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: Guo S J, Wu J H. Generalized Hopf bifurcation in delay dierential equations (in Chinese). Sci Sin Math, 2012,42(2): 91{105, doi: 10.1360/012010-1047
: 2012 42 2 : 91 105www.springerlink.com math.scichina.com
Hopf , , 410082; Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, Canada M3J 1P3
E-mail: [email protected], [email protected]
: 2010-12-31; : 2011-12-23; * (: 10971057) (: 10JJ1001) NSERC
Lyapunov-Schmidt , , k , , , van der Pol.
Lyapunov-Schmidt Hopf van der Pol MSC (2000) 34K18, 92B20
1
> 0, C = C([; 0);Rn) [; 0] Rn , 2 C kk = sup660 j()j, C = C([; 0);Rn) Banach . , t0 2 R, > 0, x : [t0 ; t0 + ]! Rn , t 2 [t0; t0 + ], xt 2 C xt() = x(t+ ), 2 [; 0].
_x(t) = L()xt + f(; xt); (1.1)
2 R, xt 2 C . , L() : C ! Rn 2 R , f 2Cl(RC ;Rn) l , f(; 0) = 0, 0 (1.1), f(; )
Frechet = 0 Df(; 0) . , ,.,
= 0 . Hopf,,. 1942, Hopf [1] Hopf. Hopf Chafee [2] 1971 . 1977 , Chow Mallet-Paret [3] Hopf. Hopf ( [4, 5]), Hopf : , ; ,
, ; , ,
-
: Hopf
, . , , . Hopf , . , 2 . ,
, 2 . . , Faria Magalhaes [6], Liu [7] . ,
. , Wu [8] Hopf . , Guo [9],Guo Lamb [10] Hopf , .
,, van der Pol,, Hopf . ,
Hopf Hopf. , Lyapunov-Schmidt Hopf S1- ().
, Hopf . , , 1 k. Hopf . [11{16], , . , , , Hopf, [17,18]. Caprino [19] Vanderbauwhede [20] , Lyapunov-Schmidt Hopf , , . , ., van der Pol .
: 2 ; 3 Banach Lyapunov-Schmidt ; 4 Hopf ; van der Pol .
2
, (1.1) :
_x(t) = L()xt: (2.1)
2 C , x(;; ) (2.1) x0(;; ) = , T(t) :C ! C xt(;; ) = T(t). L() T(t) . Hale Verduyn Lunel [4] , n n (; ) : [; 0]! Rn2 ,
L()' =
Z 0
d(; )'(); ' 2 C .
92
-
: 42 2
fT(t) : t > 0g , A : C ! C
A = _; 2 Dom(A) = f 2 C : _ 2 C ; _(0) = L()g:
A , (A) = f 2 C : (; )v = 0 v 2 Cn n f0g g,
(; ) = Idn Z 0
ed(; ).
, Idm (m ) m ; IdC C ; j(; ) (j ) (; ) j- ; (; ) (; ) . ,0(; ) = (; ).
(1.1) Hopf :(NS) A0 i!, k > 1 dimCKer((A0i!Id)j) = min(j; k) j 2 N . , A0 i! .
(NS) i! 1, k. , k = 1 , i! , Hopf . k > 1 , k Ker((A0 i!Id)k). ,
Cn; = Ker((A0 i!Id)k) Ran((A0 i!Id)k):
, Ker((A0i!Id)k) A0Ker((A0i!Id)k) Ker((A0i!Id)k). f'1; : : : ; 'kg Ker((A0i!Id)k) , j = 2; 3; : : : ; k
(A0 i!Id)'1 = 0; (A0 i!Id)'j = 'j1: (2.2)
, : 2.1 'j(t) =Pj1s=0 1s! tsujsei!t, j = 1; 2; : : : ; k, uj 2 Cn (j = 1; 2; : : : ; k)
j1Xs=0
1
s!s(0; i!)ujs = 0; (2.3)
s(; ) (; ) s , 0(; ) = (; ). (A0 i!Id)'1 = 0, '1(t) = ei!tu1, u1 2 Cn (0; i!)u1 = 0.
j = 2; 3; : : : ;m (m 6 k) 2.1 . (A0 i!Id)'m+1 = 'm , 'm+1
_'(t) i!'(t) = 'm(t): (2.4)
L(0)' i!'(0) = 'm(0): (2.5) (2.4) '(t) = 'm+1(t), um+1 2 Cn. (2.5)
0 =mXs=0
1
s!
