Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 169214, 12 pages
http://dx.doi.org/10.1155/2013/169214
Research Article
Convergence and Stability of the Split-Step 𝜃-Milstein Method
for Stochastic Delay Hopfield Neural Networks
Qian Guo,1,2 Wenwen Xie,1 and Taketomo Mitsui3
1
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
Division of Computational Science, E-Institute of Shanghai Universities, 100 Guilin Road, Shanghai 200234, China
3
Department of Mathematical Sciences, Faculty of Science and Engineering, Doshisha University, Kyoto 610-0394, Japan
2
Correspondence should be addressed to Qian Guo; qguo@shnu.edu.cn
Received 8 December 2012; Accepted 26 February 2013
Academic Editor: Chengming Huang
Copyright © 2013 Qian Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A new splitting method designed for the numerical solutions of stochastic delay Hopfield neural networks is introduced and
analysed. Under Lipschitz and linear growth conditions, this split-step 𝜃-Milstein method is proved to have a strong convergence
of order 1 in mean-square sense, which is higher than that of existing split-step 𝜃-method. Further, mean-square stability of the
proposed method is investigated. Numerical experiments and comparisons with existing methods illustrate the computational
efficiency of our method.
1. Introduction
Hopfield neural networks, which originated with Hopfield in
the 1980s [1], have been successfully applied in many areas
such as combinatorial optimization [2, 3], signal processing
[4], and pattern recognition [5, 6]. In the last decade, neural
networks in the presence of signal transmission delay and
stochastic perturbations, also named as stochastic delay Hopfield neural networks (SDHNNs), have gained considerable
research interest (see, e.g., [7–9] and the references therein).
It is noticed that, so far, most works on SDHNNs focus mainly
on the stability analysis of the analytical solutions, including
mean-square exponential stability [7], global asymptotic
stability [9], and so forth. Not only simulation is an important
tool to explore interesting dynamics of kinds of Hopfield
neural networks (HNNs) (see, e.g., [10] and the references
therein), but also parameter estimation in dynamical systems based on HNNs (see, e.g., [11]) needs to solve HNNs
numerically. Moreover, because most of SDHNNs do not
have explicit solutions, the numerical analysis of SDHNNs
recently stirred some initial research attention. For example,
Li et al. [12] investigated the exponential stability of the Euler
method and the semi-implicit Euler method for SDHNNs.
Rathinasamy [13] introduced a split-step 𝜃-method (SST)
for SDHNNs and analysed the mean-square stability of this
method, and the SST is only given for the commensurable
delay case. To the best of our current knowledge, the authors
mainly discussed the stability of numerical solutions for
stochastic Hopfield neural networks with discrete time delays
but skipped the details of convergence analysis.
The split-step Euler method for stochastic differential
equations (SDEs) was proposed by Higham et al. [14], further,
the splitting Euler-type algorithms have been derived for
stochastic delay differential equations (SDDEs) [15, 16]. In
this paper, we will present a splitting method with higher
order convergence for SDHNNs. To be specific, we will go
into detail about the convergence analysis and comparing the
stability with split-step 𝜃-method given in [13].
The rest of this paper is organized as follows. In Section 2,
we recall the stochastic delay neural networks model and
present a split-step 𝜃-Milstein method. In Section 3, we derive
the convergence results of the split-step 𝜃-Milstein method
for the model. In Section 4, the numerical stability analysis is
performed. In Section 5, some numerical examples are given
to confirm the theory. In the last Section, we draw some
conclusions.
2
Abstract and Applied Analysis
2. Model and the Split-Step 𝜃-Milstein Method
2.1. Model. Consider the stochastic delay Hopfield neural
networks of the form
dx (𝑡) = [−𝐶x (𝑡) + 𝐴f (x (𝑡)) + 𝐵g (z (𝑡))] dt
(1)
+ Φ (x (𝑡)) d𝑊 (𝑡) ,
where x(𝑡) = [𝑥1 (𝑡), . . . , 𝑥𝑛 (𝑡)]𝑇 ∈ R𝑛 is the state vector associated with the 𝑛 neurons, z(𝑡) = [𝑥1 (𝑡 − 𝜏1 ), . . . , 𝑥𝑛 (𝑡 − 𝜏𝑛 )]𝑇 ,
the diagonal matrix 𝐶 = diag(𝑐1 , 𝑐2 , . . . , 𝑐𝑛 ) has positive
entries, and 𝑐𝑖 represents the rate at which the 𝑖th unit
will reset its potential to the resting state in isolation when
discounted from the network and the external stochastic
perturbation. The matrices 𝐴 = (𝑎𝑖𝑗 )𝑛×𝑛 and 𝐵 = (𝑏𝑖𝑗 )𝑛×𝑛
are the connection weight matrix and the discretely delayed
connection weight matrix, respectively. Furthermore, the
vector functions f(x(𝑡)) = [𝑓1 (𝑥1 (𝑡)), . . . , 𝑓𝑛 (𝑥𝑛 (𝑡))]𝑇 and
g(z(𝑡)) = [𝑔1 (𝑥1 (𝑡 − 𝜏1 )), . . . , 𝑔𝑛 (𝑥𝑛 (𝑡 − 𝜏𝑛 ))]𝑇 denote the
neuron activation functions with the conditions 𝑓𝑖 (0) =
0, 𝑔𝑖 (0) = 0 for all positive 𝜏𝑗 .
On the initial segment [−𝜏, 0] the state vector satisfies
x(𝑡) = 𝜓(𝑡), where 𝜓(𝑡) = [𝜓1 (𝑡), . . . , 𝜓𝑛 (𝑡)]𝑇 is a given
function in 𝐶([−𝜏, 0], R𝑛 ) and 𝜏 stands for max1≤𝑖≤𝑛 {𝜏𝑖 }.
Moreover, Φ(x(𝑡)) = diag(𝜙1 (𝑥1 (𝑡)), . . . , 𝜙𝑛 (𝑥𝑛 (𝑡))) is a
diagonal matrix with 𝜙𝑖 (0) = 0 and 𝑊(𝑡) = [𝑊1 (𝑡), . . . ,
𝑊𝑛 (𝑡)]𝑇 ∈ R𝑛 is an 𝑛-dimensional Wiener process defined
on the complete probability space (Ω, F, {F𝑡 }𝑡≥0 , P) with a
filtration {F𝑡 }𝑡≥0 satisfying the usual conditions (i.e., it is
increasing and right continuous while F0 contains all P-null
sets).
Let 𝑓𝑖 and 𝑔𝑖 be functions in 𝐶2 (𝐷; R) ⋂ L2 ([0, 𝑇]; R)
and 𝜙𝑖 be in 𝐶1 (𝐷; R) ∩ L2 ([0, 𝑇]; R). Here 𝐶𝑙 (𝐷; R) denotes
the family of continuously 𝑙-times differentiable real-valued
function defined on 𝐷, while L𝑙 ([0, 𝑇]; R) denotes the
family of all real-valued measurable {F𝑡 }-adapted stochastic
𝑇
processes {𝑓(𝑡)}𝑡∈[0,𝑇] such that ∫0 |𝑓(𝑡)|𝑙 𝑑𝑡 < +∞.
2.2. Numerical Scheme. We define the mesh with a uniform
step-size ℎ (0 < ℎ < 1) on the interval [0, 𝑇]; that is, 𝑡𝑘 =
𝑘 ⋅ ℎ (𝑘 = 0, 1, . . . , 𝐾) and 𝑇 = 𝐾ℎ.
Let Δ𝑊𝑖𝑘 = 𝑊𝑖 (𝑡𝑘+1 ) − 𝑊𝑖 (𝑡𝑘 ) denote the increment of the
Wiener process. The split-step 𝜃-Milstein (SSTM) scheme for
the solution of SDEs (1) is given by
𝑌𝑘 = 𝑋𝑘 + [− (1 − 𝜃) 𝐶𝑋𝑘 − 𝜃𝐶𝑌𝑘 + 𝐴f (𝑋𝑘 ) + 𝐵g (𝑍𝑘 )] ℎ,
1̂ 𝑘
(𝑌 ) [Δ𝑊𝑘 ∘ Δ𝑊𝑘 − 1ℎ] ,
𝑋𝑘+1 = 𝑌𝑘 + Φ (𝑌𝑘 ) Δ𝑊𝑘 + Φ
2
(2)
𝑘
where the merging parameter 𝜃 satisfies 0 ≤ 𝜃 ≤ 1, 𝑋 =
[𝑋1k , . . . , 𝑋𝑛𝑘 ]𝑇 is an approximation to x(𝑡𝑘 ), and for 1 ≤ 𝑞𝑗 ∈
Z+
𝜓𝑗 (𝑡𝑘 − 𝜏𝑗 ) , 𝑡𝑘 − 𝜏𝑗 ≤ 0,
𝑍𝑗𝑘 = { 𝑘−𝑞𝑗
0 < 𝑡𝑘 − 𝜏𝑗 ∈ [𝑡𝑘−𝑞𝑗 , 𝑡𝑘−𝑞𝑗 +1 ) .
