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1 Journal of Differential Equations (994), 0, Arnold diffusion and oscillatory solutions in the planar three-body problem Zhihong Xia Center for Dynamical Systems and Nonlinear Studies Georgia Institute of Technology, Atlanta, Georgia Abstract () (V.I.Arnold, 964, Dokl. Acad. Nauk SSSR 56, 9) (2) (escape) (3) (pseudo Arnold difusion) (4). [2] n (Weinstein[20], Arnold[4]) 2 n [2] Holmes and Marsden[9] Lim[0] [2] ( [2], Holmes and Marsden[9], Robinson[7] ) Xia[2]

2 6 Sitnikov Sitnikov[9] m 3 =0 Alekseev[] m 3 > 0 McGehee[] Moser[3] Sitnikov Easton and McGehee[8] - - Easton[6] - Robinson[6] Easton[7] KAM 7 ( S 3 ) S 3 S 3 S 3 S 3 S 3 (exact) KAM S 3 S 3 (pseudo Arnold difusion) 7 2. m,m 2 m 3 3 2

3 r, r 2, r 3 m,m 2 m 3 h C Robinson[6] Easton[6] [6] q R 2 m m 2 Q R 2 m m 2 m 3 q = r 2 r, Q = ( r 3 m r + m 2 r 2 m + m 2 ) = Mμ r 3. M = m + m 2 + m 3 μ = m + m 2 p R 2 P R 2 q Q Q = k P, Ṗ = Mk Q 3 Q + O( Q 3 ), q = k2 p, ṗ = μk 2 q 3 q + O( Q 3 ), () k = m 3 μm k 2 = m m 2 μ ω( α) t ( )q Q P m 3 Q = α = Mμ Q = αx 2 s Q = αx 2 s P = k αys + k αx 2 ρis. is s ρ m 3 m 3 () s q = zs p = k 2 μ ws, q p z w( ) ẋ = 2 x3 y, ẏ = μx 4 + O(x 6 ), ż = μ w + O(x 4 ), ẇ = μ 2 z 3 z + O(x 4 ), ṡ = x 4 ρis, ρ = O(x 6 ). (2) C = m 3 ρ + k2 2 2μi (zw wz), h = 2 m 3y k3 2 μ 2 ww + m m 2 z + O(x 2 ). 3

4 (2) s s m 3 0 ρ ρ C m 3 =0 ρ m m 2 z =0 Levi-Civita z =2μξ 2, w = μh ηξ, K =4(ξξ + ηη)μξξ, K ẋ = 2 Kx3 y, ẏ = Kμx 4 + O(x 6 ), ξ = η + O(x 4 ), η = ξ + O(x 4,y 2 ), ρ = O(x 6 ). (3) h = H = 2 m 3y 2 k2 3(ξξ + ηη)+o(x2 ), C = m 3 ρ + k2 2 μ hi (ξη ηξ). x =0 H = h<0 y x =0 y S 3 Hopf ω ( α ) t ( ) x(t) y(t) ω ( α ) x =0 y =0 3 S 3 W s (S 3 )( W u (S 3 )) x>0 S 3 (normally hyperbolic) (3) x y x 3 W s (S 3 ) W u (S 3 ) Robinson 2.(Robinson[6]). (a) (3) {x 0} S 3 (x =0 )C W s (S 3 ) {x>0} W u (S 3 ) {x>0} (b) π : S 3 S 3 / = S 2 S 3 / = S 2 Hopf α : W u (S 3 ) S 2 W u (S 3 ) S 3 α π ω : W s (S 3 ) S 2 α ω C ( {x >0} ) 4

5 (graph transform) Robinson[6] 3. H ε (q, p, x, y) =F (q, p)+g(x, y)+εh (q, p, x, y, t). (4) (q, p, x, y) 2(n +) M q p x =(x,x 2,...,x n ) y =(y,y 2,...,y n ) ε H t 2π - (θ,θ 2,...,θ n,i,i 2,...,I n ) (4) H 0 = F (q, p)+g(x, y) =F (q, p)+ H ε = H 0 (q, p, I)+εH (q, p, θ, I, t). n G i (I i ), i= (5) H θ,θ 2,...,θ n t 2π Ω i (I i )= G i(i i ) I i 0, (4) (5) {(q, p, θ, I, t) t = t 0 [0, 2π)} Σ t 0 Π :Σ t 0 Σ t 0 Σ t 0 Π Σ t 0 ε =0 (5) F (q, p) (q 0,p 0 ) γ =(q(t), p(t)) H 0 Σ t 0 n T (h,h 2,...,h n ) n I i = h i =, i =, 2,...,n t = t 0, θ i [0, 2π), i =, 2,...,n q = q 0, p = p 0. (6) T (h,h 2,...,h n ) (n +) I i = h i i =, 2,...,n t = t 0, θ i S q = q(s), p = p(s), s R, n T (h,h 2,...,h n ) 5