Z 0
(t+ )sd(0; )um+1sei! i!tsum+1s
m1Xs=0
1
s!tsums
= (0; i!)um+1 +m1Xs=0
Z 0
(t+ )s+1
(s+ 1)!d(0; )umsei! i! t
s+1
(s+ 1)!ums t
s
s!ums
93
-
: Hopf
= (0; i!)um+1 m1Xs=0
s+1Xl=0
tl
(s+ 1 l)!l!s+1l(0; i!)ums
= mXs=0
1
s!s(0; i!)um+1s
m1Xs=0
sXl=0
tl+1
(s l)!l!sl(0; i!)ums
= mXs=0
1
s!s(0; i!)um+1s
m1Xl=0
tl+1
l!
m1Xs=l
1
(s l)!sl(0; i!)ums
= mXs=0
1
s!s(0; i!)um+1s
m1Xl=0
tl+1
l!
ml1Xs=0
1
s!s(0; i!)umls
= mXs=0
1
s!s(0; i!)um+1s:
, ., :
( ;') = T(0)'(0)
Z 0
Z 0
T( )d(0; )'()d;
2 Cn; def= C([0; ];Cn), ' 2 Cn; . , Cn . A0 A0
(A )() =
8>:d ()=d; 2 (0; ] ;Z 0
dT(0; ) (); = 0 :
, Ker((A0 + i!Id)k) f 1; : : : ; kg
(A0 + i!Id) k = 0; (A0 + i!Id) j = j+1; (2.6)
j = 1; 2; : : : ; k 1. 2.1, : 2.2 j(t) =Pkjs=0 1s! (t)svj+sei!t, j = 1; 2; : : : ; k, vj 2 Cn (j = 1; 2; : : : ; k)
kjXs=0
1
s!vTj+ss(0;i!) = 0: (2.7)
'k =2 Ran(A0 i!Id) k 2 Ran(A0 + i!Id), ( k; 'k) = 6= 0. , ( j ; 's) = 0 j 6= s , ( j ; 'j) = j = 1; 2; : : : ; k . ,
=k1Xs=0
vT1+s
(t)ss!
Idn +
Z 0
(t)s+1 (t+ )s+1(s+ 1)!
d(0; )ei!u1
=
k1Xs=0
vT1+s
(t)ss!
Idn sX
m=0
(t)mm!(s+ 1m)!
Z 0
s+1md(0; )ei!u1
=
k1Xs=0
sXm=0
(t)mm!(s+ 1m)!v
T1+ss+1m(0; i!)u1
=k1Xm=0
k1Xs=m
(t)mm!(s+ 1m)!v
T1+ss+1m(0; i!)u1
94
-
: 42 2
=
k1Xm=0
(t)mm!
k1Xs=m
1
(s+ 1m)!vT1+ss+1m(0; i!)u1
=
k1Xm=0
(t)mm!
kmXs=1
1
s!vTm+ss(0; i!)u1
=kX
s=1
1
s!vTs s(0; i!)u1 +
k1Xm=1
(t)mm!
kmXs=1
1
s!vTm+ss(0; i!)u1
=kX
s=1
1
s!vTs s(0; i!)u1
k1Xm=1
(t)mm!
vTm(0; i!)u1
=kX
s=1
1
s!vTs s(0; i!)u1:
3 Lyapunov-Schmidt
, (1.1) Lyapunov-Schmidt 2! ., 2 (1; 1), x(t) = u((1 + )t), (1.1) :
(1 + ) _u(t) = L()ut; + f(; ut;);
ut;() = u (t+ (1 + )), 2 [; 0]. , C(R;Rn) Banach C! ( C1!), () 2! - . , C! S1
Banach , :
u(t) = u(t+ ); 2 S1:
, = expfig 2 S1 , . C! h; i : C! C! ! R, :
hv; ui = !2
Z 2=!0
vT(t)u(t)dt; (3.1)
u; v 2 C!. F : C1! R2 ! C!