𝑋𝑗 ,
(3)
̂ 𝑘)
=
Moreover,
we
adopt
the
symbols
Φ(𝑌
󸀠
𝑘
𝑘
󸀠
𝑘
𝑘
𝑘
diag(𝜙1 (𝑌1 )𝜙1 (𝑌1 ), . . . , 𝜙𝑛 (𝑌𝑛 )𝜙𝑛 (𝑌𝑛 )) and Δ𝑊 = [Δ𝑊1𝑘 ,
. . . ,Δ𝑊𝑛𝑘 ]𝑇 , the Hadamard product Δ𝑊𝑘 ∘ Δ𝑊𝑘 means
[(Δ𝑊1𝑘 )2 , . . . ,(Δ𝑊𝑛𝑘 )2 ]𝑇 , and 1 = [1, . . . , 1]𝑇 ∈ R𝑛 . When
𝑡𝑘 ≤ 0, we define 𝑋𝑘 = 𝜓(𝑡𝑘 ).
Then scheme (2) can be written in equivalent form as
𝑌𝑘 = 𝑋𝑘 − (𝐼 + 𝜃𝐶ℎ)−1 𝐶ℎ𝑋𝑘
+ (𝐼 + 𝜃𝐶ℎ)−1 ℎ [𝐴f (𝑋𝑘 ) + 𝐵g (𝑍𝑘 )] ,
(4a)
1̂ 𝑘
𝑋𝑘+1 = 𝑌𝑘 + Φ (𝑌𝑘 ) Δ𝑊𝑘 + Φ
(𝑌 ) [Δ𝑊𝑘 ∘ Δ𝑊𝑘 − 1ℎ] .
2
(4b)
Substituting (4a) into (4b), we have a stochastic explicit
single-step method with an increment function Λ(𝜉, 𝜂,
ℎ, Δ𝑊𝑘 ) ∈ R𝑛 ; that is,
𝑋𝑘+1 = 𝑋𝑘 + Λ (𝑋𝑘 , 𝑍𝑘 , ℎ, Δ𝑊𝑘 ) .
(5)
3. Order and Convergence Results for SSTM
In this section we consider the global error of SSTM (2) as
applied to SDHNNs (1) with initial condition. In what follows,
‖ ⋅ ‖ denotes Euclidean norm in R𝑛 .
For convergence purpose we make the following standard
assumptions.
̂ satisfy the Lipschitz
Assumption 1. Assume that f, g, Φ, and Φ
condition
󵄨
󵄨󵄨
󵄨󵄨𝑓𝑖 (𝑎)−𝑓𝑖 (𝑏)󵄨󵄨󵄨 ≤ 𝛼𝑖 |𝑎−𝑏| ,
󵄨󵄨
󵄨
󵄨󵄨𝑔𝑖 (𝑎)−𝑔𝑖 (𝑏)󵄨󵄨󵄨 ≤ 𝛽𝑖 |𝑎−𝑏| ,
(𝑎, 𝑏 ∈ R and 𝑡 ∈ [0, 𝑇])
󵄨󵄨
󵄨
󵄨󵄨𝜙i (𝑎)−𝜙𝑖 (𝑏)󵄨󵄨󵄨 ≤ 𝛾𝑖 |𝑎−𝑏| ,
󵄨󵄨 󸀠
󵄨
󵄨󵄨𝜙𝑖 𝜙𝑖 (𝑎)−𝜙𝑖 𝜙𝑖󸀠 (𝑏)󵄨󵄨󵄨 ≤ 𝜅𝑖 |𝑎−𝑏| ,
󵄨
󵄨
(6)
for every 𝑖 and the linear growth condition
󵄩2
󵄩̂
󵄩2
󵄩
(x) ⋅ 1󵄩󵄩󵄩󵄩
‖f(x)‖2 ∨ 󵄩󵄩󵄩g(x)󵄩󵄩󵄩 ∨ ‖Φ (x) ⋅ 1‖2 ∨ 󵄩󵄩󵄩󵄩Φ
̃ (1 + ‖x‖2 ) ,
≤𝐿
∀x ∈ R𝑛 ,
(7)
̃ is a positive constant and ∨ is the maximal operator.
where 𝐿
We also define 𝐿 𝑖 as 𝐿 𝑖 = max{𝛼𝑖 , 𝛽𝑖 , 𝛾𝑖 , 𝜅𝑖 } (𝑖 = 1, 2, . . . , 𝑛).
We also need the following assumption on the initial
condition.
Assumption 2. Assume that the initial function 𝜓(𝑡) is Lipschitz continuous from [−𝜏, 0] to R𝑛 , that is, there is a positive
constant 𝐿 𝜓 satisfying
󵄩
󵄩󵄩
󵄩󵄩𝜓 (𝑡) − 𝜓 (𝑠)󵄩󵄩󵄩 ≤ 𝐿 𝜓 (𝑡 − 𝑠)
if − 𝜏 ≤ 𝑠 < 𝑡 < 0.
(8)
Abstract and Applied Analysis
3
Now we give the definition of local and global errors.
Definition 1. Let x(𝑡) denote the exact solution of (1). The
local approximate solution x̃(𝑡𝑘+1 ) starting from x(𝑡𝑘 ) by
SSTM (2) given by
̃ (𝑡𝑘 ) , ℎ, Δ𝑊𝑘 ) ,
x̃ (𝑡𝑘+1 ) := x (𝑡𝑘 ) + Λ (x (𝑡𝑘 ) , 𝑍
Proof. If 𝑡𝑘 − 𝜏𝑗 ≤ 0 and 𝑠 − 𝜏𝑗 ≤ 0, under Assumption 2 we
have
̃𝑗 (𝑡𝑘 )]2
𝐸[𝑥𝑗 (𝑠 − 𝜏𝑗 ) − 𝑍
2
= 𝐸[𝜓𝑗 (𝑠 − 𝜏𝑗 ) − 𝜓𝑗 (𝑡𝑘 − 𝜏𝑗 )]
(9)
̃ 𝑘 ) denotes the evaluation of (3) using the exact
where 𝑍(𝑡
solution, yields the difference
𝛿𝑘+1 := x (𝑡𝑘+1 ) − x̃ (𝑡𝑘+1 ) .
(10)
≤ 𝐿 𝜓 ℎ2 .
If 𝑡𝑘 − 𝜏𝑗 ≤ 0 and 𝑠 − 𝜏𝑗 > 0, with (13) we obtain
̃𝑗 (𝑡𝑘 )]2
𝐸[𝑥𝑗 (𝑠 − 𝜏𝑗 ) − 𝑍
Then the local error of SSTM is defined by ‖𝛿𝑘+1 ‖, whereas its
global error means ‖𝜖𝑘 ‖ where 𝜖𝑘 := x(𝑡𝑘 ) − 𝑋𝑘 .
2
= 𝐸[𝑥𝑗 (𝑠 − 𝜏𝑗 ) − 𝜓𝑗 (𝑡𝑘 − 𝜏𝑗 )]
Definition 2. If the global error satisfies
󵄩 󵄩 2
𝐸 (󵄩󵄩󵄩󵄩𝜖𝑘 󵄩󵄩󵄩󵄩) ≤ Γℎ2𝑝
∀ℎ ∈ (0, ℎ0 )
≤ 2𝐸[𝑥𝑗 (𝑠 − 𝜏𝑗 ) − 𝑥𝑗 (0)]
(11)
with positive constants ℎ0 and Γ and a finite 𝑝, then we say
that the order of mean-square convergence accuracy of the
method is 𝑝. Here 𝐸 is the expectation with respect to P.
We then give the following lemmas that are useful in
deriving the convergence results.
