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Persistence of elliptic invariant tori for Hamiltonian systems
Nonlinear Analysis 45 (2001) 241 – 260
www.elsevier.nl/locate/na
Persistence of elliptic invariant tori for Hamiltonian systems Huang Qingdao∗ , Cong Fuzhong, Li Yong Department of Mathematics, Jilin University, Changchun, Jilin 130023, People’s Republic of China Received 20 January 1999; accepted 7 September 1999
Keywords: Elliptic invariant tori; KAM iteration; Measure estimate; Nonstandard Hamiltonian system
1. Introduction and result In 1984, Parasyuk [6] considered the existence of invariant tori for Hamiltonian systems with distinct numbers of action-angle variables. Such systems are usually called nonstandard Hamiltonian systems. Next, Herman [3] also considered similar problems. Thereafter, there is an increasing interest in the persistence problem for nonstandard Hamiltonian systems. Recently, Cong and Li [1] gave a KAM theorem for a nonstandard Hamiltonian system. Motivated by above works, we construct symplectic constructure for Hamiltonian functions to study the persistence of elliptic invariant tori for a nonstandard Hamiltonian system. Let G ⊂ Rl be bounded and connected closed domain, and T n = Rn =2Z n usual n-dimension tori. Let l + n be even, and D ⊂ R2m some bounded open subset containing the origin. Consider an n + l + 2m-dimensional complex symplectic manifold (G × T n × D; !2 ), where !2 (· ; E!1 ) = !1 (·); !1 is a 1-form on G × T n × D, I (x; y) 0 E= ; 0 J ∗
Corresponding author.
0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 9 ) 0 0 4 0 7 - 1
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J is the usual 2m-dimensional symplectic matrix, I is an analytic Hamiltonian homeomorphism from the space of 1-form to vector Celds on G×T n ; deCne the corresponding Poisson bracket {· ; ·} as follows: for all smooth functions f1 ; f2 on G × T n × D, let {f1 ; f2 } = df1 (Edf2 ) = !2 (Edf2 ; Edf1 ): Let y ∈ G; x ∈ T n . Assume !2 is invariant relative to the quasi-periodic motions on T n , and {yi ; yj } = 0;
i; j = 1; : : : ; l:
So the coeEcients of 2-form !2 and the matrix I are independent of the angle variable x. In this paper, we consider the nearly integrable Hamiltonian system N + P, where N is an unperturbed system of the following form: N ≡ h(y) + F(y); y + O3 (z);
(1.1) 2m
2 = 12 (zi2 +zi+m );
and P is a perturbation of N , where y ∈ G; z ∈ R ; y 1 ≤ i ≤ m; · ; · denotes the usual inner product, O3 (z) denotes the sum of the power of the order at least three about z. According to Lemma 2 in the appendix and (1.1), for all given c1 ∈ G; c2 ∈ Rm + , the system y˙ x˙ = E grad T N (y; z) (1.2) z˙ admits n + m-dimensional invariant tori, {(y; x; z): x ∈ T n ; y = c1 ; y = c2 } and n-dimensional invariant tori {(c1 ; x; 0): x ∈ T n }, where the latter are the lowerdimensional stable invariant tori of (1.2) with the frequency !(c1 ). By Lemma 2 in the appendix, for y ∈ G, we have that (0; : : : ; 0; !1 (y); : : : ; !n (y))T = I (y)grad T(y; x) N (y; 0);
l
where grad T(y; x) denotes the gradient of a function in the variable (y; x). Assume the following conditions hold: (H1 ) I (y); N (y; z); P(y; x; z) are real analytic functions on G × T n × D; (H2 ) !(y), for y ∈ G satisCes @! = r ≤ min{l; n}; rank @y @ ! l rank !; : ∀ ∈ Z+ ; | | ≤ n − r + 1 = n; @y
(1.3) (1.4)
where @ !=@y =(@| | !1 =@y ; : : : ; @| | !n =@y )T ; | |= 1 +· · ·+ l . Moreover, we assume that there exists 0 ¿ 0 such that |F(y); l | ≥ 0 ;
y ∈ G;
0 = l ∈ Rm :
(1.5)
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243
Put G = {y: Re y ∈ G; |Im y| ≤ };
={x: Re x ∈ T n ; |Im x| ≤ };
B = {z ∈ C 2m : |z| ≤ }; where for a vector, | · | denotes its maximum norm in components; for a function, | · | denotes its supremum norm. The main result of this paper is the following. Theorem A. Assume (H1 ) and (H2 ) hold. Then there exist a constant M0 ¿ 0 and a nonempty Cantor set G∗ ⊂ G such that |P| ≤ M0 implies that the Hamiltonian system N + P admits an n-dimensional invariant torus Ty ; y ∈ G∗ , whose frequency vector (!∞ (y); F∞ (y)) satis4es max{|!∞ (y) − !(y)|;
|F∞ (y) − F(y)|} ¡ M01=4 ;
where !∞ denotes the tangential frequency and F∞ denotes the normal frequency. Moreover, mesl (G\G∗ ) = O(M01=8(n+$+4)(n−r+1) ); where $ ¿ (n + l)(n − r + 1) + 1 is a constant. Remark 1. If m = 1; l = n, Moser [4,5] gave the following nondegeneracy condition: @F @! @y det @y = 0: !
F
Remark 2. Theorem A corresponds to the result of Eliasson [2] when r = n = l. There, the following condition is required: for all k ∈ Z m \{0}; |k| = |k1 | + · · · + |km | ≤ 3 −1 @F @! : k; F(y) = k; !(y) @y @y Eliasson’s result implies that !∞ (y) is parallel to !(y). In our case, such a result seems impossible. Remark 3. If l ¿ n, that is, the dimension of action variables is larger than one of angle variables, then Theorem A gives a lower-dimensional invariant torus with dimension at most 12 (l + n); if l ¡ n, that is, the dimension of action variables is less than one of angle variables, Theorem A gives a higher-dimensional invariant torus with dimension at most 12 (l + n); if m = 0, Theorem A corresponds to the result of Cong and Li [1] and Parasyuk [6].
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In the following, we give some notations throughout the paper. Fix y0 ∈ G and D( ; s) = {(y; x; z): Re x ∈ T n ; |Im x| ¡ ; |y − y0 | ¡ s2 ; |z| ¡ s}: Let P be an analytic function on D( ; s); [P] denotes the averaging value over the torus T n+m . Write P = P0 + P 1 + · · · ; where Pk (y; x; z) =
( ;()∈Z+l+2m
1 @ @( P(y; x; 0)(y − y0 ) z ( : !(! @y @z (
(1.6)
2| |+|(|=k
We use the abbreviation Pˆ = P − (P0 + P1 + P2 + P3 ):
(1.7)
According to condition (H1 ), it holds that there exists a constant C0 ¿ 0 such that on G × ×B , 2 @F @h @ h @I (1.8) max |I (y)|; ; |F(y)| ; |h(y)|; @y ; @y2 ≤ C0 : @y @y In the following, we use Ck to denote a positive constant which only depends of C0 ; $; ; m; n; l. By (1.7) and Cauchy’s integral formula, if |P| ≤ M on D( ; s), then on D( ; 78 s), we have |P0 | ≤ M;
|Pi | ≤ Ci M;
i = 1; 2; : : : ; 5:
(1.9)
This paper is arranged as follows: in Section 2, we give an important proposition about a small divisor problem; in Section 3, we give an outline of the proof of Theorem A; in Section 4 we give a measure estimate for the persisting tori; in Section 5, we give a KAM iteration and the proof of Theorem A; in Section 6 we illustrate Theorem A; and in Section 7 we list some preliminary lemmas. 2. A small divisor problem Take y0 ∈ G, and consider @h N2 = (y0 ); y − y0 + F(y0 ); y : @y By Lemma 2 in the appendix, we have (0; : : : ; 0; !1 (y); : : : ; !n (y))T = I (y)grad T(y; x) N2 (y; 0):
l
Assume ! and F satisfy |k; !(y0 ) + l; F(y0 ) | ≥ )(|k| + |l|)−$ ;
0 = (k; l) ∈ Z n+m ;
and (1.5), where $ ≥ (n + l)(n − r + 1) − 1 is a constant.
