1
56.2
3.
!? > 0;b > 0,p;q (1??
Oq? 2.ó?Z?F
dx
dt = 1x?x
pyq; dy
dt = b(x
pyq?y);
TM
MD,
P ?è3(x(t);y(t)) · (1?;0)???Z?F￥
,3i)
×??.
3:TMDu = x? 1?;v = y5eZ?FM?1
du
dt =u?(u+
1
)
pvq; dv
dt = b((u+
1
)
pvq?v);
L??s￥"
?1
A =
0
@
0
0?1
1
A
￥
?+?;?1 (1μ
L
yN?? ?2.2?eZ?F￥?3(x(t);y(t)) · (1?;0)ví×
?b
4. I
n/

?Z?F
dx
dt = (A+B(t))x; (1)
dx
dt = Ax; (2)
?A1è
′ ?,B(t)1t? 0
￥ ?? ?′f
, O
@Hq
Z +1
0
jB(t)jdt < 1;
¨? ?2.2￥￡
üZE￡
ü ?(2)￥
μ3?t? 0
Hμ?,5(1)￥
μ3?t? 0
H9μ?.
￡
ü:
n5y1(1)(2)?
^L?Z?F O·H￥"
? ??,yN
ì￥
μ3￥Kvi
uW (1t 2 (?1;+1).
C
!Φ(t)
^Z?F(2)￥
@Φ(0) = E￥'3 ?b?è
M^
T(1)
@′H
qx(0) = x0￥31
x(t) = Φ(t)x0 +
Z t
0
Φ(t)B(?)x(?)d?,(3)
?L
!iè
K > 0
P¤?t? 0
H
jΦ(t)j? K;
yN?(3)??t? 0
H
jx(t)j? Kjx0j+K
Z t
0
jB(?)jjx(?)jd?:
?Gronwall??
T¤?t? 0
H
jx(t)j? Kjx0jexp(K
Z t
0
jB(?)jd?):
2
?L
!
Z +1
0
jB(t)jdt =?K < 1;
yN?t? 0
H
jx(t)j? Kjx0jexp(K
Z +1
0
jB(t)jdt) = Kjx0jeK?K < 1:
'(1)￥
μ3?t? 0
H9μ?.
6.
!fi,fl,,–,??
^?
,x? 0,y? 0, pZ?F
dx
dt =?fix+flx
2? xy; dy
dt =?–y +?xy
￥
μ?è3i)
×??.
3: p3}
Z?F
fix+flx2? xy = 0;?–y +?xy = 0
¤eZ?Fμ ???3
I, (x(t);y(t)) · (0;0); II, (x(t);y(t)) · (fifl;0);
III, (x(t);y(t)) · (–?; fl– fi ):
?3I, L??s￥"
?1
0
@
fi 0
0?–
1
A
￥
?+?fi;?– (1μ
L
yNví×?b
?3II, L??s￥"
?1
0
@
fi?fi fl
0?– + fi?fl
1
A
à
μB??+??fiyN?×?b
?3III, L??s￥"
?1
0
@
fl–

–
fl–?fi?
0
1
A
7
= (fl–? )2? 4–(fl–?fi?)?,
 < 0
H
￥+??1B
aˉ
1;2 = fl–2? § 12pi;
 = 0
H
￥+??1
L×?fl–2?; > 0
H
￥+??1Ms
L?
1;2 = fl–2? § 12p?:
? ?? f ?/
à
μB?+??μ?
L?yN?×?.
3
56.3
7.
!fi;fl; ;–?
^?
,fl?fi– < 0,f
f(y) ?? V±,f(0) = 0 O?y 6= 0
Hμyf(y) > 0. ?¨
? ?
V = 12Ax2 +B
Z y
0
f(u)du
￥Liapunovf
)
Z?F
dx
dt =?fix+flf(y);
dy
dt = x?–f(y)
￥
,3￥×??.
3:/Lyapunovf
V(x;y) = 12 x2 +fl
Z y
0
f(u)du
^??￥  ??
1
dV
dt = x(?fix+flf(y))+flf(y)( x?–f(y))
=?fi x2 +2fl xf(y)?fl–(f(y))2:
?? 
Y
T? = 4fl2 2?4fl fi– = 4fl (fl?fi–) < 0,yNdVdt1?μf
,?? ?3.1?eZ?￥
,3ví×?.