EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 4, 2020, 794-806
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Robust Exponential Stability of Recurrent Neural
Networks with Deviating Argument and Stochastic

Disturbance

Zha Mingxin1, Si Wenxiao1, Xie Tao1,∗

1 Hubei Normal University, College of Mathematics and Statistics, Huangshi, China

Abstract. It is well known that deviating argument and stochastic disturbance may derail the
stability of recurrent neural networks (RNNs). This paper discusses the robustness of global expo-
nential stability (GES) of RNNs accompanied with deviating argument and stochastic disturbance.
For a given global exponentially stable RNNs, it is interesting to know how much the length of the
interval of piecewise function and the interference intensity so that the disturbed system may still
be exponentially stable. The available upper boundary of the range of piecewise variables and the
interference intensity in the disturbed RNNs to keep GES are the solutions of some transcenden-
tal equations. Finally, some examples are provided to demonstrate the efficacy of the inferential
results.

2020 Mathematics Subject Classifications: 34A34, 34A50, 58A10, 93C10

Key Words and Phrases: Robustness, recurrent neural networks, deviating argument, stochastic
disturbance

1. Introduction

As a kind of nonlinear dynamic system, the stability of recurrent neural networks
(RNNs) mainly rely on its parameter allocation [5, 7, 18, 27]. It is known that arbitrary
perturbations and various time delays in the process of neuron activation may led to
instability or oscillation of RNNs [3, 4, 12, 14].

Piecewise argument, unifying the advanced and hysteretic arguments [13, 21, 26], is
one of the nonlinear non-smooth actuators that play an important role in the operation
of a nonlinear system [2, 22–25]. For example, in order to describe the stationary state of
the wire length temperature, the nonlinear dynamic model with deviation parameters may
be used for better fitting. Stochastic disturbance, which is hardly avoided in some real
applications considered in some nonlinear systems, nontrivially generalizes the classical

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i4.3848

Email addresses: xietao1294@sina.com (Xie Tao),
chamingxin@163.com (Z. Mingxin), 2673430651@qq.com (S. Wenxiao),

https://www.ejpam.com 794 c© 2020 EJPAM All rights reserved.



Z. Mingxin, X. Tao, S. Wenxiao / Eur. J. Pure Appl. Math, 13 (4) (2020), 794-806 795

deterministic process [6, 9–11, 15]. In terms of stability analysis, there are a variety
of unique stability theories and research results, which mainly include robust analysis,
dissipative analysis and impulse control. In addition, discrete time delay and fuzzy systems
have also attracted much attention [1, 8, 19, 21]. For a stable RNNs, it is significant
to describe how much the length of the interval of piecewise argument and interference
intensity of the perturbed RNNs can withstand without losing stability.

This paper characterizes the robustness of RNNs with piecewise argument and arbi-
trary disturbance. For globally exponential stable RNNs, the available upper boundary of
the range of deflection arguments and the interference intensity in the perturbed RNNs to
preserve globally exponential stability are the solutions of some transcendental equations.
Roughly speaking, the contributions of this paper include: (i) The relationship between
piecewise deviation variables and system solutions in recurrent neural networks is con-
structed; (ii) Some mathematical inequalities are used to enlarge the upper bound of the
interval of piecewise arguments and the upper bound of random disturbances; (iii) Devel-
oping some approaches to analyze recurrent neural networks in the piecewise of deviating
arguments.

The structure of the paper is outlined as follows. Section 2 gives preliminaries and
model description. Main results are presented in Section 3. Several illustrative examples
are given in Section 4. Some concluding remarks are presented in Section 5.

2. Preliminaries

Throughout this paper, let < be the set of real numbers and <+ be the set of positive
real numbers. <n denotes the n-dimensional vectors space over < and <n×m be the set
of n ×m matrices over <, where n,m are included in the set N of nature numbers. The
Euclidean norm in <n and <n×m are expressed by the same symbol ‖·‖. Specifically, for an
n-dimensional vector v = (v1, v2, . . . , vn)T ∈ <n (T denotes the transpose of the vector),
‖v‖ = (

∑n
i=1 v

2
i )

1/2, and for a matrix A ∈ <n×n, ‖A‖ = sup{‖Ay‖
∣∣ ‖y‖ = 1, y ∈ <n}.