|l| ≤ 3
(2.1)
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
245
Assume F is a real analytic function deCned on D( ; s) and with at most an order of three. Proposition 1. The equation @’ @’ @N2 = F − [F]; ; !(y0 ) + ;J @x @z @z
[’] = 0
(2.2)
has a unique real analytic solution ’; whose order is as same as that of F; and |’|D( −); s) ≤
C6 |F|D( ; s) : )n+$+2
Proof. BrieMy, denote D = D( ; s). We take a transformation √ xˆi = xi ; 2zi = (1 + −1)(zˆi − zˆi+m ); √ yˆ i = yi ; 2zi+m = (1 − −1)(zˆi + zˆi+m ):
(2.3)
Obviously, transformation (2.3) is a symplectic transformation. So, √ yi = − −1zˆi zˆi+m : We also denote it by z. Then F can be expressed as follows:
√ F(y; x; z) = C(k)exp( −1k; x ) (y − y0 ) z ( ; 2| |+|(|=j
(2.4)
k∈Z n
where ( = (( ; ( ) ∈ Z+2m ; ∈ Z+n ; 0 ≤ j ≤ 3. Using Cauchy’s integral formula we get |C(k)(y − y0 ) z ( | ≤ C7 e− |k| |F|D : DeCne ,(k) =
√ − −1(k; ! + ( − ( ; F )−1
(2.5) (k; ( − ( ) = 0; (k; ( − ( ) = 0:
0
Putting (2.4) into (2.2), we have
√ ’(y; x; z) = ,(k)C(k)exp( −1k; x ) (y − y0 ) z ( : 2| |+|(|=j
Obviously, [’] = 0. Let
√ ’ ; ( = ,(k)C(k)exp( −1k; x )(y − y0 ) z ( : k∈Z n
Then applying (2.5), (2.1) and (1.5), we obtain
|’ ; ( | ≤ |,(k)||C(k)(y − y0 ) z ( |e|k|( −)) k∈Z n
≤ C7
k∈Z n
(2.6)
k∈Z n
1 |F|D e|k|) |k; ! + ( − ( ; F |
(2.7)
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
≤ C7
3$ |F|D + C7 0
k∈Z n \{0}
|k|$ |F|D e|k|) )
∞
≤
2n j n−1 j $ C8 |F|D |F|D + C7 )ej) 0 j=1
∞ 1 C8 C9 |F|D ≤ |F|D + n+$+2 0 ) j 2+[$]−$ j=1
C10 |F|D ; (2.8) )n+$+2 where [$] denotes the integral part of $. Then by (2.4) and (2.6) – (2.8), we have the estimate of the proposition. ≤
3. Outline of the proof of Theorem A Let y0 ∈ G, and rewrite N as follows: @h (y0 ); y − y0 + F(y0 ); y + Nˆ (y; z); N (y; z) = h(y0 ) + @y 1 @2 h Nˆ (y; z) = (y )(y − y ); y − y t 0 0 2! @y2 @F + (y. )(y − y0 ); y + O3 (z); @y
(3.1)
(3.2)
where yt = y0 + t(y − y0 ); y. = y0 + .(y − y0 ); 0 ≤ t; . ≤ 1. Corresponding to (3.1) and (3.2), denote N = N0 + N2 + Nˆ :
(3.3)
Then
N0 = [N0 ] = h(y0 );
N2 =
@h (y0 ); y − y0 @y
+ F(y0 ); y :
(3.4)
By (1.8), on D( ; s1 ), |N2 | ≤ /10 (s1 )2 ;
|Nˆ | ≤ /20 (s1 )3 ;
s1 ≤ s;
(3.5)
where /10 and /20 ¿ 0 are constants. We will prove Theorem A by applying rapidly convergent iteration method. First we choose rapidly convergent sequences Mi+1 = Mi9=8 ; i+1 = i − 6)i ;
)i = Mi1=8(n+$+4) ;
si = Mi1=8 ;
i = 0; 1; 2; : : : ;
where M0 is the positive constant satisfying the following inequalities (A)−(I ); 0 = .
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
Let Dji
7−j si = D i − j)i ; 8 |y| ¡
7−j si 8
2
247
(y; x; z): Re x ∈ T n ; |Im x| ¡ i − j)i ;
=
7−j si ; |z| ¡ 8
; 0 ≤ j ≤ 6:
Assume the Hamiltonian function N + P is taken into N i + P i by a symplectic transformation on D0i , such that i
N i = N0i + N2i + Nˆ ;
N0i = [N0i ];
(3.6i )
i (y0 ); y − y0 + Fi (y0 ); y ; N2i = N21
(3.7i )
i |Nˆ |D( i ;s1 ) ≤ /2i (s1 )3 ; s1 ≤ si ;
|N2i |D( i ;s1 ) ≤ /1i (s1 )2 ;
(3.8i )
|P i | ≤ Mi :
(3.9i )
Let Oi = {y ∈ G: |k; !i (y) + l; Fi (y) | ≥ )i (|k| + |l|)−$ ; 0 = (k; l) ∈ Z n+m ; |l| ≤ 3}; where {0; : : : ; 0; !1 (y0 ); : : : ; !n (y0 )}T = I (y0 )grad T(y; x) N2i (y; 0)|y=y0 . When y0 ∈ Oi , one
l
can construct a symplectic transformation Qi such that (N i + P i ) ◦ Qi = N i+1 + P i+1 : We shall prove N i+1 and P i+1 satisfying (3:6i+1 ) − (3:9i+1 ), where N 0 = N; P 0 = P. If ∞ y0 ∈ Ok = 1; k=0
then the above iteration procedure can continue. Denote [P2i ] = Z i (y0 ); y − y0 + Qi (y0 ); y :
(3.10i )
According to (1.9), (3:10), the deCnition of si and Cauchy’s estimate, we have C11 |Z i |D1i ; |Qi |D1i ≤ 2 |P2i |D0i ≤ C12 Mi3=4 : si Put i+1 i = N21 + Z i; N21
We can prove that on |!i+1 − !0 | ≤
Fi+1 = Fi + Qi :
(3.11i )
D1i+1 ,
i
k=0
|!k+1 − !k |Dk ≤ 1
i
k=0
|I ||grad T(y; x) N2k (y; 0)|Dk
1
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
=
i
|I ||Z k |Dk ≤ C0 C12 1
k=0
|Fi+1 − F0 | ≤
i
≤ C12
k=0
|Fk+1 − Fk |Dk =
i
1
k=0 i
k=0
Mk3=4 ¡
i
Mk3=4 ¡ 00 ;
(3.12i )
|Qk |Dk
k=0
1
00 : 3
(3.13i )
Here we have used the inequality 32 00 1 00 ; M0 ¡ min ; ; 2 2(C12 + 1) 6(C0 C12 + 1)
(A)
where 00 will be given in Section 4. Hence, |!i+1 | ≤ |!0 | + |!i+1 − !0 | ≤ C02 + 00 ;
(3.14i )
00 ; 3
(3.15i )
|Fi+1 | ≤ |F0 | + |Fi+1 − F0 | ≤ C0 + where !0 = !; F0 = F.
4. Estimate of the measure in Theorem A Let B ⊂ Rn be a bounded connected closed domain, g : B → Rn a C n−r+1 - continuously diRerentiable vector-valued function, such that on B @g = r; rank @y @ g n rank g; : ∈ Z+ ; | | ≤ n − r + 1 = n: (4.1) @y Proposition 2. If g˜ : B → Rn ; g:B T → R are C n−r+1 -continuously di6erentiable functions; and g satis4es (4:1); then there exists 0 ¿ 0 such that whenever max{|g|; T |g|} ˜ ¡ 0 ; the set ˜ + g(y)| T ≥ )|k|−$ ; k ∈ Z n \{0}} B0 = {y ∈ B : |k; g(y) + g(y) is a positive measure set; and mesn (B\B0 ) ≤ 4)1=(n−r+1) ; where 4 ¿ 0 is a constant. T and {DVrT g(y) : ∀V ∈ Rn } Proof. According to Xu et al. [8], {@( g=@y( : ∀(; |(| = r} rT rT are equivalent, where rT ¿ 0 is an integer, DV g(y) = d =dt rTg(y + Vt)|t=0 . By the
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
249
nondegeneracy condition and the Cnite covering theorem, there exist n integers 1 ≤ r1 ; r2 ; : : : ; rn ≤ n − r + 1 and n vectors V1 ; V2 ; : : : ; Vn ∈ Rn such that rank{DVr11 g(y); DVr22 g(y); : : : ; DVrnn g(y)} = n;
y ∈ G:
(4.2)
Denote A(y) = (DVr11 g(y); DVr22 g(y); : : : ; DVrnn g(y)). By (4.1) there exists 41 ¿ 0 such that for all (y; V ) ∈ B × S, |A(y)V | ≥ 41 ;
(4.3)
where S = {V ∈ Rn : |V1 | + · · · + |Vn | = 1}. Obviously, for k ∈ Z n \{0}, ri D g(y); k ≥ 41 ; y ∈ B; i = 1; 2; : : : ; n: Vi |k|
(4.4)
Let f(y) = k; g(y) + g(y) ˜ + g(y): T According to (4.4), there exists 1 ¿ 0, such that when |g|C n−r+1 ¡ 1 , 1 ri 1 ri k D f(y) = Dri g(y) + Dri g(y); + DVi g(y) T ≥ 42 ¿ 0; Vi ˜ |k| Vi Vi |k| |k| where 42 ¿ 0 is a constant. Let RkVi = {t: |f(y + tVi )| ≤ )|k|−$ ;
y ∈ B;
y + tVi ∈ B};
Rk = {y ∈ B: |f(y)| ≤ )|k|−$ }: According to Proposition 1, mes(RkVi ) ≤ 43 )1=(n−r+1) |k|−$−1 ; where 43 ¿ 0 is a constant; moreover, mes(Rk ) ≤ 43 (diam B)n−1 )1=(n−r+1) |k|−$−1 ; where “diam” denotes the diameter of the set. Hence mes(B0 ) = mesB\ Rk k∈Z n \{0}
≥ mes(B) − 43 (diam B)n−1 )1=(n−r+1)
|k|−$−1
k∈Z n \{0}
≥ mes(B) − 44 (diam B)n−1 )1=(n−r+1)
∞
j=1
1 j 2+$−n
≥ mes(B) − 45 )1=(n−r+1) ; where 44 ; 45 ¿ 0 are constants. This completes the proof.