F (u; ; ) = (1 + ) _u(t) + L()ut; + f(; ut;): (3.2)
, (1.1) ! ! . , F (u; ; )=0 (1.1) 2!(1+) -. F S1- , 2S1
F (u; ; ) = F ( u; ; ):
L F , , Lu = _u + L(0)ut. , KerL _u = L(0)ut u(t) = u(t+ 2! ) . L :
Lu = _u+Z 0
dT(0; )u(t );
95
-
: Hopf
, u; v 2 C1!, hv;Lui = hLv; ui. (NS)
KerL = fRe(z'1); z 2 Cg; KerL = fRe(z k); z 2 Cg:
2 C!, P Q
P = 2Refhv1ei!t; i'1g; Q = 2Refhvkei!t; i'kg:
, P Q S1- , KerL = RanP , RanL = KerQ. F (u; ; ) = 0 :8
-
: 42 2
(IQ)F ((z; ; ); ; ) 0 (4.2)
. z = 0 (4.1) (4.2) hv1ei!t; z(0; ; )i = 1
(IQ)Fu(0; ; ) z(0; ; ) = 0: (4.3)
, z(0; 0; 0) = '1. (3.6) zh(jzj2; ; ) = h k; F ((z; ; ); ; )i. z = 0
h(0; ; ) = h k; Fu(0; ; ) z(0; ; )i: (4.4)
. 4.1 h(0; 0; 0) = vTk(0; i!)u1, h(0; 0; ) = (i!)k +O(jjk+1). (4.4) h(0; 0; 0) = 0. = 0 (4.4)
h(0; 0; 0) = h k; Fu(0; 0; 0) z(0; 0; 0)i
= vTk
Z 0
d(0; )'1()
= vTk(0; i!)u1:
(4.3) , = 0 vjei!t ,
0 = hvjei!t; Fu(0; 0; ) z(0; 0; )i
=
(1 + )i!vje
i!t +
Z 0
dT(0; )vjei!(t); z(0; 0; )
=
(1 + )i!vje
i!t +1Xs=0
(i!)ss!
Z 0
sdT(0; )vjei!(t); z(0; 0; )
= k1X
s=0
(i!)ss!
Ts (0;i!)vjei!t; z(0; 0; )+O(jjk):
hT(0;i!)vjei!t; z(0; 0; )i = k1X
s=1
(i!)ss!
Ts (0;i!)vjei!t; z(0; 0; )+O(jjk): (4.5)
, (4.4)
h(0; 0; ) = kX
s=1
(i!)ss!
Ts (0;i!)vkei!t; z(0; 0; )+O(jjk+1):
(4.5) k1Xl=1
(i!)lT (0;i!)vklei!t; z(0; 0; )
97
-
: Hopf
= k1X
l=1
k1Xs=1
(i!)l+ss!
Ts (0;i!)vklei!t; z(0; 0; )
= k1X
s=1
k1+sXm=s+1
(i!)ms!
Ts (0;i!)vkm+sei!t; z(0; 0; )
= k1X
s=1
kXm=s+1
(i!)ms!
Ts (0;i!)vkm+sei!t; z(0; 0; )+O(jjk+1)
= kX
m=2
m1Xs=1
(i!)ms!
Ts (0;i!)vkm+sei!t; z(0; 0; )+O(jjk+1)
= kX
l=2
l1Xs=1
(i!)ls!
Ts (0;i!)vkl+sei!t; z(0; 0; )+O(jjk+1):
, (2.7)
h(0; 0; ) =
k1Xl=1
l1Xs=0
(i!)ls!
Ts (0;i!)vkl+sei!t; z(0; 0; )+O(jjk+1)
=
k1Xl=2
l1Xs=1
(i!)ls!
Ts (0;i!)vkl+sei!t; z(0; 0; )
+
k1Xl=1
(i!)lT (0;i!)vklei!t; z(0; 0; )+O(jjk+1)
=
k1Xl=2
l1Xs=1
(i!)ls!
Ts (0;i!)vkl+sei!t; z(0; 0; )
kX
l=2
l1Xs=1
(i!)ls!
