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Page 6 of 15 Sun et al. Complex Eng Syst 2022;2:17 I http://dx.doi.org/10.20517/ces.2022.48

where ( ) = ( − ). Under the LLC, system (14) has a unique maximal global solution on [− , ),
∞

denoted as ( ), where is the explosion time. Then, we prove = ∞ a.s. Thus, it is necessary to define
∞ ∞
the stopping time sequence. Let 0 be a constant sufficiently large to satisfy 0 > | (0)|. For any integer
≥ 0, we define the stopping time sequence as follows:

, = inf{ ∈ [0, ), | ( )| ⩾ }.
∞
Clearly , , increases as → ∞ and therefore we set ∞, := lim , . Observe that ∞, ≤ ∞, a.s. Thus,
→∞
∞, = ∞, a.s., which yields ∞, = ∞ a.s. From the It ˆ formula and condition (9), it is easily proven that

( ∧ , ) Λ( ∧ , , ( ∧ , ), )
∫
∧ ,

= 0 Λ( 0 , ( 0 ), ) + [ Λ( , ( ), ) + LΛ( , ( ), )]
0
∫
∧ ,

≤ 0 Λ( 0 , ( 0 ), ) + [ 2 | ( )| − 1 ( ( )) + 2 ( ( ))
0

− 3 | ( )| + 4 | ( )| ] .
By Lemma 1, we have
∫
∧ ,

( ( − ))
0
∫
∧ ,

≤ ¯ ( ( ))
0 −
∫ ∫
0 ∧ ,

≤ ¯ ( ( )) + ¯ ( ( )) , (15)
0 − 0
and
∫
∧ ,

| ( − )|
0
∫
∧ ,

≤ ¯ | ( )|
0 −
∫ ∫
0 ∧ ,

≤ ¯ | ( )| + ¯ | ( )| . (16)
0 − 0
Hence,
( ∧ , ) Λ( ∧ , , ( ∧ , ), )
∫ ∫
∧ , ∧ ,

≤ − ( 3 − 4 ¯ − 2 ) | ( )| − ( 1 − 2 ¯ ) ( ( )) ,
0 0
where
( ) ( ∫ )
0
= sup 0 Λ( 0 , , ) + 2 ¯ sup ( )
0
[ 0 − , 0 ] [ 0 − , 0 ] 0 −
( ∫ )
0

+ 4 ¯ sup 0 | |
[ 0 − , 0 ] 0 −
is a finite constant. Applying (10) and (11) from Assumption 3, we can deduce that
( ∧ , ) Λ( ∧ , , ( ∧ , ), ) ≤ .