(see [7])
(4.5)
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
Now let us discuss the estimate of the measure in theorem A. Denote ∞ G∗ = Oi : i=0
(1) l = n. By the nondegeneracy condition (H2 ) and Proposition 2, there exists 00 ¿ 0, such that when max{|!|; ˜ |!|} ¡ 00 , mesn (G\G∗ ) ≤ C13 )1=(n−r+1) ;
(4.6)
where GX = {y ∈ G: |k; ! + ! ˜ + !| ≥ )|k|−$ ; 0 = k ∈ Z n }. By (3:12)i , (3:13)i and (4.6), for every , ∈ Z m ; 0 ≤ |,| ≤ 3, mesn (G\Oi; , ) ≤ C13 )i1=(n−r+1) ;
(4.7)
where Oi; , = {y ∈ G: |k; !i + ,; Fi | ≥ )i |k|−$ ; 0 = k ∈ Z n }: Because of
Oi =
Oi; , ;
,∈Z m ;0≤|,|≤3
by (4.7) we can obtain mesn (G\Oi ) ≤ C14 )i1=(n−r+1) :
(4.8)
Moreover, mesn (G\G∗ ) ≤
∞
mesn (G\Oi ) ≤ C14
i=0
∞
)i1=(n−r+1)
i=0
≤ 2C14 )01=(n−r+1) = O(M01=8(n−r+1)(n+$+4) ):
(4.9)
(2) l ¡ n. Denote yˆ = (y; yl+1 ; : : : ; yn ); !( ˆ y) ˆ = !(y); !ˆ i (y) ˆ = !i (y); G = G × [1; 2] × · · · × [1; 2]. According to the nondegeneracy condition (H2 ),
n−l
rank
@!ˆ @yˆ
= r; yˆ ∈ G ;
@ !ˆ : ∀ ; 0 ¡ | | ≤ n − l + 1 = n; yˆ ∈ G : rank !; ˆ @yˆ
(4.10)
We deCne Oi = Oi × [1; 2] × · · · × [1; 2];
n−l
G∗
= G∗ × [1; 2] × · · · × [1; 2] :
n−l
(4.11)
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251
Then mesn (G \G∗ ) = O(M01=8(n−r+1)(n+$+4) ): By Fubini’s theorem, mesl (G\G∗ ) = O(M01=8(n−r+1)(n+$+4) ):
(4.12)
T there exist integers (3) l ¿ n. By the nondegeneracy condition, for every y ∈ G, i1 ; : : : ; ir ; : : : ; il , 1 ≤ ij ≤ l; j = r; ij = ir and multi-indices 1 ; : : : ; n−r ; | i | ≥ 2; i = 1; : : : ; n − r, such that @! @! rank = r; (4.13) ;:::; @yi1 @yir rank
1
n−r
@! @! @ ! @ ! ;:::; ; 1 ; : : : ; n−r @yi1 @yir @y @y
= n:
(4.14)
Let !(y) ˜ = (!(y); yin+1 ; : : : ; yil ). Then @!˜ @!˜ rank = r; ;:::; @yi1 @yir rank
1
(4.15) n−r
@!˜ @ !˜ @!˜ @!˜ @!˜ @ !˜ ; 1 ; : : : ; n−r ; ;:::; ;:::; @yi1 @yir @y @yin+1 @yil @y
1
n−r
@ ! @! @ ! @! @y ; : : : ; @y ; @y 1 ; : : : ; @y n−r ; =rank i1 ir 0;
∗ = l;
(4.16)
El−n
where El−n denotes l − n-dimensionial unit matrix. Without loss of generality, we assume (4.15) and (4.16) hold on G. DeCne !˜ i (y) = (!i (y); yin+1 ; : : : ; yil ); T y T −$ ; Oi = {y ∈ G : |k; !i+1 (y) + k; T + ,; Fi (y) | ≥ )i (|k| + |k|) T ∈ Z l ; 0 ≤ |,| ≤ 3; , ∈ Z m }; 0 = (k; k) where yT = (yin+1 ; : : : ; yl ); kT = (kn+1 ; : : : ; kl ); $ ≥ (n + l)(n − k + 2) − 1. Let G∗ =
∞
Oi :
i=0
Similar to case 1, for the dimension l, by (4.15), (4.16) and Proposition 2, we can prove that mesl (G\G∗ ) = O(M01=8(n−r+1)(n+$+4) ):
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
Obviously, G∗ ⊃ G∗ , so we have mesl (G\G∗ ) = O(M01=8(n−r+1)(n+$+4) ):
(4.17)
By (4.9), (4.12) and (4.17), we complete the estimate of the measure in Theorem A. 5. KAM iteration and proof of Theorem A We only prove one iteration cycle. That is, if we assume (3:6k ) and (3:9k ) hold for k, we need to prove that they also hold for k + 1. For simplicity, we omit “k” and rewrite “k + 1” as “+”. We apply Eliasson’s reductive iteration scheme [2]. Now we construct the functions S = S0 + S1 + S2 + S3 and R = R0 + R2 + Rˆ such that grad T(y; x) S; I (y0 )grad T(y; x) N + grad Tz S; J grad Tz N = P − R; [S] = 0;
[R0 + R2 ] = R0 + R2 :
(5.1) 1
t
Let Q be phase Mow determined by EdS. Denote Q = Q . Then Q is a symplectic transformation. Applying d=dtF ◦ Qt = {F; S} ◦ Qt and Taylor’s formula, we can obtain 1 + (N + P) ◦ Q = N + {tP + (1 − t)R; S} ◦ Qt dt 0
+grad T(y; x) S; (I (y) − I (y0 ))grad T(y; x) N ; N + = N + R;
P+ =
0
1
(5.2)
{tP + (1 − t)R; S} ◦ Qt dt
+grad T(y; x) S; (I (y) − I (y0 ))grad T(y; x) N :
(5.3)
In a general KAM iteration method, iteration and estimate of the measure are two independent processes, and it is necessary to Cx the action variables in the iteration process. Because we consider a nonstandard elliptic Hamiltonian system, whose Hamiltonian vector Celd is I (y)grad T H (y; x; z), we need to modify Eliasson’s iteration scheme (its vector is J grad T H (y; x; z)) in Cxing y = y0 . Resolve (5.1) as follows: grad T(y; x) S0 ; I (y0 ) grad T(y; x) N2 + grad Tz S0 ; J grad Tz N2 = P0 − R0 ; [S0 ] = 0;
[R0 ] = R0 ;
grad T(y; x) S1 ; I (y0 )grad T(y; x) N2 + grad Tz S1 ; J grad Tz N2 = P1 ;
(5.4) (5.5)
grad T(y; x) S2 ; I (y0 ) grad T(y; x) N2 + grad Tz S2 ; J grad Tz N2 = P2 −grad T(y; x) S0 ; I (y0 )grad T(y; x) N4 + grad Tz S0 ; J grad Tz N4 − R2 ; [S2 ] = 0;
[R2 ] = R2 ;
(5.6)
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
253
grad T(y; x) S3 ; I (y0 )grad T(y; x) N2 + grad Tz S3 ; J grad Tz N2 = P3 − grad T(y; x) S0 ; I (y0 )grad T(y; x) N5 + grad Tz S0 ; J grad Tz N5 −grad T(y; x) S1 ; I (y0 )grad T(y; x) N4 + grad Tz S1 ; J grad Tz N4 :
(5.7)
Put R0 = [P0 ];
R2 = [P2 ];
(5.8)
Rˆ = Pˆ − grad T(y; x) S0 ; I (y0 )grad T(y; x) (Nˆ − N4 − N5 ) +grad Tz S0 ; J grad Tz (Nˆ − N4 − N5 ) −grad T(y; x) S1 ; I (y0 )grad T(y; x) (Nˆ − N4 ) +grad Tz S1 ; J grad Tz (Nˆ − N4 ) − grad T(y; x) S2 ; I (y0 )grad T(y; x) Nˆ +grad Tz S2 ; J grad Tz Nˆ − grad T(y; x) S3 ; I (y0 )grad T(y; x) Nˆ +grad Tz S3 ; J grad Tz Nˆ :
(5.9)
Applying (3:9), (5.4), (5.8), (5.5), (1.9) and Proposition 1, we have C16 C15 (5.10) M; |S1 |D1 ≤ n+$+2 M: ) )n+$+2 According to (1.9), (5.8), (3:8), (5.10) and Cauchy’s estimate, we can obtain |S0 |D1 ≤
|P2 + grad T(y; x) S0 ; I (y0 )grad T(y; x) N4 + grad Tz S0 ; J grad Tz N4 + R2 |D2 ≤ 2|P2 |D2 + |grad T(y; x) S0 ; I (y0 )grad T(y; x) N4 | + |grad Tz S0 ; J grad Tz N4 | ≤ C17 /2 M ≤ C18 M
(C18 = (3/20 + 1)C17 );
where we have used /2 ¡ 3/20 + 1. Indeed, applying Proposition 1, we have C6 C19 ≤ n+$+2 M: )n+$+2 ) For N4 and N5 similar to (1.9) and applying (3:8), we can obtain |S2 |D3 ≤ C18 M
(5.11)
|N4 |D0 ≤ C4 /2 ( 78 s)4 ≤ C20 ( 78 s)4
(C20 = (3/20 + 1)C4 );
(5.12)
|N5 |D0 ≤ C5 /2 ( 78 s)4 ≤ C21 ( 78 s)4
(C21 = (3/20 + 1)C5 ):
(5.13)
Applying (1.9), (5.10), (5.12), (5.13) and Cauchy’s integral formula, we have |P3 − grad T(y; x) S0 ; I (y0 )grad T(y; x) N5 + grad Tz S0 ; J grad Tz N5 −grad T(y; x) S1 ; I (y0 )grad T(y; x) N4 + grad Tz S1 ; J grad Tz N4 |D2 ≤ C3 M + C22 M + C23 M ≤ C24 M;
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By Lemma 1, we can obtain |S3 |D3 ≤
C25 C6 C24 M ≤ n+$+2 M: )n+$+2 )
(5.14)
Using (5.9), (1.9), (5.10) – (5.14), (3:8) and Cauchy’s integral formula, we derive ˆ D4 ≤ C26 M + C27 M + C28 M + C29 M + C30 M ≤ C31 M: |R|
(5.15)
By (5.10), (5.11) and (5.14), we can obtain |S|D3 ≤ C32 M:
(5.16)
According to Lemma 2, grad T(y; x) S; (I (y) − I (y0 ))grad T(y; x) N = (0; : : : ; 0; grad x S)T ; (I (y) − I (y0 ))(grad y N2 ; 0; · · · ; 0)f
n
l
+ grad T(y; x) S; (I (y) − I (y0 ))grad T(y; x) (N − N2 ) : Using (5.16), (1:18) and Cauchy’s integral formula implies grad T(y; x) S; (I (y) − I (y0 ))grad T(y; x) N @I @(N − N2 ) 8C32 M @I 8C32 M 2 @N2 + ≤ n+$+4 • s )n+$+4 @y @y @y @y ) ≤ C40
M M • s2 + C41 N +$+4 • s2 ≤ (C40 + C41 )Ms): )n+$+4 )
By (5.3), (5.15), (5.16) and Cauchy’s integral formula, we can obtain |P + |D4 ≤ |{P; S}|D4 + |{R; S}|D4 |grad T(y; x) S; (I (y) − I (y0 ))grad T(y; x) N | ≤
C33 M2 2 s )n+$+2
+
C34 M2 2 s )n+$+2
+ (C40 + C41 )Ms)
≤ (C35 + C40 + C41 )Ms): So, by the choice of s and ), if 0 ¡ M0 ¡ [2(C35 + C40 + C41 )]−8(n+$+4) ;
(B)
then |P + | ≤ 12 M+ ¡ M+ :
(5.17)
Using (5.3) – (5.7) and (3:10) we can obtain ˆ N + = (N0 + [P0 ]) + (N21 + Z; y − y0 + F + Q; y ) + (Nˆ + R) +
= N0+ + N2+ + Nˆ :
(5.18)
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255
We need to prove N + satisCes (3:8+ ). In fact, by (3.5) and (5.15), we can obtain 3 1 C31 + ˆ ˆ ˆ (5.19) |N |D6 ≤ |N |D6 + |R|D6 ≤ /2 s + 2n+2$+6 M: 8 ) Put M0 such that 0 ¡ M0 ¡
/20 8C31 + 1
8 ;
(C)
then + |Nˆ |D6 ≤ (/2 + /20 1=8 )( 18 s)3 :
Put
/2+
= /2 +
/20 M 1=8 .