Moreover, E(·) is represented as mathematical expectation.
In this paper, we consider the following recurrent neural network with deviating argu-

ment and stochastic disturbance:{
dy(t) = [−Ay(t) +Bf(y(t)) +Df(y(β(t)))]dt+ σy(t)dω(t), t ≥ t0 ≥ 0,

y(t0) = y0 ∈ <n.
(1)

where β(t) is called deviating argument which is a piecewise function defined as β(t) = ηk,
if t ∈ [αk, αk+1), k ∈ N, the real-value sequences {αk}, {ηk}, k ∈ N satisfy αk < αk+1,
αk ≤ ηk ≤ αk+1 and αk+1 − αk ≤ α for arbitrary k ∈ N with αk → +∞ as k → +∞
and α > 0. y(t) = (y1(t), . . . , yn(t))T ∈ <n is the state vector of the system (1). A =
diag{a1, . . . , an} ∈ <n×n is the self-feedback connection weight matrix, B ∈ <n×n is the
connection matrix, f(y(t)) = (f1(y1(t)), . . ., fn(yn(t)))T ∈ <n represents an activation
function. D ∈ <n×n is the connection matrix related with the deviating function β(t). σ
denotes the intensity of the stochastic disturbance and ω(t) represents Brownian motion



Z. Mingxin, X. Tao, S. Wenxiao / Eur. J. Pure Appl. Math, 13 (4) (2020), 794-806 796

on a complete probability space (Ω, F, P ) with a natural filtration {Ft}t≥t0≥0 generated
by {ω(s) : t0 ≤ s ≤ t}. It can be referred to [15] for the basic knowledge of stochastic
differential equations.

Without the deviating argument and stochastic disturbance, then the model (1) de-
grades into the following classical recurrent neural networks:{

ẋ(t) = −Ax(t) +Bf(x(t)), t ≥ t0 ≥ 0,

x(t0) = x0 ∈ <n.
(2)

Remark 1. The concept of deviating argument was first introduced in [17] to describe
some alternately advanced and retarded systems. Actually, if the system (1) is defined on
the interval [αk, αk+1), k ∈ N, if αk ≤ t < ηk, then (1) is an advanced system, and if
ηk < t < αk+1, (1) is a retarded system.

In order to state the main results, we need the following assumption:
(A1) There exist a nonnegative constant k ∈ < such that for arbitrary u, v ∈ <

‖f(u)− f(v)‖ ≤ k‖u− v‖, (3)

and f(0) = 0.
Under the assumption (A1), it is clear that x = 0 is a trivial state of system (2) and

y = 0 is also a trivial state of system (1). Moreover, the system (2) has a unique state
x(t; t0, x0) for any t0 and x0, and (1) also has a unique solution y(t; t0, y0) satisfying initial
conditions [16].

The exponential stability of system (1) is defined as follows.

Definition 1. ([16]) The state vector x(t; t0, y0) in (2) is globally exponentially stable, if
for arbitrary t0 ∈ <+ and x0 ∈ <+, there exist u, v ∈ <+, such that the inequality

‖x(t; t0, x0)‖ ≤ v‖x0‖exp{−u(t− t0)}

holds for all t ≥ t0 ≥ 0.

For stochastic differential equations, various definitions of exponential stability are
proposed, two of the most important ones are given in the following.

Definition 2. ([16]) The state vector y(t; t0, y0) in (1) is almost surely exponentially stable
if for any t0 ∈ <+ and y0 ∈ <+, there exist u, v ∈ <+ such that

‖y(t; t0, y0)‖ ≤ v‖y0‖exp{−u(t− t0)}, t ≥ t0 ≥ 0

holds almost surely for all t ≥ t0 ≥ 0.