Lemma 3 (see also [17]). Let the linear growth condition
(7) hold, and the initial function 𝜓(𝑡) is assumed to be
F0 -measurable and right continuous. And one puts 𝐸𝜓 :=
𝐸(sup−𝜏≤𝑡≤0 ‖𝜓(𝑡)‖2 ) < ∞. For any given positive 𝑇, there exist
positive numbers Γ𝜓 and Γ2 such that the solution of (1) satisfies
𝐸 ( sup ‖x (𝑠)‖2 ) ≤ Γ𝜓 ,
−𝜏≤𝑠≤𝑇
𝐸(‖x (𝑡) − x (𝑠)‖)2 ≤ Γ2 (𝑡 − 𝑠)
(13)
2
≤ 2 (Γ2 + 𝐿 𝜓 ℎ) ℎ.
If 𝑡𝑘 − 𝜏𝑗 > 0 and 𝑠 − 𝜏𝑗 > 0, we assume 𝑡𝑘 − 𝜏𝑗 ∈ [𝑡𝑘−𝑞𝑗 , 𝑡𝑘−𝑞𝑗 +1 )
without loss of generality. Hence,
̃𝑗 (𝑡𝑘 )]2
𝐸[𝑥𝑗 (𝑠 − 𝜏𝑗 ) − 𝑍
2
= 𝐸[𝑥𝑗 (𝑠 − 𝜏𝑗 ) − 𝑥𝑗 (𝑡𝑘−𝑞𝑗 )]
2
(14)
by using inequality (13).
Lemma 5. Let x(𝑡) denote the exact solution of (1). One
assumes conditions (6) and (7). Then for the local intermediate value y(𝑡𝑘 ) := x(𝑡𝑘 ) − (𝐼 + 𝜃𝐶ℎ)−1 𝐶ℎx(𝑡𝑘 ) + (𝐼 +
̃ 𝑘 ))], one has the estimation
𝜃𝐶ℎ)−1 ℎ[𝐴f(x(𝑡𝑘 )) + 𝐵g(𝑍(𝑡
(15)
(19)
Proof. The difference between the 𝑖th components of x(𝑡𝑘 )
and y(𝑡𝑘 ) leads to
󵄨2
󵄨󵄨
󵄨󵄨𝑥𝑖 (𝑡𝑘 ) − 𝑦𝑖 (𝑡𝑘 )󵄨󵄨󵄨
=
Lemma 4. For 𝑠 ∈ [𝑡𝑘 , 𝑡𝑘 + ℎ], one has
Here the constant Γ𝜏 is independent of step-size ℎ.
2
≤ 4Γ2 ℎ
from (12).
󵄩
̃ (𝑡𝑘 )󵄩󵄩󵄩󵄩2 ) ≤ Γ𝜏 ℎ.
𝐸 (󵄩󵄩󵄩󵄩z (𝑠) − 𝑍
󵄩
(18)
≤ 2𝐸[𝑥𝑗 (𝑠 − 𝜏𝑗 ) − 𝑥𝑗 (𝑡𝑘 − 𝜏𝑗 )]
󵄩2
󵄩
𝐸 (󵄩󵄩󵄩x(𝑡𝑘 ) − y(𝑡𝑘 )󵄩󵄩󵄩 ) ≤ Γ3 ℎ2 .
The Jensen inequality derives
−𝜏≤𝑠≤𝑇
(17)
+ 2𝐸[𝜓𝑗 (0) − 𝜓𝑗 (𝑡𝑘 − 𝜏𝑗 )]
(12)
holds.
𝐸 ( sup ‖x (𝑠)‖) ≤ √Γ𝜓
2
+ 2𝐸[𝑥𝑗 (𝑡𝑘−𝑞𝑗 ) − 𝑥𝑗 (𝑡𝑘 − 𝜏𝑗 )]
where the constant Γ𝜓 is independent of step-size ℎ but
dependent on 𝑇. Moreover, for any 0 ≤ 𝑠 < 𝑡 ≤ 𝑇, 𝑡 − 𝑠 < 1,
the estimation
(16)
2
≤ 𝐿 𝜓 (𝑠 − 𝑡𝑘 )
𝑛
ℎ2
(1 + 𝑐𝑖 𝜃ℎ)
2
[𝑐𝑖 𝑥𝑖 (𝑡𝑘 ) − ∑𝑎𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡𝑘 ))
𝑗=1
[
𝑛
2
̃𝑗 (𝑡𝑘 ))] ,
− ∑𝑏𝑖𝑗 𝑔𝑗 (𝑍
𝑗=1
]
(20)
4
Abstract and Applied Analysis
whose expectation, together with 𝑐𝑖 > 0, 𝑓𝑖 (0) = 𝑔𝑖 (0) = 0,
the Lipschitz condition (6), and the estimation (12), gives
= ∫
𝑡𝑘
󵄩2
󵄩
𝐸 (󵄩󵄩󵄩x(𝑡𝑘 ) − y(𝑡𝑘 )󵄩󵄩󵄩 )
𝑛
+∫
3ℎ2
≤∑
𝑖=1 (1
𝑐𝑖 [𝑥𝑖 (𝑡𝑘 ) − 𝑥𝑖 (𝑠)] d𝑠 −
+ 𝑐𝑖 𝜃ℎ)
2
∑𝑎𝑖𝑗 [𝑓𝑗 (𝑥𝑗 (𝑠)) − 𝑓𝑗 (𝑥𝑗 (𝑡𝑘 ))]
𝑛
𝑗=1
(21)
+
󵄨̃
󵄨󵄨2 ]
+ 𝑛 ∑ 𝑏𝑖𝑗2 𝛽𝑗2 𝐸󵄨󵄨󵄨󵄨𝑍
𝑗 (𝑡𝑘 )󵄨󵄨󵄨
𝑗=1
]
3Γ𝜓 ∑ (𝑐𝑖2
𝑖=1
+
𝑗=1
̃𝑗 (𝑡𝑘 ))] d𝑠
+ ∑ 𝑏𝑖𝑗 [𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 )) − 𝑔𝑗 (𝑍
𝑛
𝑛
𝑐𝑖2 ℎ2 𝜃
𝑥 (𝑡 )
1 + 𝑐𝑖 𝜃ℎ 𝑖 𝑘
𝑡𝑘+1 𝑛
𝑡𝑘
𝑛
󵄨
󵄨2
󵄨
󵄨2
× [𝑐𝑖2 𝐸󵄨󵄨󵄨𝑥𝑖 (𝑡𝑘 )󵄨󵄨󵄨 + 𝑛 ∑𝑎𝑖𝑗2 𝛼𝑗2 𝐸󵄨󵄨󵄨󵄨𝑥𝑗 (𝑡𝑘 )󵄨󵄨󵄨󵄨
𝑗=1
[
≤
𝑡𝑘+1
𝑛
𝑛 ∑ 𝑎𝑖𝑗2 𝛼𝑗2
𝑗=1
+
𝑛
𝑐𝑖 𝜃ℎ2 [ 𝑛
̃𝑗 (𝑡𝑘 ))]
∑ 𝑎𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡𝑘 )) + ∑𝑏𝑖𝑗 𝑔𝑗 (𝑍
1 + 𝑐𝑖 𝜃ℎ 𝑗=1
𝑗=1
]
[
+∫
𝑛
𝑡𝑘+1
𝑡𝑘
𝑛 ∑ 𝑏𝑖𝑗2 𝛽𝑗2 ) ℎ2 .
𝑗=1
−∫
𝜙𝑖 (𝑥𝑖 (𝑠)) d𝑊𝑖 (𝑠) − ∫
𝑡𝑘+1
𝑡𝑘
𝑡𝑘+1
𝜙𝑖 (𝑦𝑖 (𝑡𝑘 )) d𝑊𝑖 (𝑠)
𝑠
∫ 𝜙𝑖󸀠 (𝑦𝑖 (𝑡𝑘 )) 𝜙𝑖 (𝑦𝑖 (𝑡𝑘 )) d𝑊𝑖 (𝑟) d𝑊𝑖 (𝑠) ,
𝑡𝑘
𝑡𝑘
(25)
Now we discuss local error estimates.