(5.20)
Then
/2i+1 = /20 + /20 M01=8 + /20 M11=8 + · · · + /20 Mi1=8 1 M01=8 ≤ 3/20 + 1; ≤ /20 1 + 1 − 1=2 where we have used the inequality 0 ¡ M0 ¡ ( 12 )64 :
(D)
Obviously, by (D), we can obtain s+ ¡ 18 s: So +
|Nˆ |D( + ;s+ ) ≤ /2+ (s )3 ;
s ≤ s+ :
(5.21)
According to (5.16) and Cauchy’s integral estimate, we have @S ≤ C36 M; @S ≤ C37 M; @x D4 @y )n+$+2 s2 )n+$+2 D4 @S ≤ C38 M: @z s)n+$+2 D4 If 0 ¡ M0 ¡ (C0 + 1)−8 (C36 + C37 + C38 )−8 ;
(E)
then by the deCnation of Q + 1 y y Q: x = x+ + Egrad T S ◦ Qt (y+ ; x+ ; z + ) dt; + 0 z z (C36 + C35 + C37 )(C0 + 1) M ¡ M 3=8 ¡ s+ : s2 )n+$+2 Therefore Q is well deCned on D( + ; s+ ), and |Q − id|D5 ≤
(5.22)
Q: D( + ; s+ ) → D5 :
(5.23)
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
By (3:8+ ); (3:14) and (3:15), we can obtain /1+ ≤ C02 + C0 + 2 00 ; that is |N2+ |D( + ;s ) ≤ /1+ (s )2 ;
s ≤ s+ :
(5.24)
According to (5.17) – (5.19), (5.22) and (5.23), we prove (3:6)–(3:9) hold for (i + 1)th step. One cycle is complete. ∞ Put y0 ∈ G∗ = i=0 Oi . DeCne Wi = Q0 ◦ Q1 ◦ · · · ◦ Qi−1 . Obviously, if 64(n+$+4) 1 8(n+$+4) ; (F) 0 ¡ M0 ¡ min ; 2 24 then ∞ = − 6
∞
1 )i ¿ : 2
i=0
Let D∞ = {x: Re x ∈ T n ; |Im x| ≤ ∞ } × {y = y0 ; z = 0}: By (5.22) and (D), on D∞ , |Wi − id| ≤
i
|Qk − id|Dk ≤ 5
k=1
i
Mi3=8 ¡ 2M03=8 :
k=1
Therefore {Wi } is uniformly convergent on D∞ . Similar to (3:14) and (3:15), if 32 1 1 1 ; (G) ; ; 0 ¡ M0 ¡ min 2 2C12 2C0 C12 then |!∞ (y) − !(y)| ¡ M01=4 ;
|F∞ (y) − F(y)| ¡ M01=4 ;
y ∈ G∗ ;
where !∞ (y) = limi→∞ !i (y); F∞ (y) = limi→∞ Fi (y). Applying (5.22) and putting 0 ¡ M0 ¡ (C39 + 1)−24 ; we can obtain @Qi @(y; x; z) − Id
D6i
≤
(H)
C39 3=8 M ¡ Mi1=24 : si2 )i i
(5.25)
Therefore i−1 ∞ k !
@Wi 1 3 1=24 ≤ (1 + M ) ¡ = ; k @(y; x; z) 3 2 D∞ k=0
where we have used the inequality 132 1 : 0 ¡ M0 ¡ 2
k=0
(I)
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
257
So {@Wi =@(y; x; z)} is convergent on D∞ . Thus, we complete the proof of Theorem A. 6. Some examples In this section we give some examples to illustrate our theorem. Example 1. We consider the following unperturbed system: y˙ = I (y)grad T(y; x) h(y) + I (y)grad T(y; x) F(y); y ; x˙ z˙i = FTi (y; x)zi+2 ; z˙i+2 = −Fi (y; x)zi ;
i = 1; 2;
(6.1)
2 ), and where y ∈ R1 ; x ∈ R3 ; yi = 12 (zi2 + zi+2 0 y y −1 2 y + y y −1 0 I(y) = ; y 1 0 0
1
0
0
√ √ F = ( 2; 3);
h(y) = y:
0
Then I (y) is symplectic and there exists a 0 ¿ 0 such that √ √ |k1 2 + k2 3| ¿ 0 ; 0 ¡ |k| ≤ 3: Hence,
2y + 1 @!(y) = rank 1 = 1 = l; rank @y 0 2 y +y @!(y) @2 !(y) rank !(y); = rank ; y @y @y2 1
2y + 1 1 0
2 0 = 3 = n: 0
Obviously, !(y); F(y; x) satisfy conditions (H1 )– (H2 ) in Theorem A, hence the higherdimensional invariant torus of system (6.1), $y0 = {(y; x; 0; 0) | y = y0 ; x ∈ T 3 } persists under a small perturbation P. Example 2. We discuss the above system (6.1), where y = (y1 ; y2 ) ∈ R2 ; x = 2 (x1 ; x2 ; x3 ; x4 ) ∈ R4 ; yi = 12 (zi2 + zi+2 ), and √ √ h(y) = y1 + y2 ; F(y; x) = ( 2; 3);
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
0
0 3 y + y2 1 1 I(y) = y2 1 y1 1
0
y12
y12
y1
0
0
0
−1
0
y1
−1
0
0
1
0
0
1
0
0
0
0
0
0
0
Obviously I (y) is symplectic. Thus
−1
0 0 : 0 0 0
0
0 3 y1 + y12 T (0; !(y))T = I(y)grad (y; x) h(y) = : y2 1 y1 + 1 1 Also
rank
@!(y) @y
rank !(y);
= rank
= rank
3y12 + 2y1 2y1 1 0
@!(y) @2 !(y) @3 !(y) ; ; @y @y2 @y3
3 y1 + y12 y2 1
y1 + 1 1
0
0 = 1 ¡ 2 = l; 0 0
3y12 + 2y1
6y1 + 2
2y1
2
1
0
0
0
6 0 0 0
= 4 = n:
Here F is as well as in Example 1. Hence, !(y); F(y) satisfy the conditions (H1 ) – (H2 ) in Theorem A, and the higher dimensionian invariant torus $y0 ={(y; x; 0; 0) | y =y0 ; x ∈ T 4 } of system (6.1) persists under the small perturbation P. Example 3. We consider system (6.1) with y1 3 1 2 x ∈ R1 ; y = y2 ∈ R ; h(y) = 2 y3 ; y3
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
and
0
1 I (y) = 0 0
−1
0
0
0
0
0
0
1
0
259
0 : −1 0
In this case m = 0; l = 3; n = 1. By Theorem A, most of the one-dimensional tori persist under small perturbations. Acknowledgements The authors thank the refree for helpful comments. Appendix Lemma 1 (Xu et al. [8]). Assume J ⊂ R is a bounded closed interval; g is a continuously di6erentiable function on J . Let Jh = {x ∈ J : |g(x)| ¡ h};
h ¿ 0:
If there exists a d ¿ 0 such that for some i ≥ 1; |g(i) (x)| ≥ d;
x ∈ J;
then mes(Jh ) ≤ Ch1=i ; where C is a positive constant. Lemma 2 (Parasyuk [6]). Consider the Hamiltonian system z˙ = I (y)grad T H (z);
z = (y; x):
If H (z) = H (y); then the equation takes the following form: y˙ = 0;
x˙ = !(y);
where !(y) = (!1 (y); : : : ; !n (y))T . References [1] F.Z. Cong, Y. Li, Existence of higher dimensional invariant tori for Hamiltonian systems, J. Math. Anal. Appl. 222 (1998) 255–267. [2] L.H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. 4 15 (1988) 115–147. [3] M. Herman, Examples de Mots hamiltoniens dont aucune perturbation en topologie c∞ n’a d’orbites pZeriodiques sur un ouvert de surfaces d’Zenergies, C.R. Acad. Sci. Paris, Ser. I Math. 312 (1991) 989–994.