Definition 3. ([16]) The state vector y(t; t0, y0) in (1) is mean square exponentially stable
if for any t0 ∈ <+ and y0 ∈ <+, there exist u, v ∈ <+ such that

E‖y(t; t0, y0)‖2 ≤ v‖y0‖2exp{−u(t− t0)}, t ≥ t0 ≥ 0

holds for all t ≥ t0 ≥ 0.



Z. Mingxin, X. Tao, S. Wenxiao / Eur. J. Pure Appl. Math, 13 (4) (2020), 794-806 797

It is shown that almost surely exponentially stable implies mean square exponentially
stable, but the converse is not true [15]. However, under Assumption (A1), we have the
following lemma.

Lemma 1. ([15]) If (A1) holds, then almost surely exponential stability could be derived
from mean square exponential stability.

Lemma 2. ([20]) For any xi > 0 , 0 < λi < 1 and λ1 + · · ·+ λn = 1 we have

(x1 + · · ·+ xn)2 ≤ x2
1

λ1
+ · · ·+ x2

n

λn
. (4)

Next, to get the main result, we need another assumption:

(A2) : γ̄(α) =
(3α2

λ2
(‖A‖2 + k2‖B‖2) +

ασ2

λ3

)( 1

λ1
+

3α

λ2
k2‖D‖2

)
× exp

{3α2

λ2
(‖A‖2 + k2‖B‖2) +

ασ2

λ3

}
+

3

λ2
α2k2‖D‖2 < 1.

Lemma 3. If Assumptions (A1) and (A2) hold, then for any t ∈ <+, the solution y(t) of
(1) satisfies

E‖y(β(t))‖2 ≤ λE‖y(t)‖2 (5)

where the coefficient
λ = λ−1

1 (1− γ̄(α))−1. (6)

Proof. For any t ≥ t0, by the definition of β(t), there exist sequences {αk} and {ηk}
(k ∈ N) such that

β(t) = ηk ∈ [αk, αk+1), t ∈ [αk, αk+1).

For t ≥ ηk, from (1) we have

y(t) = y(ηk) +

∫ t

ηk

[
−Ay(s) +Bf(y(s)) +Df(y(ηk))

]
ds+

∫ t

ηk

σy(s)dω(s). (7)

By applying Lemma 2 in the case n = 3, Assumption (A1), the properties of mathematical
expectation E(·) and Cauchy-Schwarz inequality we have

E‖y(t)‖2 = E‖y(ηk) +

∫ t

ηk

[
−Ay(s) +Bf(y(s)) +Df(y(ηk))

]
ds+

∫ t

ηk

σy(t)dω(t)‖2

≤ 1

λ1
E‖y(ηk)‖2 +

1

λ2
E‖
∫ t

ηk

[
−Ay(s) +Bf(y(s)) +Df(y(ηk))

]
ds‖2

+
1

λ3
E‖
∫ t

ηk

σy(t)dω(t)‖2

≤ 1

λ1
E‖y(ηk)‖2 +

1

λ2
E‖
∫ t

ηk

1×
[
−Ay(s) +Bf(y(s)) +Df(y(ηk))

]
ds‖2



Z. Mingxin, X. Tao, S. Wenxiao / Eur. J. Pure Appl. Math, 13 (4) (2020), 794-806 798

+
1

λ3
E‖
∫ t

ηk

σy(t)dω(t)‖2

≤ 1

λ1
E‖y(ηk)‖2 +

1

λ2
E‖
∫ t

ηk

1ds×
∫ t

ηk

[
−Ay(s) +Bf(y(s)) +Df(y(ηk))

]
ds‖2

+
1

λ3
E‖
∫ t

ηk

σy(t)dω(t)‖2

≤ 1

λ1
E‖y(ηk)‖2 +

3α

λ2

[
E

∫ t

ηk

(
‖A‖2 + k2‖B‖2

)
‖y(s)‖2ds

+ E

∫ t

ηk

k2‖D‖2‖y(ηk)‖2ds
]

+
1

λ3
E

∫ t

ηk

σ2‖y(s)‖2ds

≤ (
1

λ1
+

3α

λ2
k2‖D‖2)E‖y(ηk)‖2 + [

3α

λ2
(‖A‖2 + k2‖B‖2) +

σ2

λ3
]

∫ t

ηk

E‖y(s)‖2ds.