Theorem 6. When one assumes Assumptions 1 and 2 and the
conditions of Lemma 3, there exist positive constants Γ0 and Γ1 ,
such that
󵄩
󵄩
max 󵄩󵄩󵄩𝐸 (𝛿𝑘+1 )󵄩󵄩󵄩󵄩 ≤ Γ0 ℎ2 ,
(22)
0≤𝑘≤𝐾−1 󵄩
󵄩2
󵄩
max 𝐸 (󵄩󵄩󵄩󵄩𝛿𝑘+1 󵄩󵄩󵄩󵄩 ) ≤ Γ1 ℎ3
(23)
0≤𝑘≤𝐾−1
where Δ𝑊𝑖𝑘
𝑡𝑘+1
∫𝑡
𝑘
Proof. The Itô integral form of the 𝑖th component of (1) on
[𝑡𝑘 , 𝑡] implies
𝑥𝑖 (𝑡) − 𝑥𝑖 (𝑡𝑘 )
𝑠
𝐸 (𝛿𝑖𝑘+1 )
= 𝐸 (∫
𝑡𝑘+1
−
𝑛
+ ∑𝑏𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑠 − 𝜏𝑗 ))] d𝑠
𝑗=1
]
𝑡𝑘
𝑐𝑖 ℎ
= 𝑥𝑖 (𝑡𝑘+1 ) − 𝑥𝑖 (𝑡𝑘 ) +
𝑥 (𝑡 )
1 + 𝑐𝑖 𝜃ℎ 𝑖 𝑘
𝑛
𝑛
ℎ
̃𝑗 (𝑡𝑘 ))]
[ ∑ 𝑎𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡𝑘 )) + ∑ 𝑏𝑖𝑗 𝑔𝑗 (𝑍
1 + 𝑐𝑖 𝜃ℎ 𝑗=1
𝑗=1
]
[
𝑘
󸀠
− 𝜙𝑖 (𝑦𝑖 (𝑡𝑘 )) Δ𝑊𝑖 − 𝜙𝑖 (𝑦𝑖 (𝑡𝑘 ))
2
𝑡𝑘
𝑡𝑘
𝑗=1
𝑡𝑘+1 𝑛
𝑠−𝜏𝑗
𝑡𝑘
𝑡𝑘−𝑞𝑗
+ 𝐸 (∫
∑𝑏𝑖𝑗 ∫
𝑗=1
𝑔𝑗󸀠 (𝑥𝑗 (𝑟)) 𝜇𝑗 (𝑥𝑗 (𝑟))
1
+ 𝑔𝑗󸀠󸀠 (𝑥𝑗 (𝑟)) 𝜙𝑗2 (𝑥𝑗 (𝑟)) d𝑟d𝑠)
2
By utilizing the previous identity, the 𝑖th component of the
difference 𝛿𝑘+1 introduced in Definition 1 can be calculated
as
−ℎ
𝑠
∑𝑎𝑖𝑗 ∫ 𝑓𝑗󸀠 (𝑥𝑗 (𝑟)) 𝜇𝑗 (𝑥𝑗 (𝑟))
(26)
𝑡𝑘
× 𝜙𝑖 (𝑦𝑖 (𝑡𝑘 ))
𝑡𝑘+1 𝑛
1
+ 𝑓𝑗󸀠󸀠 (𝑥𝑗 (𝑟)) 𝜙𝑗2 (𝑥𝑗 (𝑟)) d𝑟d𝑠)
2
+ ∫ 𝜙𝑖 (𝑥𝑖 (𝑠)) d𝑊𝑖 (𝑠) .
2
(Δ𝑊𝑖𝑘 )
𝑐𝑖2 ℎ2 𝜃
𝐸 (𝑥𝑖 (𝑡𝑘 ))
1 + 𝑐𝑖 𝜃ℎ
(24)
𝑡
−
𝑠
𝑐𝑖 ∫ −𝜇𝑖 (𝑥𝑖 (𝑟)) d𝑟d𝑠)
+ 𝐸 (∫
𝑛
𝑡
=
∫𝑡 d𝑊𝑖 (𝑟)d𝑊𝑖 (𝑠).
𝑘
Taking expectations of both sides of (25),
= ∫ [−𝑐𝑖 𝑥𝑖 (𝑠) + ∑𝑎𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑠))
𝑡𝑘
𝑗=1
[
𝛿𝑖𝑘+1
𝑘
𝑡𝑘
as ℎ ↓ 0.
2
𝑡
∫𝑡 𝑘+1 d𝑊𝑖 (𝑠) and ((Δ𝑊i𝑘 ) − ℎ)/2
=
+
𝑐𝑖 𝜃ℎ2 [ 𝑛
∑𝑎 𝐸 (𝑓𝑗 (𝑥𝑗 (𝑡𝑘 )))
1 + 𝑐𝑖 𝜃ℎ 𝑗=1 𝑖𝑗
[
𝑛
̃𝑗 (𝑡𝑘 )))]
+ ∑𝑏𝑖𝑗 𝐸 (𝑔𝑗 (𝑍
𝑗=1
]
by (24) and Itô formula, where 𝜇𝑖 (𝑥𝑖 (𝑟)) := −𝑐𝑖 𝑥𝑖 (𝑟) +
∑𝑛𝑗=1 𝑎𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑟)) + ∑𝑛𝑗=1 𝑏𝑖𝑗 𝑔𝑗 (𝑥𝑗 (𝑟 − 𝜏𝑗 )). Under conditions
of this theorem, we have |𝐸(𝛿𝑘+1 )| ≤ Γ̃0 ℎ2 by (14), Jensen
𝑖
Abstract and Applied Analysis
5
inequality |𝐸(𝑋)| ≤ 𝐸(|𝑋|), triangle inequality, and properties of definite integral. Then we have max0≤𝑘≤𝐾−1 ‖𝐸(𝛿𝑘+1 )‖ ≤
Γ0 ℎ2 from the relation between |𝐸(𝛿𝑖𝑘+1 )| and ‖𝐸(𝛿𝑘+1 )‖.
Now we prove (23). By Itô formula,
{
{
𝑛
{
𝑡𝑘+1
{
2 2
≤ 9 {ℎ (𝑐𝑖 + 𝑛 ∑ 𝑎𝑖𝑗 𝛼𝑗 ) ∫
⏟⏟Γ⏟⏟⏟
⏟⏟d𝑠
2ℎ
{
𝑡𝑘
{
𝑗=1
(13)
{
{
𝑛
2 2
𝑡𝑘+1
+ ℎ (𝑛 ∑ 𝑏𝑖𝑗 𝛽𝑗 ) ∫
𝑡𝑘
𝑗=1
∫
𝑡𝑘+1
𝑡𝑘
𝜙𝑖 (𝑥𝑖 (𝑠)) d𝑊𝑖 (𝑠) − ∫
𝑡𝑘+1
𝑡𝑘
𝑡𝑘+1
−∫
=∫
𝑡𝑘
4 2 2
𝑡𝑘+1
∫ 𝜙𝑖󸀠 (𝑦𝑖 (𝑡𝑘 )) 𝜙𝑖 (𝑦𝑖 (𝑡𝑘 )) d𝑊𝑖 (𝑟) d𝑊𝑖 (𝑠)
+∫
𝑡𝑘
[𝜙𝑖 (𝑥𝑖 (𝑡𝑘 )) − 𝜙𝑖 (𝑦𝑖 (𝑡𝑘 ))] d𝑊𝑖 (𝑠)
𝑡𝑘+1
+∫
𝑠
(27)
∫ 𝜙𝑖󸀠 (𝑥𝑖 (𝑟)) 𝜇𝑖 (𝑥𝑖 (𝑟))
𝑡𝑘+1
𝛾𝑗2 ⏟⏟
Γ⏟3⏟⏟
ℎ⏟2⏟d𝑠 + ∫
𝑡𝑘
(19)
2
Γ4 (𝑠 − 𝑡𝑘 ) d𝑠
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
moment bounded
}
}
}
}
̃1 ℎ3 .
