: 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: shangjguo@hnu.edu.cn, wujh@mathstat.yorku.ca

: 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