(8)

Utilizing the Gronwall-Bellman inequality to (8), we have

E‖y(t)‖2 ≤ (
1

λ1
+

3α

λ2
k2‖D‖2)E‖y(ηk)‖2exp{α[

3α

λ2
(‖A‖2 + k2‖B‖2) +

σ2

λ3
]}. (9)

By exchanging the position of y(t) and y(ηk) in (7) and similar deduction as above, we
have

E‖y(ηk)‖2 ≤
1

λ1
E‖y(t)‖2 +

3α

λ2
k2‖D‖2E‖y(ηk)‖2

+ [
3α

λ2
(‖A‖2 + k2‖B‖2) +

σ2

λ3
]

∫ t

ηk

E‖y(s)‖2ds. (10)

Substituting (9) into (10) we have

E‖y(ηk)‖2 ≤
1

λ1
E‖y(t)‖2 +

3α

λ2
k2‖D‖2E‖y(ηk)‖2

+ α[
3α

λ2
(‖A‖2 + k2‖B‖2) +

σ2

λ3
](

1

λ1
+

3α

λ2
k2‖D‖2)

× exp{α[
3α

λ2
(‖A‖2 + k2‖B‖2) +

σ2

λ3
]}E‖y(ηk)‖2

=
1

λ1
E‖y(t)‖2 + γ̄(α)E‖y(ηk)‖2. (11)

By using Assumption (A2), it follows that

E‖y(ηk)‖2 ≤
1

λ1
(1− γ̄(α))−1E‖y(t)‖2 = λE‖y(t)‖2.

For t < ηk, we can get the same result as above inequality. So, (5) holds for any
t ∈ [αk, αk+1) and k ∈ N. The proof is finished.



Z. Mingxin, X. Tao, S. Wenxiao / Eur. J. Pure Appl. Math, 13 (4) (2020), 794-806 799

Remark 2. The inequality (12) in Lemma 3 tells the relationship between the norm ‖y(ξi)‖
and ‖y(t)‖. It brings convenience for the study of the recurrent neural network system (1).

3. Main Results

In this section, we will quantitatively analyze the influence of the deviation function
and random disturbance on the global exponential stability of recurrent neural network
system (1).

Theorem 1. Let Assumptions (A1) and (A2) hold, and assume that (2) is globally expo-
nentially stable. Then (1) is mean square globally exponentially stable, which implies (1)
is almost surely exponentially stable, if |σ| < σ̄√

2
and α < min(ρ2 , ᾱ), where σ̄ is the unique

solution σ̂ in the following transcendental equation

2vexp{−uρ}+
8ρv

u

(8ρk2‖D‖2

λ̄3
(1 +

1

λ1
) +

σ̂2

λ̄4

)
exp{2ρc0} = 1 (12)

and the interval length ᾱ is a unique solution α̂ respect to the equation (13)

2vexp
{
− u(ρ− α̂)

}
+

8ρv

u

(8ρk2‖D‖2

λ̄3
(1 + λ) +

σ̂2

λ̄4

)
exp{2ρc1} = 1 (13)

where

c0 = 2ρ
(‖A‖2
λ̄1

+
k2‖B‖2

λ̄2
+

2k2‖D‖2

λ̄3

)
+ 2
(8ρk2‖D‖2

λ̄3
(1 +

1

λ1
) +

σ̂2

λ̄4

)
c1 = 2ρ

(‖A‖2
λ̄1

+
k2‖B‖2

λ̄2
+

2k2‖D‖2

λ̄3

)
+ 2
(8ρk2‖D‖2

λ̄3
(1 + λ) +

σ̂2

λ̄4

)
and λ = λ−1

1 (1− γ̂)−1, γ̂ = γ̄(α̂), ρ > ln(v)
u > 0.