≤Γ
}
}
}
}
Triangle Inequality, (6), (13), (19)}
𝑠
∫
∫ Γ5 ℎd𝑟d𝑠
𝑡𝑘
𝑡𝑘
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
(28)
Finally, it is easy to prove 𝐸(‖𝛿
𝑠
∫ [𝜙𝑖󸀠 (𝑥𝑖 (𝑟)) 𝜙𝑖 (𝑥𝑖 (𝑟)) − 𝜙𝑖󸀠 (𝑦𝑖 (𝑡𝑘 ))
𝑡𝑘
𝑡𝑘
𝑛
𝑘+1 2
1
2
+ 𝜙𝑖󸀠󸀠 (𝑥𝑖 (𝑟)) 𝜙𝑖 (𝑥𝑖 (𝑟)) d𝑟d𝑊𝑖 (𝑠)
2
+∫
𝑡𝑘+1
𝑡𝑘+1
+
𝑡𝑘
𝑡𝑘
𝑛
(6) and (12)
𝑠
𝑡𝑘
𝑡𝑘
𝑡𝑘+1
(15)
ℎ 𝑐𝑖 𝜃
Γ [𝑐𝑖2 + 𝑛 ∑ 𝑎𝑖𝑗2 𝛼j2 + 𝑛 ∑ 𝑏𝑖𝑗2 𝛽𝑗2 ]
2 𝜓
(1 + 𝑐𝑖 𝜃ℎ) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
𝑗=1
𝑗=1
[
]
+
𝜙𝑖 (𝑦𝑖 (𝑡𝑘 )) d𝑊𝑖 (𝑠)
⏟⏟Γ⏟⏟⏟
⏟⏟d𝑠
𝜏ℎ
× 𝜙𝑖 (𝑦𝑖 (𝑡𝑘 )) ] d𝑊𝑖 (𝑟) d𝑊𝑖 (𝑠) .
‖ ) ≤ Γ1 ℎ3 .
Thanks to Theorem 1 in [18], we can conclude that
󵄩 󵄩2
𝐸 (󵄩󵄩󵄩󵄩𝜖𝑘 󵄩󵄩󵄩󵄩 ) ≤ Γℎ2 ∀ℎ ∈ (0, 1) ,
(29)
that is, the mean-square order of global error of the SSTM is
1.
4. Stability of SSTM
We are concerned with the stability of SSTM solution. Since
(1) has an equilibrium solution x(𝑡) ≡ 0, we will discuss
whether the SSTM solution 𝑋𝑘 with a positive step-size can
attain a similar stability when 𝑘 goes to infinity. First we give
a sufficient condition for the exponential stability in meansquare sense of the equilibrium solution. The references [13,
19] give the condition as
From (25) and (27), we have
󵄨 𝑘+1 󵄨2
𝐸 (󵄨󵄨󵄨󵄨𝛿𝑖 󵄨󵄨󵄨󵄨 )
{
{
{
𝑡𝑘+1
{
2
𝐸[𝑥𝑖 (𝑡𝑘 ) − 𝑥𝑖 (𝑠)] d𝑠
≤ 9 {𝑐𝑖 ℎ ∫
{
𝑡𝑘
{
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
{
Cauchy−Schwartz inequality
{
𝑛
𝑛
𝑛
𝑛
𝑛
󵄨 󵄨
󵄨 󵄨
󵄨 󵄨
󵄨 󵄨
∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 + ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 ≤ 𝛼𝑖 ∑ 󵄨󵄨󵄨󵄨𝑎𝑗𝑖 󵄨󵄨󵄨󵄨 + 𝛽𝑖 ∑ 󵄨󵄨󵄨󵄨𝑏𝑗𝑖 󵄨󵄨󵄨󵄨 ,
𝑡𝑘+1
2
̈
Cauchy−Schwartz and Holder
inequality
𝑛
𝑡𝑘+1
+
(1 + 𝑐𝑖 𝜃ℎ)
𝑛𝑐𝑖2 ℎ4 𝜃2
2
󵄨
󵄨2
𝐸󵄨󵄨󵄨𝑥𝑖 (𝑡𝑘 )󵄨󵄨󵄨
2
𝑡𝑘+1
2
+∫
𝛾𝑗2 𝐸[𝑥𝑖 (𝑡𝑘 ) − 𝑦𝑖 (𝑡𝑘 )] d𝑠
𝑡𝑘
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
𝑗=1
𝑗=1
𝑗=1
𝑗=1
𝑗=1
Definition 7. A numerical method is said to be mean-square
stable (MS-stable) if there exists an ℎ0 > 0 such that any
application of the method to problem (1) generates numerical
approximations 𝑋𝑖𝑘 , which satisfies
󵄨 󵄨2
lim 𝐸 (󵄨󵄨󵄨󵄨𝑋𝑖𝑘 󵄨󵄨󵄨󵄨 ) = 0, 𝑖 = 1, 2, . . . , 𝑛,
(31)
𝑘→∞
for all ℎ ∈ (0, ℎ0 ).
It̂
o Isometry
𝑡𝑘+1
𝑠
󵄨
2 󵄨2
+∫
(𝑠−𝑡𝑘 ) ∫ 𝐸󵄨󵄨󵄨󵄨𝜙𝑖󸀠 (𝑥𝑖 (𝑟)) 𝜇𝑖 (𝑥𝑖 (𝑟))+(1/2) 𝜙𝑖󸀠󸀠 (𝑥𝑖 (𝑟)) 𝜙𝑖 (𝑥𝑖 (𝑟)) 󵄨󵄨󵄨󵄨 d𝑟d𝑠
𝑡𝑘
𝑡𝑘
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
Cauchy−Schwartz inequality and It̂
o Isometry
}
}
}
}
󸀠
󸀠
+∫
∫ 𝐸[𝜙𝑖 (𝑥𝑖 (𝑟)) 𝜙i (𝑥𝑖 (𝑟)) − 𝜙𝑖 (𝑦𝑖 (𝑡𝑘 )) 𝜙𝑖 (𝑦𝑖 (𝑡𝑘 ))] d𝑟d𝑠}
}
𝑡𝑘
⏟⏟⏟𝑡⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟}
𝑘
}
It̂
o Isometry
}
𝑠
𝑛
for every 𝑖 (𝑖 = 1, . . . , 𝑛).
𝑛
𝑛
[ ∑ 𝑎2 𝐸󵄨󵄨󵄨󵄨𝑓𝑗 (𝑥𝑗 (𝑡𝑘 ))󵄨󵄨󵄨󵄨2 + ∑ 𝑏2 𝐸󵄨󵄨󵄨󵄨𝑔𝑗 (𝑍
̃𝑗 (𝑡𝑘 ))󵄨󵄨󵄨󵄨2 ]
𝑖𝑗 󵄨
𝑖𝑗 󵄨
󵄨
󵄨
(1 + 𝑐𝑖 𝜃ℎ) 𝑗=1
𝑗=1
[
]
𝑡𝑘+1
𝑗=1
𝑛
(30)
̈
Cauchy−Schwartz and Holder
inequality
𝑐𝑖4 ℎ4 𝜃2
𝑗=1
𝑛
2
̃𝑗 (𝑡𝑘 )] d𝑠
+ 𝑛 ∑ 𝑏𝑖𝑗2 𝛽𝑗2 ℎ ∫
𝐸[𝑥𝑗 (𝑠 − 𝜏𝑗 ) − 𝑍
𝑡𝑘
𝑗=1
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
+
𝑗=1
𝑛
󵄨 󵄨
󵄨 󵄨
󵄨 󵄨
󵄨 󵄨
−2𝑐𝑖 + ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 + ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 + 𝛼𝑖 ∑ 󵄨󵄨󵄨󵄨𝑎𝑗𝑖 󵄨󵄨󵄨󵄨 + 𝛽𝑖 ∑ 󵄨󵄨󵄨󵄨𝑏𝑗𝑖 󵄨󵄨󵄨󵄨 + 𝛾𝑖2 < 0
+ 𝑛 ∑ 𝑎𝑖𝑗2 𝛼𝑗2 ℎ ∫
𝐸[𝑥𝑗 (𝑠) − 𝑥𝑗 (𝑡𝑘 )] d𝑠
𝑡𝑘
𝑗=1
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2
Theorem 8. Assume (6), (30), and (1 − 𝜃)𝑐𝑖 ℎ ≤ 1 are satisfied;
then the SSTM (2) are mean-square stable if ℎ ∈ (0, ℎ0 ), where
min {1, ℎ𝑖 }
{
{1≤𝑖≤𝑛
ℎ0 = {
1
{ min {1,
,ℎ }
1≤𝑖≤𝑛
−
𝜃) 𝑐𝑖 𝑖
(1
{
for 𝜃 = 1,
for 𝜃 ∈ [0, 1) .
