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Sun et al. Complex Eng Syst 2022;2:17 I http://dx.doi.org/10.20517/ces.2022.48 Page 11 of 15

[4]
From the well-known Borel-Cantelli lemma , it follows that for almost all ∈ Ω, there exists a positive inte-
ger 0 = 0 ( ) such that

sup | ( )| ≤ −0.25 ˆ .
≤ ≤ +1
Therefore, for almost all ∈ Ω,

1 0.25 ˆ
ln(| ( )|) ≤ − , ∈ [ , + 1], ≥ 0 .
( + 1)
Then, we can obtain

1

lim sup ln(| ( )|) ≤ −0.25 ˆ < 0 . .
→∞
which is the required assertion in (29). Thus, the proof is completed.
So far, we can conclude that under Assumptions 1-3, system (1) is not only th moment exponentially stable
but also almost surely exponentially stable.

Remark 5 In general, for a stochastic nonlinear system, the th moment exponential stability does not imply
almost surely exponential stability without any imposed conditions. However, this result can be ensured using the
PGC (6). Similar arguments can be found in [4,13] .

Remark 6 Inthispaper, thehighlynonlinearSSSswithtime-varyingdelaysareconsidered, inwhichtheswitching
signal is deterministic and differs from those considered in [13,16,29–32] . In the current study on stochastic systems
with Markovian switching [13,16,29–32] , matrix theory is an efficient tool for achieving stochastic stability. How-
ever, this method is not valid for SSS (1) because a deterministic switching signal rather than the Markovian
switching signal is involved in (1). In this paper, a new stability analysis based on the ADT method coupled with
the MLF approach is developed for SSSs. In our proof, the Lyapunov functions do not need to be specified initially,
which increases the flexibility for the choice of Lyapunov functions in practice.

4. NUMERICAL EXAMPLE
In this section, a numerical example is presented to validate the derived results. Consider the following highly
nonlinear SSS with a time-varying delay:

( ) = ( ) ( , ( ), ( − ( ))) + ( ) ( , ( ), ( − ( ))) ( ), (30)

1
3
where the time-varying delay = 1 + | sin(10 )|, the initial data ( ) = = 0.1 with − ≤ ≤ 0, and
2 4 4
1 2 1 1 2
3
1 ( , , ) = − − + , 1 ( , , ) = + ,
3 4 4
1 4 3 1 2 1 2
2 ( , , ) = − + − + , 2 ( , , ) = .
3 3 3 4
. It is not difficult to verify that Assumption 1 holds
6
In addition, we set Λ( , , 1) = and Λ( , , 2) = 11 6
12
4
3
1
with 1 = , = , and ¯ = , and 1 , 2 , 1 , 2 satisfy Assumption 2. Then, we have 1 = 11 , 2 = 1 and
2 4 3 12
1 = 2 = , which satisfy (7) and (8). A direct computation yields
11
10
1 30 1 1
5 3 2 4 2 2
L ( , , , 1) = 6 (− − + ) + ( + )
3 2 4 4
407 8 169 8 93 6 67 6
≤ − + − + .
112 112 28 56