Ts (0;i!)vkl+sei!t; z(0; 0; )+O(jjk+1)
= (i!)k k1X
s=1
1
s!Ts (0;i!)vsei!t; z(0; 0; )
+O(jjk+1)
= (i!)k +O(jjk+1):
h1(0; 0; 0) h(u; ; ) (u; ; ) = (0; 0; 0) u . h1(0; 0; 0), (1.1) f(;') = 0 ' Taylor ,
f(0; ') =1
2B(';') + 1
6E(';'; ') + o(k'k3); (4.6)
' 2 Cn; , B(; ) C(; ; ) f(0; ) , 2- 3- . ,
h1(0; 0; 0) =@3
@z2@zg(0; 0; 0):
[21] , h1(0; 0; 0) = h k; E('1; '1; '1)i+ 2h k;B('1;W11)i+ h k;B('1;W20)i; (4.7)
W11 W20 (z; 0; 0) Taylor zz z22 . (4.2) W20 = L1(IQ)B('1; '1); W11 = L1(IQ)B('1; '1):
98
-
: 42 2
B('1; '1), B('1; '1) 2 RanL, (IQ) B('1; '1) B('1; '1) . ,
LW20 + B('1; '1) = 0; LW11 + B('1; '1) = 0:, RanW KerP ,
hv1ei!t;W20i = 0; hv1ei!t;W11i = 0:
,W20 =
1(0; 2i!)B('1; '1); W11 = 1(0; 0)B('1; '1): (4.8) (4.7) 4 , h(r2; ; ) = 0 :
(i!)k vTk(0; i!)u1+ r2h1(0; 0; 0) + h:o:t: = 0:
6= 0,
k A+Br2 + h:o:t: = 0; (4.9)
A =
vTk(0; i!)u1(i!)k ; B =
h1(0; 0; 0)
(i!)k : (4.10)
(4.9) ,
k RefAg+ r2RefBg+ h:o:t: = 0 (4.11)
ImfAg+ r2ImfBg+ h:o:t: = 0: (4.12) ImfAg 6= 0,
ImfikvTk(0; i!)u1g 6= 0; (4.13), (4.12) :
= (r2; ) :=ImfBgImfAg r
2 +O(r4; ): (4.14)
(4.11) ,
k = 0r2 +O(r4; ); (4.15)
0 =
Im(BA)
Im(A): (4.16)
= " kpr2, (4.15) "kr2 = 0r2 +O(r4; "),
"k = 0 +O(r2; "): (4.17)
k 0 6= 0, r = 0 , (4.17) " = kp0. , r 0, . , " = "(r) (4.17) "(0) = kp0 .
99
-
: Hopf
(r) = kpr2"(r), (r) = (r2; (r)), (r2; ) (4.14) , h(r2; ; ) = 0 , (r; (r); (r)), r = 0 (0; 0; 0), r > 0 .
k 0 < 0, r = 0, (4.17), jrj,. , k 0 > 0, r = 0, (4.17) " = kp0, r 0
, . , " = "(r), ": [0;1) ! R , "(0) = kp0. (r) =
pr2"(r), (r) = (r2; (r)), (r2; ) (4.14) , h(r2; ; ) = 0 , (r; (r); (r)), r = 0 (0; 0; 0), r > 0 . , .
4.1 A, B 0 (4.10) (4.16) , (NS) (4.13) , :(i) k 0 6= 0, (0; 0; 0) (u; ; ), (1.1) 2(1+)! - , ImfAgImfBg > 0 ( ImfAgImfBg < 0), > 0 ( < 0) , .
(ii) k 0 < 0, (0; 0; 0) (u; ; ), (1.1) 2
(1+)! - .(iii) k 0 > 0, (0; 0; 0) (u; ; ), (1.1)
2(1+)! - , ImfAgImfBg > 0 ( ImfAgImfBg < 0), > 0 ( < 0) , .
k = 1 (NS) , A () () (0) = i!, C1- u() u(0) = u1, (; ())u() 0
.
[(; ()) + 0()1(; ())]u() + (; ())u0() = 0:
, [(0; i!) +
0(0)1(0; i!)]u1 +(0; i!)u0(0) = 0:
vT1(0; i0) = 0 vT11(0; i!)u1 = ( 1; '1) = , vT1(0; i!)u1+0(0) = 0. (4.10) A B:
A =0(0)i!