(32)
6
Abstract and Applied Analysis
𝑛
Here ℎ𝑖 is the smallest positive root of the cubic equation with
respect to 𝑧 given by
3
2
A𝑖 𝑧 + B𝑖 𝑧 + C𝑖 𝑧 + D𝑖 = 0,
+ ∑ {2ℎ (1 − (1 − 𝜃) 𝑐𝑖 ℎ) 𝑋𝑖𝑘
𝑗=1
× [𝑎𝑖𝑗 𝑓𝑗 (𝑋𝑗𝑘 ) + 𝑏𝑖𝑗 𝑔𝑗 (𝑍𝑗𝑘 )]}
(33)
𝑛
where the coefficients mean
2
A𝑖 =
B𝑖 =
𝑛
𝑛
𝜅𝑖2
󵄨 󵄨
󵄨 󵄨
(−(1 − 𝜃)𝑐𝑖 + ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 + ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 ) ,
2
𝑗=1
𝑗=1
𝛾𝑖2 (−(1
Taking expectations of both sides of (35), we can get
2
𝑛
𝑛
󵄨 󵄨
󵄨 󵄨
− 𝜃)𝑐𝑖 + ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 + ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 )
𝑗=1
󵄨
󵄨2
𝐸 (󵄨󵄨󵄨󵄨𝑋𝑖𝑘+1 󵄨󵄨󵄨󵄨 )
𝑗=1
󵄨 󵄨2
󵄨
󵄨2
= 𝐸 (󵄨󵄨󵄨󵄨𝑌𝑖𝑘 󵄨󵄨󵄨󵄨 ) + ℎ𝐸 (󵄨󵄨󵄨󵄨𝜙𝑖 (𝑌𝑖𝑘 )󵄨󵄨󵄨󵄨 )
𝑛
𝑛
󵄨 󵄨
󵄨 󵄨
+ 𝜅𝑖2 ( ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 + ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 ) − 𝜅𝑖2 (1 − 𝜃) 𝑐𝑖 ,
𝑗=1
𝑗=1
+
2
𝑛
𝑛
󵄨 󵄨
󵄨 󵄨
C𝑖 = (−(1 − 𝜃)𝑐𝑖 + ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 + ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 )
𝑗=1
(34)
𝑛
𝑛
󵄨 󵄨
󵄨 󵄨
+ 2𝛾𝑖2 (− (1 − 𝜃) 𝑐𝑖 + ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 + ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 )
−
𝑐𝑖2 𝜃2
ℎ2 󵄨󵄨 󸀠 𝑘
󵄨2
𝐸 (󵄨󵄨󵄨𝜙𝑖 (𝑌𝑖 ) 𝜙𝑖 (𝑌𝑖𝑘 )󵄨󵄨󵄨󵄨 )
2
≤ (1 + 𝛾𝑖2 ℎ +
𝑗=1
𝑗=1
𝑛
+ 2ℎ2 [ ∑𝑎𝑖𝑗 𝑓𝑗 (𝑋𝑗𝑘 )] [∑ 𝑏𝑖𝑗 𝑔𝑗 (𝑍𝑗𝑘 )] .
] [𝑗=1
]
[𝑗=1
(36)
(37)
ℎ2 2
󵄨 󵄨2
𝜅 ) 𝐸 (󵄨󵄨󵄨󵄨𝑌𝑖𝑘 󵄨󵄨󵄨󵄨 ) .
2 𝑖
Together with (36), we have
𝑗=1
󵄨
󵄨2
𝐸 (󵄨󵄨󵄨󵄨𝑋𝑖𝑘+1 󵄨󵄨󵄨󵄨 )
𝜅2
+ 𝑖,
2
𝑛
𝑛
𝑗=1
𝑗=1
≤ (1 + 𝛾𝑖2 ℎ +
󵄨 󵄨
󵄨 󵄨
D𝑖 = 2 (−𝑐𝑖 + ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 + ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 ) + 𝛾𝑖2 .
≤
for 𝑖 = 1, 2, . . . , 𝑛.
Squaring on both sides of (4b) and (4a), we have
+ [𝜙𝑖󸀠 (𝑌𝑖𝑘 ) 𝜙𝑖 (𝑌𝑖𝑘 )] [
[
2
(Δ𝑊𝑖𝑘 ) − ℎ
2
+ [(𝜙𝑖 (𝑌𝑖𝑘 )) Δ𝑊𝑖𝑘 + 𝑌𝑖𝑘 ]
2
× 𝜙𝑖󸀠 (𝑌𝑖𝑘 ) 𝜙𝑖 (𝑌𝑖𝑘 ) [(Δ𝑊𝑖𝑘 ) − ℎ]
2
󵄨2
2󵄨
(1 + 𝜃𝑐𝑖 ℎ) 󵄨󵄨󵄨󵄨𝑌𝑖𝑘 󵄨󵄨󵄨󵄨 = [(1 − (1 − 𝜃) 𝑐𝑖 ℎ) 𝑋𝑖𝑘 ]
+ℎ
[
2
𝑛
∑ 𝑎𝑖𝑗 𝑓𝑗 (𝑋𝑗𝑘 )]
𝑗=1
]
𝑛
2
(1 + 𝜃𝑐𝑖 ℎ)
𝑛
𝑛
󵄨 󵄨
󵄨 󵄨2
󵄨 󵄨
+ ℎ2 (∑ 󵄨󵄨󵄨𝑎𝑖𝑟 󵄨󵄨󵄨 𝛼𝑟 ) ( ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 𝐸 (󵄨󵄨󵄨󵄨𝑋𝑗𝑘 󵄨󵄨󵄨󵄨 ))
]
𝑟=1
(35)
𝑗=1
𝑛
𝑛
󵄨 󵄨
󵄨 󵄨2
󵄨 󵄨
+ ℎ2 (∑ 󵄨󵄨󵄨𝑏𝑖𝑟 󵄨󵄨󵄨 𝛽𝑟 ) ( ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 𝐸 (󵄨󵄨󵄨󵄨𝑍𝑗𝑘 󵄨󵄨󵄨󵄨 ))
𝑟=1
𝑗=1
𝑛
󵄨
󵄨
+ ∑ℎ 󵄨󵄨󵄨1 − (1 − 𝜃) 𝑐𝑖 ℎ󵄨󵄨󵄨
𝑗=1
+ 2𝑌𝑖𝑘 𝜙𝑖 (𝑌𝑖𝑘 ) Δ𝑊𝑖𝑘 ,
2[
2
]
1 + 𝛾𝑖2 ℎ + (ℎ2 /2) 𝜅𝑖2
{
󵄨 󵄨2
2
× {[1 − (1 − 𝜃) 𝑐𝑖 ℎ] 𝐸 (󵄨󵄨󵄨󵄨𝑋𝑖𝑘 󵄨󵄨󵄨󵄨 )
{
2
2
󵄨󵄨 𝑘+1 󵄨󵄨2
󵄨󵄨𝑋𝑖 󵄨󵄨 = (𝑌𝑖𝑘 ) + [𝜙𝑖 (𝑌𝑖𝑘 ) Δ𝑊𝑖𝑘 ]
󵄨
󵄨
2
ℎ2 2
󵄨 󵄨2
𝜅 ) 𝐸 (󵄨󵄨󵄨󵄨𝑌𝑖𝑘 󵄨󵄨󵄨󵄨 )
2 𝑖
+ ℎ2 [∑ 𝑏𝑖𝑗 𝑔𝑗 (𝑍𝑗𝑘 )]
]
[𝑗=1
2
󵄨 󵄨
󵄨 󵄨2
󵄨 󵄨2
× [󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 [𝐸 (󵄨󵄨󵄨󵄨𝑋𝑖𝑘 󵄨󵄨󵄨󵄨 ) + 𝐸 (󵄨󵄨󵄨󵄨𝑋𝑗𝑘 󵄨󵄨󵄨󵄨 )]
󵄨 󵄨
󵄨 󵄨2
󵄨 󵄨2
+ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 [𝐸 (󵄨󵄨󵄨󵄨𝑋𝑖𝑘 󵄨󵄨󵄨󵄨 ) + 𝐸 (󵄨󵄨󵄨󵄨𝑍𝑗𝑘 󵄨󵄨󵄨󵄨 )]]
𝑛
󵄨 󵄨
+ 2ℎ2 ( ∑ 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 )
𝑗=1
𝑛
󵄨 󵄨
󵄨 󵄨2
󵄨 󵄨2 }
× ( ∑ 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 ) [𝐸 (󵄨󵄨󵄨󵄨𝑋𝑗𝑘 󵄨󵄨󵄨󵄨 ) + 𝐸 (󵄨󵄨󵄨󵄨𝑍𝑗𝑘 󵄨󵄨󵄨󵄨 )]} .