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[4] J. Moser, On the theory of quasi-periodic motions, SIAM Rev. 8 (1966) 145–172. [5] J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967) 136–176. [6] I.O. Parasyuk, On preservation of multidimensional invariant tori of Hamiltonian systems, Ukrain. Mat. Zh. 36 (1984) 467– 473. [7] J. P[oschel, On the elliptic lower dimensional tori in Hamiltonian systems, Math. Z. 202 (1989) 559–608. [8] J.X. Xu, J.G. You, Q.J. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z. 226 (1997) 375–387.
www.elsevier.nl/locate/na
Persistence of elliptic invariant tori for Hamiltonian systems Huang Qingdao∗ , Cong Fuzhong, Li Yong Department of Mathematics, Jilin University, Changchun, Jilin 130023, People’s Republic of China Received 20 January 1999; accepted 7 September 1999
Keywords: Elliptic invariant tori; KAM iteration; Measure estimate; Nonstandard Hamiltonian system
1. Introduction and result In 1984, Parasyuk [6] considered the existence of invariant tori for Hamiltonian systems with distinct numbers of action-angle variables. Such systems are usually called nonstandard Hamiltonian systems. Next, Herman [3] also considered similar problems. Thereafter, there is an increasing interest in the persistence problem for nonstandard Hamiltonian systems. Recently, Cong and Li [1] gave a KAM theorem for a nonstandard Hamiltonian system. Motivated by above works, we construct symplectic constructure for Hamiltonian functions to study the persistence of elliptic invariant tori for a nonstandard Hamiltonian system. Let G ⊂ Rl be bounded and connected closed domain, and T n = Rn =2Z n usual n-dimension tori. Let l + n be even, and D ⊂ R2m some bounded open subset containing the origin. Consider an n + l + 2m-dimensional complex symplectic manifold (G × T n × D; !2 ), where !2 (· ; E!1 ) = !1 (·); !1 is a 1-form on G × T n × D, I (x; y) 0 E= ; 0 J ∗
Corresponding author.
0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 9 ) 0 0 4 0 7 - 1
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J is the usual 2m-dimensional symplectic matrix, I is an analytic Hamiltonian homeomorphism from the space of 1-form to vector Celds on G×T n ; deCne the corresponding Poisson bracket {· ; ·} as follows: for all smooth functions f1 ; f2 on G × T n × D, let {f1 ; f2 } = df1 (Edf2 ) = !2 (Edf2 ; Edf1 ): Let y ∈ G; x ∈ T n . Assume !2 is invariant relative to the quasi-periodic motions on T n , and {yi ; yj } = 0;
i; j = 1; : : : ; l:
So the coeEcients of 2-form !2 and the matrix I are independent of the angle variable x. In this paper, we consider the nearly integrable Hamiltonian system N + P, where N is an unperturbed system of the following form: N ≡ h(y) + F(y); y + O3 (z);
(1.1) 2m
2 = 12 (zi2 +zi+m );
and P is a perturbation of N , where y ∈ G; z ∈ R ; y 1 ≤ i ≤ m; · ; · denotes the usual inner product, O3 (z) denotes the sum of the power of the order at least three about z. According to Lemma 2 in the appendix and (1.1), for all given c1 ∈ G; c2 ∈ Rm + , the system y˙ x˙ = E grad T N (y; z) (1.2) z˙ admits n + m-dimensional invariant tori, {(y; x; z): x ∈ T n ; y = c1 ; y = c2 } and n-dimensional invariant tori {(c1 ; x; 0): x ∈ T n }, where the latter are the lowerdimensional stable invariant tori of (1.2) with the frequency !(c1 ). By Lemma 2 in the appendix, for y ∈ G, we have that (0; : : : ; 0; !1 (y); : : : ; !n (y))T = I (y)grad T(y; x) N (y; 0);
l
where grad T(y; x) denotes the gradient of a function in the variable (y; x). Assume the following conditions hold: (H1 ) I (y); N (y; z); P(y; x; z) are real analytic functions on G × T n × D; (H2 ) !(y), for y ∈ G satisCes @! = r ≤ min{l; n}; rank @y @ ! l rank !; : ∀ ∈ Z+ ; | | ≤ n − r + 1 = n; @y
(1.3) (1.4)
where @ !=@y =(@| | !1 =@y ; : : : ; @| | !n =@y )T ; | |= 1 +· · ·+ l . Moreover, we assume that there exists 0 ¿ 0 such that |F(y); l | ≥ 0 ;
y ∈ G;
0 = l ∈ Rm :
(1.5)
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
243
Put G = {y: Re y ∈ G; |Im y| ≤ };
={x: Re x ∈ T n ; |Im x| ≤ };
B = {z ∈ C 2m : |z| ≤ }; where for a vector, | · | denotes its maximum norm in components; for a function, | · | denotes its supremum norm. The main result of this paper is the following. Theorem A. Assume (H1 ) and (H2 ) hold. Then there exist a constant M0 ¿ 0 and a nonempty Cantor set G∗ ⊂ G such that |P| ≤ M0 implies that the Hamiltonian system N + P admits an n-dimensional invariant torus Ty ; y ∈ G∗ , whose frequency vector (!∞ (y); F∞ (y)) satis4es max{|!∞ (y) − !(y)|;
|F∞ (y) − F(y)|} ¡ M01=4 ;
where !∞ denotes the tangential frequency and F∞ denotes the normal frequency. Moreover, mesl (G\G∗ ) = O(M01=8(n+$+4)(n−r+1) ); where $ ¿ (n + l)(n − r + 1) + 1 is a constant. Remark 1. If m = 1; l = n, Moser [4,5] gave the following nondegeneracy condition: @F @! @y det @y = 0: !
F
Remark 2. Theorem A corresponds to the result of Eliasson [2] when r = n = l. There, the following condition is required: for all k ∈ Z m \{0}; |k| = |k1 | + · · · + |km | ≤ 3 −1 @F @! : k; F(y) = k; !(y) @y @y Eliasson’s result implies that !∞ (y) is parallel to !(y). In our case, such a result seems impossible. Remark 3. If l ¿ n, that is, the dimension of action variables is larger than one of angle variables, then Theorem A gives a lower-dimensional invariant torus with dimension at most 12 (l + n); if l ¡ n, that is, the dimension of action variables is less than one of angle variables, Theorem A gives a higher-dimensional invariant torus with dimension at most 12 (l + n); if m = 0, Theorem A corresponds to the result of Cong and Li [1] and Parasyuk [6].
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
In the following, we give some notations throughout the paper. Fix y0 ∈ G and D( ; s) = {(y; x; z): Re x ∈ T n ; |Im x| ¡ ; |y − y0 | ¡ s2 ; |z| ¡ s}: Let P be an analytic function on D( ; s); [P] denotes the averaging value over the torus T n+m . Write P = P0 + P 1 + · · · ; where Pk (y; x; z) =
( ;()∈Z+l+2m
1 @ @( P(y; x; 0)(y − y0 ) z ( : !(! @y @z (
(1.6)
2| |+|(|=k
We use the abbreviation Pˆ = P − (P0 + P1 + P2 + P3 ):
(1.7)
According to condition (H1 ), it holds that there exists a constant C0 ¿ 0 such that on G × ×B , 2 @F @h @ h @I (1.8) max |I (y)|; ; |F(y)| ; |h(y)|; @y ; @y2 ≤ C0 : @y @y In the following, we use Ck to denote a positive constant which only depends of C0 ; $; ; m; n; l. By (1.7) and Cauchy’s integral formula, if |P| ≤ M on D( ; s), then on D( ; 78 s), we have |P0 | ≤ M;
|Pi | ≤ Ci M;
i = 1; 2; : : : ; 5:
(1.9)
This paper is arranged as follows: in Section 2, we give an important proposition about a small divisor problem; in Section 3, we give an outline of the proof of Theorem A; in Section 4 we give a measure estimate for the persisting tori; in Section 5, we give a KAM iteration and the proof of Theorem A; in Section 6 we illustrate Theorem A; and in Section 7 we list some preliminary lemmas. 2. A small divisor problem Take y0 ∈ G, and consider @h N2 = (y0 ); y − y0 + F(y0 ); y : @y By Lemma 2 in the appendix, we have (0; : : : ; 0; !1 (y); : : : ; !n (y))T = I (y)grad T(y; x) N2 (y; 0):
l
Assume ! and F satisfy |k; !(y0 ) + l; F(y0 ) | ≥ )(|k| + |l|)−$ ;
0 = (k; l) ∈ Z n+m ;
and (1.5), where $ ≥ (n + l)(n − r + 1) − 1 is a constant.