; B =h1(0; 0; 0)
i! :
, (4.13) Ref0(0)g 6= 0
sgnfImfAgImfBgg = sgnfRef0(0)gRefh1(0; 0; 0)gg:
,sgnf0g = sgnfRef0(0)gImf0(0)h1(0; 0; 0)gg:
, 4.1 Hopf . 4.1 k = 1 (NS) , Ref0(0)g 6= 0, (1.1) , x = 0 , . ,
(i) Ref0(0)gRefh1(0; 0; 0)g : Ref0(0)gRefh1(0; 0; 0)g < 0 ( > 0), > 0 ( < 0) , , , Ref0(0)gRefh1(0; 0; 0)g < 0 ( > 0) ,
();
100
-
: 42 2
(ii) Ref0(0)gImf0(0)h1(0; 0; 0)g: Ref0(0)gImf0(0)h1(0; 0; 0)g() , () 2! .
5 van der Pol van der Pol , :
x "(1 x2) _x+ x = 0; (5.1)
x , " > 0 . Balthasar van der Pol [22] , van del Por
x "(1 x2) _x+ x = f(x(t )) (5.2)
, [23, 24]. , f(x(t )) x . Atay [23] " 1 , (5.2) ; Wei Jiang [24] (5.2) Hopf , 1 : 1 . , Hopf .
, f : R! R C3- ,
f(0) = f 00(0) = 0, f 0(0) = , f 000(0) = . (5.3)
(5.2) x = 0 :
2 "+ 1 = e : (5.4)
, Hopf :
H = f("; ; ) : 1 !2 = cos !; "! = sin !; ! 2 R n f0gg:
H (5.2) 1 : 1 Hopf . (5.4) ,
2 " = e : (5.5) = i! (! > 0) (5.4) (5.5) , 8>>>>>>>>>>:
1 !2 = cos !;"! = sin !;
" = cos !;
2! = sin !:
(5.7)
101
-
: Hopf
8>>>>>:" = 2;
(1 !2) = ";! = ;
(5.8)
x = x = tanx . , (5.8) = p2 + 2, " = 2=p2 + 2, ! = =
p2 + 2. fng1n=1 x = tanx ,
"n =2p
2 + 2n; n =
p2 + 2n; n =
2
(2 + 2n) cos n; !n =
np2 + 2n
; (5.9)
n 2 N . . 5.1 ("; ; ) = ("n; n; n) , n 2 N, (5.4) i!n. van der Pol (5.2) :
_x = y; _y = x+ f(x(t )) "(x2 1)y: (5.10)
(5.10) :_x = y; _y = x+ x(t ) + "y (5.11)
("; ; ; ) =
24 11 e "
35 : ("0; 0; 0) 2 H n f("n; n; n)g1n=1, !0 > 0 8>>>>>:
1 !20 = 0 cos 0!0;"0!0 = 0 sin !0;
2i!0 "0 + 00ei!00 6= 0:(5.12)
(5.4) i!0, det("0; 0; 0;i!0) = 0. u1 = (1; i!0)T v1 =(i!0 "0; 1)T ("0; 0; 0; i!0)u1 = 0 vT1("0; 0; 0; i!0) = 0, = vT11(0; i!0)u1 =2i!0 "0 + 00ei!00 6= 0. , (5.10) (1.1) (4.6) ,
B('; ) =24 0
0
35 ; E('; ; ) =24 0C2('; ; )
35 ;, ' = ('1; '2)T, = ( 1; 2)T, = (1; 2)T 2 C([0; 0];R2),
C2('; ; ) = '1(0) 1(0)1(0) 2"0'1(0) 1(0)2(0)2"0'1(0)1(0) 2(0) 2"01(0) 1(0)'2(0):
h1(0; 0; 0) = ei!002i!0"0. " ( , ), (; ) = (0; 0) ( ("; ) =("0; 0), ("; ) = ("0; 0)), 0("0) = i!0= ( 0(0) = i!00ei!00=, 0(0) = ei!00=),
sgnfRe[0("0)]g = sgnf2 0"0g;
102
-
: 42 2
sgnfRe[0(0)]g = sgnf"20 + 2!20 2g;sgnfRe[0(0)]g = sgnf0[020 "0(1 + !20)]g:
sgnfRe[h1(0; 0; 0)]g = sgn
0[0
20 "0(1 + !20)] + 2"0!20(0"0 2)
sgnfIm[0("0)h1(0; 0; 0)]g = sgnf0(1 !20)g;sgnfIm[0(0)h1(0; 0; 0)]g = sgnf0 + 2"20!20g;sgnfIm[0(0)h1(0; 0; 0)]g = sgnf0(1 !20)g:
4.1 . 5.1 ("0; 0; 0) 2 H n f("n; n; n)g1n=1, !0 > 0 (5.12), :
(1) (; ) = (0; 0), (1.1) x = 0 " .
