i} = \frac{1}{{25}}$, ${\bar I_i} = \frac{1}{{150}}$, ${\tau ^*} = \frac{1}{3}$, $\omega = \frac{\pi }{8}$.

By calculating we have: $\eta = \mathop {\min }\limits_{1 \leqslant i \leqslant 2} \{ 2{\beta _i} - \sum\limits_{j = 1}^2 {(\left| {{a_{ij}}} \right| + \left| {{b_{ij}}} \right|){\text{ }}} {l_j} - \sum\limits_{j = 1}^2 {(\left| {{a_{ji}}} \right| + \frac{{\left| {{b_{ji}}} \right|}}{{1 - {\tau ^*}}})} {\text{ }}{l_i}\} = 2.8884375 > 0$, so Theorem 3 holds, the system (13) has a $\frac{\pi }{8}$-anti-periodic solution with Mittag-Leffler stabilization.

On the other hand, giving the transient change of $({x_1}(t), {y_1}(t))$ and $({x_2}(t), {y_2}(t))$ for system (13) by numerical simulation, as shown in the figures below (see Figures 1 and 2).

We gain a $\frac{\pi }{8}$-anti-periodic solution from the figures, it is consistent with the conclusion of theorems.

5.   Conclusions

We study the dynamic behavior of fractional-order neural networks with time-varying delays in the article. Frist deriving the relationship between fractional-order integrals of state functions with and without time delay through the division of time interval and the properties of fractional-order calculus, the research method is innovative. Moreover, constructing the sequence solution of the system function which converges to a continuous function uniformly with the Arzela-Asoli theorem. In addition, giving the sufficient conditions the Mittag-Leffler stabilization, boundedness, and the existence of anti-periodic solutions for systems. Finally, the conclusion is feasible by a numerical simulation. Similarly, we can use the theoretical basis of this article to study the Mittag-Leffler stabilization of anti-periodic solutions of fractional-order Cohen-Grossberg neural networks and inertial Cohen-Grossberg neural networks, and so on.

Acknowledgments

Funding: Scientific Research Project of Shaoxing University Yuanpei College (No. KY2021C04).

Conflict of interest

The authors declare no conflict of interest.