|l| ≤ 3
(2.1)
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
245
Assume F is a real analytic function deCned on D( ; s) and with at most an order of three. Proposition 1. The equation @’ @’ @N2 = F − [F]; ; !(y0 ) + ;J @x @z @z
[’] = 0
(2.2)
has a unique real analytic solution ’; whose order is as same as that of F; and |’|D( −); s) ≤
C6 |F|D( ; s) : )n+$+2
Proof. BrieMy, denote D = D( ; s). We take a transformation √ xˆi = xi ; 2zi = (1 + −1)(zˆi − zˆi+m ); √ yˆ i = yi ; 2zi+m = (1 − −1)(zˆi + zˆi+m ):
(2.3)
Obviously, transformation (2.3) is a symplectic transformation. So, √ yi = − −1zˆi zˆi+m : We also denote it by z. Then F can be expressed as follows:
√ F(y; x; z) = C(k)exp( −1k; x ) (y − y0 ) z ( ; 2| |+|(|=j
(2.4)
k∈Z n
where ( = (( ; ( ) ∈ Z+2m ; ∈ Z+n ; 0 ≤ j ≤ 3. Using Cauchy’s integral formula we get |C(k)(y − y0 ) z ( | ≤ C7 e− |k| |F|D : DeCne ,(k) =
√ − −1(k; ! + ( − ( ; F )−1
(2.5) (k; ( − ( ) = 0; (k; ( − ( ) = 0:
0
Putting (2.4) into (2.2), we have
√ ’(y; x; z) = ,(k)C(k)exp( −1k; x ) (y − y0 ) z ( : 2| |+|(|=j
Obviously, [’] = 0. Let
√ ’ ; ( = ,(k)C(k)exp( −1k; x )(y − y0 ) z ( : k∈Z n
Then applying (2.5), (2.1) and (1.5), we obtain
|’ ; ( | ≤ |,(k)||C(k)(y − y0 ) z ( |e|k|( −)) k∈Z n
≤ C7
k∈Z n
(2.6)
k∈Z n
1 |F|D e|k|) |k; ! + ( − ( ; F |
(2.7)
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
≤ C7
3$ |F|D + C7 0
k∈Z n \{0}
|k|$ |F|D e|k|) )
∞
≤
2n j n−1 j $ C8 |F|D |F|D + C7 )ej) 0 j=1
∞ 1 C8 C9 |F|D ≤ |F|D + n+$+2 0 ) j 2+[$]−$ j=1
C10 |F|D ; (2.8) )n+$+2 where [$] denotes the integral part of $. Then by (2.4) and (2.6) – (2.8), we have the estimate of the proposition. ≤
3. Outline of the proof of Theorem A Let y0 ∈ G, and rewrite N as follows: @h (y0 ); y − y0 + F(y0 ); y + Nˆ (y; z); N (y; z) = h(y0 ) + @y 1 @2 h Nˆ (y; z) = (y )(y − y ); y − y t 0 0 2! @y2 @F + (y. )(y − y0 ); y + O3 (z); @y
(3.1)
(3.2)
where yt = y0 + t(y − y0 ); y. = y0 + .(y − y0 ); 0 ≤ t; . ≤ 1. Corresponding to (3.1) and (3.2), denote N = N0 + N2 + Nˆ :
(3.3)
Then
N0 = [N0 ] = h(y0 );
N2 =
@h (y0 ); y − y0 @y
+ F(y0 ); y :
(3.4)
By (1.8), on D( ; s1 ), |N2 | ≤ /10 (s1 )2 ;
|Nˆ | ≤ /20 (s1 )3 ;
s1 ≤ s;
(3.5)
where /10 and /20 ¿ 0 are constants. We will prove Theorem A by applying rapidly convergent iteration method. First we choose rapidly convergent sequences Mi+1 = Mi9=8 ; i+1 = i − 6)i ;
)i = Mi1=8(n+$+4) ;
si = Mi1=8 ;
i = 0; 1; 2; : : : ;
where M0 is the positive constant satisfying the following inequalities (A)−(I ); 0 = .
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
Let Dji
7−j si = D i − j)i ; 8 |y| ¡
7−j si 8
2
247
(y; x; z): Re x ∈ T n ; |Im x| ¡ i − j)i ;
=
7−j si ; |z| ¡ 8
; 0 ≤ j ≤ 6:
Assume the Hamiltonian function N + P is taken into N i + P i by a symplectic transformation on D0i , such that i
N i = N0i + N2i + Nˆ ;
N0i = [N0i ];
(3.6i )
i (y0 ); y − y0 + Fi (y0 ); y ; N2i = N21
(3.7i )
i |Nˆ |D( i ;s1 ) ≤ /2i (s1 )3 ; s1 ≤ si ;
|N2i |D( i ;s1 ) ≤ /1i (s1 )2 ;
(3.8i )
|P i | ≤ Mi :
(3.9i )
Let Oi = {y ∈ G: |k; !i (y) + l; Fi (y) | ≥ )i (|k| + |l|)−$ ; 0 = (k; l) ∈ Z n+m ; |l| ≤ 3}; where {0; : : : ; 0; !1 (y0 ); : : : ; !n (y0 )}T = I (y0 )grad T(y; x) N2i (y; 0)|y=y0 . When y0 ∈ Oi , one
l
can construct a symplectic transformation Qi such that (N i + P i ) ◦ Qi = N i+1 + P i+1 : We shall prove N i+1 and P i+1 satisfying (3:6i+1 ) − (3:9i+1 ), where N 0 = N; P 0 = P. If ∞ y0 ∈ Ok = 1; k=0
then the above iteration procedure can continue. Denote [P2i ] = Z i (y0 ); y − y0 + Qi (y0 ); y :
(3.10i )
According to (1.9), (3:10), the deCnition of si and Cauchy’s estimate, we have C11 |Z i |D1i ; |Qi |D1i ≤ 2 |P2i |D0i ≤ C12 Mi3=4 : si Put i+1 i = N21 + Z i; N21
We can prove that on |!i+1 − !0 | ≤
Fi+1 = Fi + Qi :
(3.11i )
D1i+1 ,
i
k=0
|!k+1 − !k |Dk ≤ 1
i
k=0
|I ||grad T(y; x) N2k (y; 0)|Dk
1
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
=
i
|I ||Z k |Dk ≤ C0 C12 1
k=0
|Fi+1 − F0 | ≤
i
≤ C12
k=0
|Fk+1 − Fk |Dk =
i
1
k=0 i
k=0
Mk3=4 ¡
i
Mk3=4 ¡ 00 ;
(3.12i )
|Qk |Dk
k=0
1
00 : 3
(3.13i )
Here we have used the inequality 32 00 1 00 ; M0 ¡ min ; ; 2 2(C12 + 1) 6(C0 C12 + 1)
(A)
where 00 will be given in Section 4. Hence, |!i+1 | ≤ |!0 | + |!i+1 − !0 | ≤ C02 + 00 ;
(3.14i )
00 ; 3
(3.15i )
|Fi+1 | ≤ |F0 | + |Fi+1 − F0 | ≤ C0 + where !0 = !; F0 = F.
4. Estimate of the measure in Theorem A Let B ⊂ Rn be a bounded connected closed domain, g : B → Rn a C n−r+1 - continuously diRerentiable vector-valued function, such that on B @g = r; rank @y @ g n rank g; : ∈ Z+ ; | | ≤ n − r + 1 = n: (4.1) @y Proposition 2. If g˜ : B → Rn ; g:B T → R are C n−r+1 -continuously di6erentiable functions; and g satis4es (4:1); then there exists 0 ¿ 0 such that whenever max{|g|; T |g|} ˜ ¡ 0 ; the set ˜ + g(y)| T ≥ )|k|−$ ; k ∈ Z n \{0}} B0 = {y ∈ B : |k; g(y) + g(y) is a positive measure set; and mesn (B\B0 ) ≤ 4)1=(n−r+1) ; where 4 ¿ 0 is a constant. T and {DVrT g(y) : ∀V ∈ Rn } Proof. According to Xu et al. [8], {@( g=@y( : ∀(; |(| = r} rT rT are equivalent, where rT ¿ 0 is an integer, DV g(y) = d =dt rTg(y + Vt)|t=0 . By the
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
249
nondegeneracy condition and the Cnite covering theorem, there exist n integers 1 ≤ r1 ; r2 ; : : : ; rn ≤ n − r + 1 and n vectors V1 ; V2 ; : : : ; Vn ∈ Rn such that rank{DVr11 g(y); DVr22 g(y); : : : ; DVrnn g(y)} = n;
y ∈ G:
(4.2)
Denote A(y) = (DVr11 g(y); DVr22 g(y); : : : ; DVrnn g(y)). By (4.1) there exists 41 ¿ 0 such that for all (y; V ) ∈ B × S, |A(y)V | ≥ 41 ;
(4.3)
where S = {V ∈ Rn : |V1 | + · · · + |Vn | = 1}. Obviously, for k ∈ Z n \{0}, ri D g(y); k ≥ 41 ; y ∈ B; i = 1; 2; : : : ; n: Vi |k|
(4.4)
Let f(y) = k; g(y) + g(y) ˜ + g(y): T According to (4.4), there exists 1 ¿ 0, such that when |g|C n−r+1 ¡ 1 , 1 ri 1 ri k D f(y) = Dri g(y) + Dri g(y); + DVi g(y) T ≥ 42 ¿ 0; Vi ˜ |k| Vi Vi |k| |k| where 42 ¿ 0 is a constant. Let RkVi = {t: |f(y + tVi )| ≤ )|k|−$ ;
y ∈ B;
y + tVi ∈ B};
Rk = {y ∈ B: |f(y)| ≤ )|k|−$ }: According to Proposition 1, mes(RkVi ) ≤ 43 )1=(n−r+1) |k|−$−1 ; where 43 ¿ 0 is a constant; moreover, mes(Rk ) ≤ 43 (diam B)n−1 )1=(n−r+1) |k|−$−1 ; where “diam” denotes the diameter of the set. Hence mes(B0 ) = mesB\ Rk k∈Z n \{0}
≥ mes(B) − 43 (diam B)n−1 )1=(n−r+1)
|k|−$−1
k∈Z n \{0}
≥ mes(B) − 44 (diam B)n−1 )1=(n−r+1)
∞
j=1
1 j 2+$−n
≥ mes(B) − 45 )1=(n−r+1) ; where 44 ; 45 ¿ 0 are constants. This completes the proof.