Proof. Denote by x(t; t0, x0) ≡ x(t) and y(t; t0, x0) ≡ y(t) the state of (2) and (1)
respectively. For t0 ≤ t ≤ t0 + 2ρ, from (1), we have

E‖y(t)− x(t)‖2 = E‖
∫ t

t0

−A(y(s)− x(s)) +B(f(y(s))− f(x(s)))

+D[f(y(β(t)))− f(x(s))]ds+

∫ t

t0

σy(s)dω(s)‖2. (14)

Let n = 4 in Lemma 2, we get

(x1 + x2 + x3 + x4)2 ≤ x2
1

λ̄1
+
x2

2

λ̄2
+
x2

3

λ̄3
+
x2

4

λ̄4
,



Z. Mingxin, X. Tao, S. Wenxiao / Eur. J. Pure Appl. Math, 13 (4) (2020), 794-806 800

where λ̄i ∈ (0, 1) and
∑4

i=1 λ̄i = 1. Applying the above inequality, Cauchy-Schwarz
inequality and Assumption (A1) on (14) and together with the properties of E(·), we have

E‖y(t)− x(t)‖2 ≤ 1

λ̄1
E‖
∫ t

t0

−A(y(s)− x(s))ds‖2 +
1

λ̄2
E‖
∫ t

t0

B(f(y(s))− f(x(s)))ds‖2

+
1

λ̄3
E‖
∫ t

t0

D[f(y(β(s)))− f(x(s))]ds‖2 +
1

λ̄4
E‖
∫ t

t0

σy(s)dω(s)‖2

≤ 2ρ

[(
‖A‖2

λ̄1
+
k2‖B‖2

λ̄2

)∫ t

t0

E‖y(s)− x(s)‖2ds

+
k2‖D‖2

λ̄3

∫ t

t0

E‖y(β(s))− x(s)‖2ds
]

+
σ2

λ̄4

∫ t

t0

E‖y(s)‖2ds

≤ 2ρ

[(
‖A‖2

λ̄1
+
k2‖B‖2

λ̄2
+

2k2‖D‖2

λ̄3

)∫ t

t0

E‖y(s)− x(s)‖2ds

+
4k2‖D‖2

λ̄3

∫ t

t0

E‖y(β(s))‖2ds
]

+

(
8ρk2‖D‖2

λ̄3
+
σ2

λ̄4

)∫ t

t0

E‖y(s)‖2ds

(15)

From the inequality (5) in Lemma 3, we have

E‖y(β(t))‖2 ≤ λE‖y(t)‖2,

where λ is given by (6). Substituting above inequality into (15), we have

E‖y(t)− x(t)‖2 ≤ 2ρ

(
‖A‖2

λ̄1
+
k2‖B‖2

λ̄2
+

2k2‖D‖2

λ̄3

)∫ t

t0

E‖y(s)− x(s)‖2ds

+

(
8ρk2‖D‖2

λ̄3
(1 + λ) +

σ2

λ̄4

)∫ t

t0

E‖y(s)‖2ds

≤
[
2ρ(
‖A‖2

λ̄1
+
k2‖B‖2

λ̄2
+

2k2‖D‖2

λ̄3
)

+ 2

(
8ρk2‖D‖2

λ̄3
(1 + λ) +

σ2

λ̄4

)]∫ t

t0

E‖y(s)− x(s)‖2ds

+
4ρv

u

(
8ρk2‖D‖2

λ̄3
(1 + λ) +

σ2

λ̄4

)
‖y0‖2 (16)

When t0 + α ≤ t ≤ t0 + 2ρ, it follows from (16) that

E‖y(t)− x(t)‖2 ≤ c2

∫ t

t0

E‖y(s)− x(s)‖2ds+ c3‖y0‖2 (17)

where

c2 = 2ρ
(‖A‖2
λ̄1

+
k2‖B‖2

λ̄2
+

2k2‖D‖2

λ̄3

)
+ 2
(8ρk2‖D‖2

λ̄3
(1 + λ) +

σ2

λ̄4

)



Z. Mingxin, X. Tao, S. Wenxiao / Eur. J. Pure Appl. Math, 13 (4) (2020), 794-806 801

c3 =
4ρv

u

(8ρk2‖D‖2

λ̄3
(1 + λ) +

σ2

λ̄4

)
Utilized the well-known Gronwall Inequality to (17), for t0 + α ≤ t ≤ t0 + 2ρ,

E‖y(t)− x(t)‖2 ≤ c3exp{2ρc2}‖y0‖2. (18)