𝑗=1
}
(38)
Abstract and Applied Analysis
7
Thus, we attain
𝑛
𝑛
󵄨
󵄨2
𝐸 (󵄨󵄨󵄨󵄨𝑋𝑖𝑘+1 󵄨󵄨󵄨󵄨 ) ≤ [𝑃 (ℎ) + ∑ 𝑄𝑗 (ℎ) + ∑𝑅𝑗 (ℎ)]
𝑗=1
𝑗=1
[
]
󵄨 󵄨2
󵄨 󵄨2
󵄨 󵄨2
× max {𝐸 (󵄨󵄨󵄨󵄨𝑋𝑖𝑘 󵄨󵄨󵄨󵄨 ) , 𝐸 (󵄨󵄨󵄨󵄨𝑋𝑗𝑘 󵄨󵄨󵄨󵄨 ) , 𝐸 (󵄨󵄨󵄨󵄨𝑍𝑗𝑘 󵄨󵄨󵄨󵄨 )} ,
1≤𝑗≤𝑛
(39)
where
𝑃 (ℎ) =
1 + 𝛾𝑖2 ℎ + (ℎ2 𝜅𝑖2 /2)
(1 + 𝜃𝑐𝑖 ℎ)
}
󵄨 󵄨
󵄨 󵄨
× (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 + 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 ) } ,
}
1+
𝑥 (𝑡)
𝑓 (𝑥1 (𝑡))
𝑥 (𝑡)
) d𝑡
𝑑 ( 1 ) = − 𝐶 ( 1 ) d𝑡 + 𝐴 (
𝑥2 (𝑡)
𝑥2 (𝑡)
𝑓 (𝑥2 (𝑡))
𝑔 (𝑥1 (𝑡 − 𝜏1 ))
) d𝑡 + Φd𝑊 (𝑡)
+ 𝐵(
𝑔 (𝑥2 (𝑡 − 𝜏2 ))
(43)
on 𝑡 ⩾ 0 with the initial condition 𝑥1 (𝑡) = 𝑡 + 1, 𝑡 ∈ [−1, 0]
and 𝑥2 (𝑡) = 𝑡 + 1, 𝑡 ∈ [−2, 0].
(1
Case 1. Let 𝑓(𝑥) = sin(𝑥), 𝑔(𝑥) = arctan(𝑥),
+ (ℎ2 𝜅𝑖2 /2)
2
+ 𝜃𝑐𝑖 ℎ)
(41)
+ (ℎ2 𝜅𝑖2 /2)
2
+ 𝜃𝑐𝑖 ℎ)
𝐶=(
𝑛
󵄨 󵄨
󵄨 󵄨
󵄨 󵄨
× {ℎ (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 + 󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 ) (∑ 󵄨󵄨󵄨𝑏𝑖𝑟 󵄨󵄨󵄨 𝛽𝑟 )
2
4 −5
𝐴=(
),
6 3
0
𝑥 (𝑡)
).
Φ=( 1
0 −√5𝑥2 (𝑡)
(44)
Case 2. Let 𝑓(𝑥) = 𝑥, 𝑔(𝑥) = tanh(𝑥),
𝛾𝑖2 ℎ
(1
20 0
),
0 20
𝐶=(
−6 4
𝐵=(
),
3 1
}
󵄨 󵄨 󵄨
󵄨
+ ℎ 󵄨󵄨󵄨1 − (1 − 𝜃) 𝑐𝑖 ℎ󵄨󵄨󵄨 (󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 ) } ,
}
1+
for the value of explicit solution of (1) at time 𝑇 = 3 and 𝑋𝜔𝐾𝑟 is
its numerical approximation along the rth sample path {𝜔𝑟 :
𝑟 = 1, 2, . . . , 2000}. We compute the numerical solution using
the split-step 𝜃-Milstein method (2) with step-size ℎ = 2−12 ,
and we will call this the “exact solution.”
𝛾𝑖2 ℎ
{ 󵄨 󵄨 𝑛 󵄨 󵄨
󵄨 󵄨
× {ℎ2 󵄨󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨󵄨 𝛼𝑗 ∑ (󵄨󵄨󵄨𝑎𝑖𝑟 󵄨󵄨󵄨 𝛼𝑟 + 󵄨󵄨󵄨𝑏𝑖𝑟 󵄨󵄨󵄨 𝛽𝑟 )
𝑗=1
{
𝑅𝑗 (ℎ) =
2
𝐾
where 𝜀 := √(1/2000) ∑2000
𝑟=1 ‖𝑋𝜔𝑟 − x(𝑇)‖ . Here x(𝑇) stands
Example 9. Consider the following two-dimensional stochastic delay Hopfield neural networks of the form
2
𝑛
{
2
󵄨
󵄨
× {[1 − (1 − 𝜃) 𝑐𝑖 ℎ] + ℎ∑ 󵄨󵄨󵄨1 − (1 − 𝜃) 𝑐𝑖 ℎ󵄨󵄨󵄨 (40)
𝑗=1
{
𝑄𝑗 (ℎ) =
split-step 𝜃-method in [13], which has strong convergence
order 0.5.
The mean-square error 𝜀 of numerical approximations at
time 𝑇 versus the step-size is depicted in log-log diagrams,
(42)
𝑟=1
󵄨
󵄨 󵄨 󵄨
+ ℎ 󵄨󵄨󵄨1 − (1 − 𝜃) 𝑐𝑖 ℎ󵄨󵄨󵄨 (󵄨󵄨󵄨󵄨𝑏𝑖𝑗 󵄨󵄨󵄨󵄨 𝛽𝑗 ) } .
Note that the assumption of Theorem implies the nonnegativity of 1 − (1 − 𝜃)𝑐𝑖 ℎ.
Proof. Obviously, when 𝑘 → +∞, 𝐸(|𝑋𝑖𝑘 |2 ) → 0 if the
inequality 𝑃(ℎ) + ∑𝑛𝑗=1 𝑄𝑗 (ℎ) + ∑𝑛𝑗=1 𝑅𝑗 (ℎ) < 1 holds, which is
equivalent to the inequality A𝑖 ℎ3 + B𝑖 ℎ2 + C𝑖 ℎ + D𝑖 < 0.
Furthermore, it is easy to prove A𝑖 > 0 and D𝑖 < 0
by virtue of (30). By Vieta’s formulas, the product of three
roots of (33) satisfies 𝑧1 𝑧2 𝑧3 = −(D𝑖 /A𝑖 ) > 0. This means
that (33) has at least one positive root. Therefore, let ℎ𝑖 denote
the smallest positive root of the equation. Moreover, we note
that at the origin the right-hand side polynomial of (33) is
negative. This completes the proof.
5. Numerical Results
Now, we apply the introduced SSTM method to two test cases
of SDHNNs in order to compare their performance with the
10 0
),
0 10
5 −1
𝐵=(
),
1 5
𝐴=(
0 0
),
0 0
0
1.5𝑥1 (𝑡)
).
Φ=(
0
−1.5𝑥2 (𝑡)
(45)
In Figure 1, SSTM is applied with 7 different step-sizes:
ℎ𝑚 = 2𝑚−12 for 𝑚 = 1, 2, . . . , 7. Two pairs of time
delays (𝜏1 , 𝜏2 ) are set to (1, 2) and (1.13, 2.31). The first
pair has common factor ℎ𝑚 ; however, the second pair is
incommensurable by ℎ𝑚 . The computation errors 𝜀 versus
step-sizes ℎ are plotted on a log-log scale and the reference
lines of slope 1 are added. It illustrates that SSTM has raised
the strong order of the split-step 𝜃-method at least to 1 for
SDHNNs [13].
Next, Table 1 shows a comparison of stability intervals
between the SST and the SSTM for (43). Two sets of the
interval in the Table are calculated through Theorem 8 in this
paper and Theorem 5.1 in [13]. It is easy to see that the stability
intervals of the two methods are similar.
We know that Theorem 5.1 in [13] and Theorem 8 in this
paper only give sufficient conditions of mean-square stability.