0[0
20 "0(1 + !20)](2 0"0) 2"0!20(2 0"0)2 < 0
( > 0), " > "0 ( < "0) ; 0(1 !20)(2 0"0) > 0 ( < 0) , () 2!0 .
(2) ("; ) = ("0; 0), (1.1) x = 0 .
0[0
20 "0(1 + !20)] + 2"0!20(0"0 2)
("20 + 2!
20 2) < 0
( > 0), > 0 ( < 0) ; ("20 + 2!20 2)(0 + 2"20!20) < 0 ( > 0) , () 2!0 .
(3) ("; ) = ("0; 0), (1.1) x = 0 .
[020 "0(1 + !20)]2 + 2"0!200(0"0 2)[020 "0(1 + !20)] < 0
( > 0), > 0 ( < 0) ; (1 !20)[020 "0(1 + !20)] > 0 ( < 0) , () 2!0 .
5.1 , det("n; n; n;i!n) = 0. , u1 = (1; i!n)T, v1 = (1; 0)T, v2 = (i!n "n; 1)T
("n; n; n; i!n)u1 = 0;
vT2("n; n; n; i!n) = 0;
vT1("n; n; n; i!n) = vT2("n; n; n; i!n):
= in,
vT2"("n; n; n; i!n)u1 = i!n;
103
-
: Hopf
vT2 ("n; n; n; i!n)u1 = i!nnein ;
vT2("n; n; n; i!n)u1 = ein :
, h1(0; 0; 0) = ein 2i!n"n. (4.10) B = (2!n"n + iein)=(n!2n),(1)nIm(B) > 0.
", , , (4.10) A A", A , A . ,
A" =1n!n
; A ="n(1 in)
2n; A =
(n + i) cos nn!2n
:
, Im(A") = 0, Im(A ) < 0, (1)nIm(A) < 0. , 4.1 .
, , (4.16) 0. ,
sgnf0g = sgnIm(h1(0; 0; 0)v
T2 ("n; n; n;i!n)u1)
= sgnfIm[inein(ein 2i!n"n)]g= sgnfn + 2"2n!2ng:
,
sgnf0g = (1)n1sgnIm(h1(0; 0; 0)v
T2 ("n; n; n;i!n)u1)
= (1)nsgnfIm[ein(ein 2i!n"n)]g= 1:
4.1, . 5.2 n 2 N, (5.2) ("; ; ) = ("n; n; n) 1 : 1 Hopf . ,
(i) ("; ) ("n; n) n , (5.2) x = 0, < 0 ( > 0), > n ( < n) .
(ii) n + 2"2n!2n < 0, ("; ) ("n; n) n , (5.2) x = 0 .
(iii) n + 2"2n!2n > 0, ("; ) ("n; n) n , (5.2) x = 0 , (1)n < 0 ( (1)n > 0), > n( < n) . .
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Generalized Hopf bifurcation in delay dierential equations
GUO ShangJiang & WU JianHong
Abstract Here we employ the Lyapunov-Schmidt procedure to investigate bifurcations in a general delaydierential equation when the innitesimal generator has, for a critical value of the parameter, a pair of non-
semisimple purely imaginary eigenvalues with multiplicity k. We derive criteria, explicitly in terms of the system's
parameter values, for the existence of bifurcating periodic solutions and for the description of the bifurcation
direction. The general result is illustrated by a detailed case study of the van del Pole oscillator.
Keywords delay dierential equation, Lyapunov-Schmidt reduction, Hopf bifurcation, van der Pol oscilla-
tor
MSC(2010) 34K18, 92B20
doi: 10.1360/012010-1047
105