Consequently, for t0 + α ≤ t ≤ t0 + 2ρ,

E‖y(t)‖2 ≤ 2E‖y(t)− x(t)‖2 + 2E‖x(t)‖2

≤ 2c3exp{2ρc2}‖y0‖2 + 2u‖y0‖2exp{−v(t− t0)}. (19)

Hence, for t0 + ρ− α ≤ t ≤ t0 + 2ρ− α,

E‖y(t)‖2 ≤
[
2c3exp{2ρc2}+ 2uexp{−v(ρ− α)}

]
‖y0‖2

= ĉ‖y0‖2 (20)

where ĉ = 2c3exp{2ρc2}+ 2uexp{−v(ρ− α)}.
From (13), combining the monotonicity of the function, we know that when α < ᾱ,

we have ĉ < 1. Therefore, when t0 − α+ ρ ≤ t ≤ t0 − α+ 2ρ, we discuss the existence of
parameter λ̄1 as follows

∂ ln c̄

∂λ̄1
=

∂ĉ

∂λ̄1
= 0

where c̄ = ĉ− 2uexp{−v(ρ− α)}

∂ ln c̄

∂λ̄1
=
∂ ln c3

∂λ̄1
+ 2ρ

∂c2
∂λ̄1

=
∂ ln c3

c3∂λ̄1
+ 2ρ

∂c2
∂λ̄1

(21)

Further more, from (21) we obtain

(a2σ
2 + a1 − a3)λ̄3

1 + (2σ4λ̄3 − k2λ̄3σ
2 − k1a2σ

2 − k1a3 − 3k1a1)λ̄2
1

+ (2k2k1σ
2 + 3k2

1)λ̄1 − (k2k
2
1σ

2 + a1k
3
1) = 0 (22)

where a1 = 16ρ2k2‖A‖2‖D‖2, a2 = −4ρσ2vλ̄3
u , a3 = 64ρ3k2|D‖2(1+λ)vλ̄3

u , k1 = (1− λ̄2 − λ̄3),
k2 = 2ρλ̄3‖A‖2

Therefore, according to (22), there exists a real solution λ̄1. For λ̄2, λ̄3 and λ̄4 can
be similarly discussed as (22), substituting λ̄1, λ̄2, λ̄3 and λ̄4 into ĉ, we know that ĉ is
a strictly monotone function. So, equation (13) exists a unique α̂ such that α̂ = ᾱ, for
λ̄1, λ̄2, λ̄3 ∈ (0, 1).

On the basis of (12) and (13), we observe that ĉ < 1, when |σ| < σ̄, α < min(ρ2 , ᾱ).

Setting τ = − ln(ĉ)
ρ , we have

E‖y(t)‖2 ≤ exp{−ρτ}‖y0‖2 (23)



Z. Mingxin, X. Tao, S. Wenxiao / Eur. J. Pure Appl. Math, 13 (4) (2020), 794-806 802

Combined with the flow and the uniqueness of solution in RNN (1), for any positive
integer m.

y(t; t0, y0) = y(t; t0 + (m− 1)ρ, y(t0 + (m− 1)ρ; t0, y0)) (24)

Thus, considering (23) and (24), for t ≥ t0 − α+mρ,

‖y(t; t0, y0)‖ = ‖y(t; t0 + (m− 1)ρ, y(t0 + (m− 1)ρ; t0, y0))‖
≤ exp(−ρτ)‖y(t0 + (m− 1)ρ; t0, y0)‖
≤ exp(−mρτ)‖y0‖

Consequently, when t > t0−α+ρ, the positive integer m that satisfies t0−α+(m−1)ρ ≤
t ≤ t0 − α+mρ,

‖y(t; t0, y0)‖2 ≤ exp(−τ(t− t0))exp(τ(ρ− α))‖y0‖2 (25)

Obviously, (25) also holds for t0 ≤ t ≤ t0 − α+ ρ. Therefore, RNN (1) is mean square
exponentially stable. From Lemma 2.3, the almost surely exponential stability of system
(1) can be fully proved.