(see [7])
(4.5)
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
Now let us discuss the estimate of the measure in theorem A. Denote ∞ G∗ = Oi : i=0
(1) l = n. By the nondegeneracy condition (H2 ) and Proposition 2, there exists 00 ¿ 0, such that when max{|!|; ˜ |!|} ¡ 00 , mesn (G\G∗ ) ≤ C13 )1=(n−r+1) ;
(4.6)
where GX = {y ∈ G: |k; ! + ! ˜ + !| ≥ )|k|−$ ; 0 = k ∈ Z n }. By (3:12)i , (3:13)i and (4.6), for every , ∈ Z m ; 0 ≤ |,| ≤ 3, mesn (G\Oi; , ) ≤ C13 )i1=(n−r+1) ;
(4.7)
where Oi; , = {y ∈ G: |k; !i + ,; Fi | ≥ )i |k|−$ ; 0 = k ∈ Z n }: Because of
Oi =
Oi; , ;
,∈Z m ;0≤|,|≤3
by (4.7) we can obtain mesn (G\Oi ) ≤ C14 )i1=(n−r+1) :
(4.8)
Moreover, mesn (G\G∗ ) ≤
∞
mesn (G\Oi ) ≤ C14
i=0
∞
)i1=(n−r+1)
i=0
≤ 2C14 )01=(n−r+1) = O(M01=8(n−r+1)(n+$+4) ):
(4.9)
(2) l ¡ n. Denote yˆ = (y; yl+1 ; : : : ; yn ); !( ˆ y) ˆ = !(y); !ˆ i (y) ˆ = !i (y); G = G × [1; 2] × · · · × [1; 2]. According to the nondegeneracy condition (H2 ),
n−l
rank
@!ˆ @yˆ
= r; yˆ ∈ G ;
@ !ˆ : ∀ ; 0 ¡ | | ≤ n − l + 1 = n; yˆ ∈ G : rank !; ˆ @yˆ
(4.10)
We deCne Oi = Oi × [1; 2] × · · · × [1; 2];
n−l
G∗
= G∗ × [1; 2] × · · · × [1; 2] :
n−l
(4.11)
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
251
Then mesn (G \G∗ ) = O(M01=8(n−r+1)(n+$+4) ): By Fubini’s theorem, mesl (G\G∗ ) = O(M01=8(n−r+1)(n+$+4) ):
(4.12)
T there exist integers (3) l ¿ n. By the nondegeneracy condition, for every y ∈ G, i1 ; : : : ; ir ; : : : ; il , 1 ≤ ij ≤ l; j = r; ij = ir and multi-indices 1 ; : : : ; n−r ; | i | ≥ 2; i = 1; : : : ; n − r, such that @! @! rank = r; (4.13) ;:::; @yi1 @yir rank
1
n−r
@! @! @ ! @ ! ;:::; ; 1 ; : : : ; n−r @yi1 @yir @y @y
= n:
(4.14)
Let !(y) ˜ = (!(y); yin+1 ; : : : ; yil ). Then @!˜ @!˜ rank = r; ;:::; @yi1 @yir rank
1
(4.15) n−r
@!˜ @ !˜ @!˜ @!˜ @!˜ @ !˜ ; 1 ; : : : ; n−r ; ;:::; ;:::; @yi1 @yir @y @yin+1 @yil @y
1
n−r
@ ! @! @ ! @! @y ; : : : ; @y ; @y 1 ; : : : ; @y n−r ; =rank i1 ir 0;
∗ = l;
(4.16)
El−n
where El−n denotes l − n-dimensionial unit matrix. Without loss of generality, we assume (4.15) and (4.16) hold on G. DeCne !˜ i (y) = (!i (y); yin+1 ; : : : ; yil ); T y T −$ ; Oi = {y ∈ G : |k; !i+1 (y) + k; T + ,; Fi (y) | ≥ )i (|k| + |k|) T ∈ Z l ; 0 ≤ |,| ≤ 3; , ∈ Z m }; 0 = (k; k) where yT = (yin+1 ; : : : ; yl ); kT = (kn+1 ; : : : ; kl ); $ ≥ (n + l)(n − k + 2) − 1. Let G∗ =
∞
Oi :
i=0
Similar to case 1, for the dimension l, by (4.15), (4.16) and Proposition 2, we can prove that mesl (G\G∗ ) = O(M01=8(n−r+1)(n+$+4) ):
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
Obviously, G∗ ⊃ G∗ , so we have mesl (G\G∗ ) = O(M01=8(n−r+1)(n+$+4) ):
(4.17)
By (4.9), (4.12) and (4.17), we complete the estimate of the measure in Theorem A. 5. KAM iteration and proof of Theorem A We only prove one iteration cycle. That is, if we assume (3:6k ) and (3:9k ) hold for k, we need to prove that they also hold for k + 1. For simplicity, we omit “k” and rewrite “k + 1” as “+”. We apply Eliasson’s reductive iteration scheme [2]. Now we construct the functions S = S0 + S1 + S2 + S3 and R = R0 + R2 + Rˆ such that grad T(y; x) S; I (y0 )grad T(y; x) N + grad Tz S; J grad Tz N = P − R; [S] = 0;
[R0 + R2 ] = R0 + R2 :
(5.1) 1
t
Let Q be phase Mow determined by EdS. Denote Q = Q . Then Q is a symplectic transformation. Applying d=dtF ◦ Qt = {F; S} ◦ Qt and Taylor’s formula, we can obtain 1 + (N + P) ◦ Q = N + {tP + (1 − t)R; S} ◦ Qt dt 0
+grad T(y; x) S; (I (y) − I (y0 ))grad T(y; x) N ; N + = N + R;
P+ =
0
1
(5.2)
{tP + (1 − t)R; S} ◦ Qt dt
+grad T(y; x) S; (I (y) − I (y0 ))grad T(y; x) N :
(5.3)
In a general KAM iteration method, iteration and estimate of the measure are two independent processes, and it is necessary to Cx the action variables in the iteration process. Because we consider a nonstandard elliptic Hamiltonian system, whose Hamiltonian vector Celd is I (y)grad T H (y; x; z), we need to modify Eliasson’s iteration scheme (its vector is J grad T H (y; x; z)) in Cxing y = y0 . Resolve (5.1) as follows: grad T(y; x) S0 ; I (y0 ) grad T(y; x) N2 + grad Tz S0 ; J grad Tz N2 = P0 − R0 ; [S0 ] = 0;
[R0 ] = R0 ;
grad T(y; x) S1 ; I (y0 )grad T(y; x) N2 + grad Tz S1 ; J grad Tz N2 = P1 ;
(5.4) (5.5)
grad T(y; x) S2 ; I (y0 ) grad T(y; x) N2 + grad Tz S2 ; J grad Tz N2 = P2 −grad T(y; x) S0 ; I (y0 )grad T(y; x) N4 + grad Tz S0 ; J grad Tz N4 − R2 ; [S2 ] = 0;
[R2 ] = R2 ;
(5.6)
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
253
grad T(y; x) S3 ; I (y0 )grad T(y; x) N2 + grad Tz S3 ; J grad Tz N2 = P3 − grad T(y; x) S0 ; I (y0 )grad T(y; x) N5 + grad Tz S0 ; J grad Tz N5 −grad T(y; x) S1 ; I (y0 )grad T(y; x) N4 + grad Tz S1 ; J grad Tz N4 :
(5.7)
Put R0 = [P0 ];
R2 = [P2 ];
(5.8)
Rˆ = Pˆ − grad T(y; x) S0 ; I (y0 )grad T(y; x) (Nˆ − N4 − N5 ) +grad Tz S0 ; J grad Tz (Nˆ − N4 − N5 ) −grad T(y; x) S1 ; I (y0 )grad T(y; x) (Nˆ − N4 ) +grad Tz S1 ; J grad Tz (Nˆ − N4 ) − grad T(y; x) S2 ; I (y0 )grad T(y; x) Nˆ +grad Tz S2 ; J grad Tz Nˆ − grad T(y; x) S3 ; I (y0 )grad T(y; x) Nˆ +grad Tz S3 ; J grad Tz Nˆ :
(5.9)
Applying (3:9), (5.4), (5.8), (5.5), (1.9) and Proposition 1, we have C16 C15 (5.10) M; |S1 |D1 ≤ n+$+2 M: ) )n+$+2 According to (1.9), (5.8), (3:8), (5.10) and Cauchy’s estimate, we can obtain |S0 |D1 ≤
|P2 + grad T(y; x) S0 ; I (y0 )grad T(y; x) N4 + grad Tz S0 ; J grad Tz N4 + R2 |D2 ≤ 2|P2 |D2 + |grad T(y; x) S0 ; I (y0 )grad T(y; x) N4 | + |grad Tz S0 ; J grad Tz N4 | ≤ C17 /2 M ≤ C18 M
(C18 = (3/20 + 1)C17 );
where we have used /2 ¡ 3/20 + 1. Indeed, applying Proposition 1, we have C6 C19 ≤ n+$+2 M: )n+$+2 ) For N4 and N5 similar to (1.9) and applying (3:8), we can obtain |S2 |D3 ≤ C18 M
(5.11)
|N4 |D0 ≤ C4 /2 ( 78 s)4 ≤ C20 ( 78 s)4
(C20 = (3/20 + 1)C4 );
(5.12)
|N5 |D0 ≤ C5 /2 ( 78 s)4 ≤ C21 ( 78 s)4
(C21 = (3/20 + 1)C5 ):
(5.13)
Applying (1.9), (5.10), (5.12), (5.13) and Cauchy’s integral formula, we have |P3 − grad T(y; x) S0 ; I (y0 )grad T(y; x) N5 + grad Tz S0 ; J grad Tz N5 −grad T(y; x) S1 ; I (y0 )grad T(y; x) N4 + grad Tz S1 ; J grad Tz N4 |D2 ≤ C3 M + C22 M + C23 M ≤ C24 M;
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
By Lemma 1, we can obtain |S3 |D3 ≤
C25 C6 C24 M ≤ n+$+2 M: )n+$+2 )
(5.14)
Using (5.9), (1.9), (5.10) – (5.14), (3:8) and Cauchy’s integral formula, we derive ˆ D4 ≤ C26 M + C27 M + C28 M + C29 M + C30 M ≤ C31 M: |R|
(5.15)
By (5.10), (5.11) and (5.14), we can obtain |S|D3 ≤ C32 M:
(5.16)
According to Lemma 2, grad T(y; x) S; (I (y) − I (y0 ))grad T(y; x) N = (0; : : : ; 0; grad x S)T ; (I (y) − I (y0 ))(grad y N2 ; 0; · · · ; 0)f
n
l
+ grad T(y; x) S; (I (y) − I (y0 ))grad T(y; x) (N − N2 ) : Using (5.16), (1:18) and Cauchy’s integral formula implies grad T(y; x) S; (I (y) − I (y0 ))grad T(y; x) N @I @(N − N2 ) 8C32 M @I 8C32 M 2 @N2 + ≤ n+$+4 • s )n+$+4 @y @y @y @y ) ≤ C40
M M • s2 + C41 N +$+4 • s2 ≤ (C40 + C41 )Ms): )n+$+4 )
By (5.3), (5.15), (5.16) and Cauchy’s integral formula, we can obtain |P + |D4 ≤ |{P; S}|D4 + |{R; S}|D4 |grad T(y; x) S; (I (y) − I (y0 ))grad T(y; x) N | ≤
C33 M2 2 s )n+$+2
+
C34 M2 2 s )n+$+2
+ (C40 + C41 )Ms)
≤ (C35 + C40 + C41 )Ms): So, by the choice of s and ), if 0 ¡ M0 ¡ [2(C35 + C40 + C41 )]−8(n+$+4) ;
(B)
then |P + | ≤ 12 M+ ¡ M+ :
(5.17)
Using (5.3) – (5.7) and (3:10) we can obtain ˆ N + = (N0 + [P0 ]) + (N21 + Z; y − y0 + F + Q; y ) + (Nˆ + R) +
= N0+ + N2+ + Nˆ :
(5.18)
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
255
We need to prove N + satisCes (3:8+ ). In fact, by (3.5) and (5.15), we can obtain 3 1 C31 + ˆ ˆ ˆ (5.19) |N |D6 ≤ |N |D6 + |R|D6 ≤ /2 s + 2n+2$+6 M: 8 ) Put M0 such that 0 ¡ M0 ¡
/20 8C31 + 1
8 ;
(C)
then + |Nˆ |D6 ≤ (/2 + /20 1=8 )( 18 s)3 :
Put
/2+
= /2 +
/20 M 1=8 .