Therefore the stability intervals given by these theorems
are only subsets of real ones. To confirm the situation, we
calculated the sample moments of the approximate solution
and plotted them along the time 𝑡. Here the sample moment 𝜂𝑖
𝑖 2
𝑖
means (1/2000) ∑2000
𝑟=1 ‖𝑋𝜔𝑟 ‖ for the numerical solution 𝑋𝜔𝑟
8
Abstract and Applied Analysis
10−1
10−1
10−2
Error 𝜀
Error 𝜀
10
−2
10−3
10−3
10−4
10−4
10−4
10−3
10−2
10−1
10−5
10−4
10−3
Step-size ℎ
Step-size ℎ
𝜃 = 0.0
𝜃 = 0.5
10−1
10−2
𝜃 = 0.0
𝜃 = 0.5
𝜃 = 1.0
Reference line
(a) Case 1 with 𝜏1 = 1, 𝜏2 = 2
𝜃 = 1.0
Reference line
(b) Case 2 with 𝜏1 = 1, 𝜏2 = 2
10−1
10−2
10−2
Error 𝜀
Error 𝜀
10−1
10−3
10−3
10−4
10−4
10−3
10−2
10−1
10−4
10−4
10−3
Step-size ℎ
𝜃 = 0.0
𝜃 = 0.5
10−1
10−2
Step-size ℎ
𝜃 = 1.0
Reference line
𝜃 = 0.0
𝜃 = 0.5
(c) Case 1 with 𝜏1 = 1.13, 𝜏2 = 2.31
𝜃 = 1.0
Reference line
(d) Case 2 with 𝜏1 = 1.13, 𝜏2 = 2.31
Figure 1: Errors versus step-sizes for the SDHNNs (43).
Table 1: Calculated intervals of step-size for mean-square stability of the numerical schemes.
Numerical scheme
Split-step 𝜃-Milstein method (𝜃 = 0.0)
Split-step 𝜃-Milstein method (𝜃 = 0.2)
Split-step 𝜃-Milstein method (𝜃 = 0.5)
Split-step 𝜃-Milstein method (𝜃 = 0.8)
Split-step 𝜃-Milstein method (𝜃 = 1.0)
Split-step 𝜃-method (𝜃 = 0.0)
Split-step 𝜃-method (𝜃 = 0.2)
Split-step 𝜃-method (𝜃 = 0.5)
Split-step 𝜃-method (𝜃 = 0.8)
Split-step 𝜃-method (𝜃 = 1.0)
Case 1
(0.0000, 0.0500]
(0.0000, 0.0625]
(0.0000, 0.1000]
(0.0000, 0.0650)
(0.0000, 0.0500)
(0.0000, 0.0500]
(0.0000, 0.0625]
(0.0000, 0.1000]
(0.0000, 0.0689)
(0.0000, 0.0540)
Case 2
(0.0000, 0.1000]
(0.0000, 0.1250]
(0.0000, 0.2000]
(0.0000, 0.5000]
(0.0000, 0.3534)
(0.0000, 0.1000]
(0.0000, 0.1250]
(0.0000, 0.2000]
(0.0000, 0.5000]
(0.0000, 0.5792)
Abstract and Applied Analysis
9
1010
108
108
106
10
4
10
2
104
102
𝜂
𝜂
106
100
100
10−2
10−2
10−4
10−4
10−6
0
10
20
30
40
Time 𝑡
50
60
70
10−6
80
0
10
20
30
40
50
60
70
80
60
70
80
Time 𝑡
ℎ = 1/10
ℎ = 1/11
ℎ = 1/12
ℎ = 1/10
ℎ = 1/11
ℎ = 1/12
(a) SST
(b) SSTM
Figure 2: Mean-square stability comparison of SST and SSTM for the SDHNNs (43), Case 1, 𝜃 = 0.
1010
108
106
105
104
100
𝜂
𝜂
102
100
10−2
10−4
10−5
10−6
10−10
0
10
20
30
40
Time 𝑡
ℎ = 1/5
ℎ = 1/6
50
60
70
80
ℎ = 1/7
ℎ = 1/8
(a) SST
10−8
0
10
20
30
40
Time 𝑡
ℎ = 1/5
ℎ = 1/6
50
ℎ = 1/7
ℎ = 1/8
(b) SSTM
Figure 3: Mean-square stability comparison of SST and SSTM for the SDHNNs (43), Case 2, 𝜃 = 0.
approximating x(𝑡𝑖 ) along the 𝑟th sample path. Figures 2, 3, 4,
5, and 6 depict the results by SST and SSTM in the log-scaled
vertical axis. All the figures can give a rough estimate of the
stability interval in each case.
6. Concluding Remarks
We introduce the split-step 𝜃-Milstein method (SSTM),
which exhibits higher strong convergence rate than the
split-step 𝜃-method (SST, see [13]) for a stochastic delay
Hopfield neural networks, and the scheme proposed in this
paper can deal with incommensurable time delays which
were not considered in [13]. We give the proof of convergence
results, which has generally been omitted in the previous
works on the same subject. By comparing the stability
intervals of step size for the SST and SSTM for a test example,
we find they exhibit similar mean-square stability.
In this paper, we have found a delay-independent
sufficient condition for mean-square stability of split-step
𝜃-Milstein method applied to nonlinear stochastic delay
10
Abstract and Applied Analysis
1015
1010
1010
105
𝜂
𝜂
105
10
100
0
10−5
10−5
10−10
0
10
20
30
40
Time 𝑡
50
60
70
80
10−10
0
10
ℎ = 1/6
ℎ = 1/7
ℎ = 1/8
20
30
40
Time 𝑡
50
60
70
80
ℎ = 1/6
ℎ = 1/7
ℎ = 1/8
(a) SST
(b) SSTM
Figure 4: Mean-square stability comparison of SST and SSTM for the SDHNNs (43), Case 1, 𝜃 = 0.2.
1010
1020
1015
105
105
𝜂
𝜂
10
10
100
100
10
10−5
−5
10−10
0
10
20
30
40
Time 𝑡
50
60
70
80
ℎ = 1/5
ℎ = 1/6
ℎ = 1/3
ℎ = 1/4
10−10
0
10
20
30
40
Time 𝑡
ℎ = 1/3
ℎ = 1/4
(a) SST
50
60
70
80
ℎ = 1/5
ℎ = 1/6
(b) SSTM
Figure 5: Mean-square stability comparison of SST and SSTM for the SDHNNs (43), Case 2, 𝜃 = 0.2.
Hopfield neural networks. Further, Figure 6 suggests that
the value of ℎ0 , the right end-point of the stability interval,
given by Theorem 5.1 in [13] and Theorem 8 in this paper
is much smaller than the true value when 𝜃 is close to
unity. In this case, we need other techniques for stability
analysis in this kind of stochastic delay differential system.
To the best of our knowledge, the works in [20, 21] put
forward good attempts. On the other hand, with respect to
stochastic delay differential equations, some other types of
stability have been successfully discussed for the Euler-type
scheme, for example, mean-square exponential stability [12],
delay-dependent stability [22], delay-dependent exponential
stability [23], and almost sure exponential stability [24].
To Milstein-type scheme, in view of more sophisticated
derivations, these issues would be challenging for future
research.
Acknowledgment
The authors would like to thank Dr. E. Buckwar and Dr.
A. Rathinasamy for their valuable suggestions. The authors
also thank the editor and the anonymous referees, whose
Abstract and Applied Analysis
11
102
100
100
10−5
10−2
10−10
10−4
𝜂
𝜂
105
10−15
10−6
10−20
10−8
10−25
10−10
10−30
0
20
40
60
Time 𝑡
SSTM, 𝜃 = 0.8, ℎ = 1
SSTM, 𝜃 = 1.0, ℎ = 1
80
100
120
10−12
0
10
20
30
40
50
60
70
80
90
100
Time 𝑡
SST, 𝜃 = 0.8, ℎ = 1
SST, 𝜃 = 1.0, ℎ = 1
(a) Case 1
SSTM, 𝜃 = 0.8, ℎ = 1
SSTM, 𝜃 = 1.0, ℎ = 1
SST, 𝜃 = 0.8, ℎ = 1
SST, 𝜃 = 1.0, ℎ = 1
(b) Case 2
Figure 6: Mean-square stability comparison of SST and SSTM for the SDHNNs (43).
careful reading and constructive feedback have improved
this work. The first author is partially supported by EInstitutes of Shanghai Municipal Education Commission (no.
E03004), National Natural Science Foundation of China
(no. 10901106), Natural Science Foundation of Shanghai (no.
09ZR1423200), and Innovation Program of Shanghai Municipal Education Commission (no. 09YZ150). The third author
is partially supported by the Grants-in-Aid for Scientific
Research (no. 24540154) supplied by Japan Society for the
Promotion of Science.
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