4. Illustrative Numerical Examples

In this section, an example is provided to illustrate the results.

Example 1. Given the two-dimensional original system{
ẋ1(t) = −x1(t)− 2f(x1(t)) + 2f(x2(t)),

ẋ2(t) = −x2(t) + 2f(x1(t))− 2f(x2(t)).
(26)

Under the vector form in (1), the matrix can be written as

A =

(
1 0
0 1

)
, B =

(
−1 1
1 −1

)
, D =

(
−1 1
1 −1

)
.

If we choose f(xj) = sin2(xj), j = 1, 2. Then, by Theorem 1 in [26], the recurrent neural
network (26) is globally exponentially stable. The system (26) with deviating argument
and stochastic disturbance is modeled by{

ẏ1(t) = −y1(t)− f(y1(t)) + f(y2(t))− f(y1(β(t))) + f(y2(β(t))) + σy1(t)dω(t),

ẏ2(t) = −y2(t) + f(y1(t))− f(y2(t)) + f(y1(β(t)))− f(y2(β(t))) + σy1(t)dω(t),
(27)

where {αk} = {k4}, {ηk} = {2k+1
8 }, k ∈ N , t ∈ [αk, αk+1), t ∈ <+. The deviating function

β(t) = ηk, σ is the interference intensity, and ω(t) is a Brownian motion.

Let ρ = 1 ≥ ln(1.2)
0.9 = 0.2026, k = 0.01, λ1 = 1

3 , λ̄i = 1
4 , substituting them into (12) and

(13), then we get σ̄ = 0.0468. From |σ| < σ̄√
2
, we know that |σ| < 0.0325. In reference



Z. Mingxin, X. Tao, S. Wenxiao / Eur. J. Pure Appl. Math, 13 (4) (2020), 794-806 803

0 2 4 6 8 10
−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

t

y

 

 
y(1)
y(2)

Figure 1: Stability behavior of system (26)

0 1 2 3 4 5
−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t

y

 

 
y(1)
y(2)

Figure 2: Stability behavior of RNN (27)

0 100 200 300 400 500
−150

−100

−50

0

50

100

150

t

y

 

 
y1
y2

Figure 3: Instability behavior of RNN (27)

[26], the upper bound of the |σ| is 0.0265, which is smaller than 0.0325. This means that
the system (27) can withstand higher intensity random disturbance than the system in
[26]. Moreover, by substituting those parameters in (13), we have ᾱ = 25.3565. Note that
α < min(ρ2 , ᾱ), then α < 0.1013. In the Example 2 of [26], α < 0.0159, which is smaller
than 0.1013. This shows that the system (27) has wider range of piecewise argument and
implies that the system (27) can withstand higher intensity of impact from time delay or
advance.

Figure 2 describes the stability performance of (27) with σ = 0.04, {αk} = { k
100},

{ηk} = {2k+1
200 }, k ∈ N. Figure 3 illustrates a degenerative performance of RNN (27) for

σ = 1, {αk} = { k
100}, {ηk} = {2k+1

200 }, k ∈ N. Certainly, in this respect, these parameters
of the conditions in Theorem (1) were no longer effective, the system will involve into
unstable.

5. Conclusion

This paper investigates global robust exponential stability of recurrent neural networks
with piecewise argument and arbitrary disturbance. The upper bound of perturbation
intensity is estimated by using inequalities and transcendental equations with restricted



REFERENCES 804

parameters. The theoretical results of this paper provide a reliable basis for the application
and design of RNNs. The authors would like to search for larger upper bounds of deviating
variables and improve the interference intensity such that the system remains stable. The
treatment methods in this paper can provide references for more flexible control systems.

Future work will extend the results to the multi-stability of bidirectional associative
memory neural networks or fuzzy neural networks in the presence of deviating arguments
and random disturbances. The main problem is the construction of the relationship be-
tween piecewise deviating variables and system solution vectors in neural networks.

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