(5.20)
Then
/2i+1 = /20 + /20 M01=8 + /20 M11=8 + · · · + /20 Mi1=8 1 M01=8 ≤ 3/20 + 1; ≤ /20 1 + 1 − 1=2 where we have used the inequality 0 ¡ M0 ¡ ( 12 )64 :
(D)
Obviously, by (D), we can obtain s+ ¡ 18 s: So +
|Nˆ |D( + ;s+ ) ≤ /2+ (s )3 ;
s ≤ s+ :
(5.21)
According to (5.16) and Cauchy’s integral estimate, we have @S ≤ C36 M; @S ≤ C37 M; @x D4 @y )n+$+2 s2 )n+$+2 D4 @S ≤ C38 M: @z s)n+$+2 D4 If 0 ¡ M0 ¡ (C0 + 1)−8 (C36 + C37 + C38 )−8 ;
(E)
then by the deCnation of Q + 1 y y Q: x = x+ + Egrad T S ◦ Qt (y+ ; x+ ; z + ) dt; + 0 z z (C36 + C35 + C37 )(C0 + 1) M ¡ M 3=8 ¡ s+ : s2 )n+$+2 Therefore Q is well deCned on D( + ; s+ ), and |Q − id|D5 ≤
(5.22)
Q: D( + ; s+ ) → D5 :
(5.23)
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
By (3:8+ ); (3:14) and (3:15), we can obtain /1+ ≤ C02 + C0 + 2 00 ; that is |N2+ |D( + ;s ) ≤ /1+ (s )2 ;
s ≤ s+ :
(5.24)
According to (5.17) – (5.19), (5.22) and (5.23), we prove (3:6)–(3:9) hold for (i + 1)th step. One cycle is complete. ∞ Put y0 ∈ G∗ = i=0 Oi . DeCne Wi = Q0 ◦ Q1 ◦ · · · ◦ Qi−1 . Obviously, if 64(n+$+4) 1 8(n+$+4) ; (F) 0 ¡ M0 ¡ min ; 2 24 then ∞ = − 6
∞
1 )i ¿ : 2
i=0
Let D∞ = {x: Re x ∈ T n ; |Im x| ≤ ∞ } × {y = y0 ; z = 0}: By (5.22) and (D), on D∞ , |Wi − id| ≤
i
|Qk − id|Dk ≤ 5
k=1
i
Mi3=8 ¡ 2M03=8 :
k=1
Therefore {Wi } is uniformly convergent on D∞ . Similar to (3:14) and (3:15), if 32 1 1 1 ; (G) ; ; 0 ¡ M0 ¡ min 2 2C12 2C0 C12 then |!∞ (y) − !(y)| ¡ M01=4 ;
|F∞ (y) − F(y)| ¡ M01=4 ;
y ∈ G∗ ;
where !∞ (y) = limi→∞ !i (y); F∞ (y) = limi→∞ Fi (y). Applying (5.22) and putting 0 ¡ M0 ¡ (C39 + 1)−24 ; we can obtain @Qi @(y; x; z) − Id
D6i
≤
(H)
C39 3=8 M ¡ Mi1=24 : si2 )i i
(5.25)
Therefore i−1 ∞ k !
@Wi 1 3 1=24 ≤ (1 + M ) ¡ = ; k @(y; x; z) 3 2 D∞ k=0
where we have used the inequality 132 1 : 0 ¡ M0 ¡ 2
k=0
(I)
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
257
So {@Wi =@(y; x; z)} is convergent on D∞ . Thus, we complete the proof of Theorem A. 6. Some examples In this section we give some examples to illustrate our theorem. Example 1. We consider the following unperturbed system: y˙ = I (y)grad T(y; x) h(y) + I (y)grad T(y; x) F(y); y ; x˙ z˙i = FTi (y; x)zi+2 ; z˙i+2 = −Fi (y; x)zi ;
i = 1; 2;
(6.1)
2 ), and where y ∈ R1 ; x ∈ R3 ; yi = 12 (zi2 + zi+2 0 y y −1 2 y + y y −1 0 I(y) = ; y 1 0 0
1
0
0
√ √ F = ( 2; 3);
h(y) = y:
0
Then I (y) is symplectic and there exists a 0 ¿ 0 such that √ √ |k1 2 + k2 3| ¿ 0 ; 0 ¡ |k| ≤ 3: Hence,
2y + 1 @!(y) = rank 1 = 1 = l; rank @y 0 2 y +y @!(y) @2 !(y) rank !(y); = rank ; y @y @y2 1
2y + 1 1 0
2 0 = 3 = n: 0
Obviously, !(y); F(y; x) satisfy conditions (H1 )– (H2 ) in Theorem A, hence the higherdimensional invariant torus of system (6.1), $y0 = {(y; x; 0; 0) | y = y0 ; x ∈ T 3 } persists under a small perturbation P. Example 2. We discuss the above system (6.1), where y = (y1 ; y2 ) ∈ R2 ; x = 2 (x1 ; x2 ; x3 ; x4 ) ∈ R4 ; yi = 12 (zi2 + zi+2 ), and √ √ h(y) = y1 + y2 ; F(y; x) = ( 2; 3);
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
0
0 3 y + y2 1 1 I(y) = y2 1 y1 1
0
y12
y12
y1
0
0
0
−1
0
y1
−1
0
0
1
0
0
1
0
0
0
0
0
0
0
Obviously I (y) is symplectic. Thus
−1
0 0 : 0 0 0
0
0 3 y1 + y12 T (0; !(y))T = I(y)grad (y; x) h(y) = : y2 1 y1 + 1 1 Also
rank
@!(y) @y
rank !(y);
= rank
= rank
3y12 + 2y1 2y1 1 0
@!(y) @2 !(y) @3 !(y) ; ; @y @y2 @y3
3 y1 + y12 y2 1
y1 + 1 1
0
0 = 1 ¡ 2 = l; 0 0
3y12 + 2y1
6y1 + 2
2y1
2
1
0
0
0
6 0 0 0
= 4 = n:
Here F is as well as in Example 1. Hence, !(y); F(y) satisfy the conditions (H1 ) – (H2 ) in Theorem A, and the higher dimensionian invariant torus $y0 ={(y; x; 0; 0) | y =y0 ; x ∈ T 4 } of system (6.1) persists under the small perturbation P. Example 3. We consider system (6.1) with y1 3 1 2 x ∈ R1 ; y = y2 ∈ R ; h(y) = 2 y3 ; y3
H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
and
0
1 I (y) = 0 0
−1
0
0
0
0
0
0
1
0
259
0 : −1 0
In this case m = 0; l = 3; n = 1. By Theorem A, most of the one-dimensional tori persist under small perturbations. Acknowledgements The authors thank the refree for helpful comments. Appendix Lemma 1 (Xu et al. [8]). Assume J ⊂ R is a bounded closed interval; g is a continuously di6erentiable function on J . Let Jh = {x ∈ J : |g(x)| ¡ h};
h ¿ 0:
If there exists a d ¿ 0 such that for some i ≥ 1; |g(i) (x)| ≥ d;
x ∈ J;
then mes(Jh ) ≤ Ch1=i ; where C is a positive constant. Lemma 2 (Parasyuk [6]). Consider the Hamiltonian system z˙ = I (y)grad T H (z);
z = (y; x):
If H (z) = H (y); then the equation takes the following form: y˙ = 0;
x˙ = !(y);
where !(y) = (!1 (y); : : : ; !n (y))T . References [1] F.Z. Cong, Y. Li, Existence of higher dimensional invariant tori for Hamiltonian systems, J. Math. Anal. Appl. 222 (1998) 255–267. [2] L.H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. 4 15 (1988) 115–147. [3] M. Herman, Examples de Mots hamiltoniens dont aucune perturbation en topologie c∞ n’a d’orbites pZeriodiques sur un ouvert de surfaces d’Zenergies, C.R. Acad. Sci. Paris, Ser. I Math. 312 (1991) 989–994.
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H. Qingdao et al. / Nonlinear Analysis 45 (2001) 241 – 260
[4] J. Moser, On the theory of quasi-periodic motions, SIAM Rev. 8 (1966) 145–172. [5] J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967) 136–176. [6] I.O. Parasyuk, On preservation of multidimensional invariant tori of Hamiltonian systems, Ukrain. Mat. Zh. 36 (1984) 467– 473. [7] J. P[oschel, On the elliptic lower dimensional tori in Hamiltonian systems, Math. Z. 202 (1989) 559–608. [8] J.X. Xu, J.G. You, Q.J. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z. 226 (1997) 375–387.