6 ε 0 Σ t 0 p = p 0,q = q 0 2n 2 N 0 ε 0 N 0 Σ t 0 N ε N ε N Σ t 0 2- N ε 2- KAM(Kolmogorov-Arnold-Moser) N ε Ω i (I i )= G i (I i )/ I i 0 T (h,h 2,...,h n ) T ε (h,h 2,...,h n ) T ε (h,h 2,...,h n ) T ε (h,h 2,...,h n ) ε ε =0 2 ε 0 (Arnold[4], Weinstein[20]) 2 [2,2,9] T ε (h,h 2,...,h n ) λ (tangle) T ε (h,h 2,..., h n ) Tε,T2 ε,...,tk ε i<k Tε i i+ Tε T ε i+ Tε i λ i j k Tε i T ε j T ε j T ε i (erratic) H 0 KAM ε KAM ε 0 6

7 (escape) 4. (reduction) m m 2 m 3 m m 2 s = e iθ, z = re iφ, θ, r, φ μ = m + m 2 = (2) x ẋ = 2 x3 y, ẏ = x 4 + x 6 ρ 2 + m 2 g (x, r, φ)+o(m 3,m 2 2), ρ = m 2 g 2 (x, r, φ)+o(m 3,m 2 2), θ = x 4 ρ, ż = w x 4 ρzi, ẇ = z 3 z x 4 ρwi + O(m 3 ). (7) g (x, r, φ) g 2 (x, r, φ) g (x, r, φ) =x 4 +2x 6 r cos φ + x 4 + x 6 r cos φ (x 4 2x 2 r cos φ +) 3/2, g 2 (x, r, φ) = x 6 r sin φ + x 6 r sin φ (x 4 2x 2 r cos φ +) 3/2. θ θ m 3 =0 m m 2 a( e r = 2 ) +ecos(φ + θ), φ + θ = φ 0 + θ 0 + a 3/2 (t +2esin t)+o(e 2 ). ( ) ( ) /2 a = 2h e = +2 hc 2 m 2 ( m 2 ) m 3 2( m 2 ) 3. h C a = h = 2 m 2( m 2 ) 7

8 e =0 2hC 2 + m 3 2 ( m 2) 3 =0 Jacobi 2 y2 + 2 x4 ρ 2 U ρ = J. (8) J Jacobi U m m 2 ) U = x 4 + m 2 x ( 2 x 2 r cos φ + + O(m 2 ( 2x 2 r cos φ + x 4 ) 3/2 2 ). e 0 m 3 0 (8) J J w = Re iφ, 2 R2 r = m 2 ( m 2 ) h + O(m 3 )= 2 + O(m 3), Rr sin(φ φ) =m 2 ( m 2 ) C + O(m 3 )= e 2 + O(m 3 ). r R 2 φ Φ r = r(φ, Φ,e)=+O(m 3,e), R = R(φ, Φ,e)=+O(m 3,e). t φ dφ dt = x4 ρ + O(m 3,e), (7) dx dφ = x 4 ρ + O(m 3,e), dy dφ = x4 + x 6 ρ 2 + m 2 g (x, r, φ) + O(m x 4 3,m 2 2 ρ,e), dj dφ = m 2eF (x, y, J, Φ,φ)+O(m 3,m 2 2,e2 ), dφ dφ =+O(m 3,m 2,e). 2 x3 y (9) F x, y, J, Φ φ F (9) m 2,e m 3 ue (φ ψ)i = e φi + R 2 r sin(φ φ)ie Φi. (0) 8

9 u u = e + O(m 3 ) u ( 0 m 3 e ) ψ ψ Φ x, y, J ψ φ dx dφ = 2 x3 y x 4 ρ + O(m 3,e), dy dφ = x4 + x 6 ρ 2 + m 2 g (x, r, φ) x 4 + O(m ρ 3,m 2 2,e), dj dφ = 2m ( ) 2ex 4 sin φ cos ψ x ρ ( 2x 2 cos φ + x 4 ) 5/2 + O(m 3,m 2 2,e 2 ), dψ dφ = x 4 ρ + O(e, m 3). () ψ (0) 0 m 3 e m 3 = e =0 ψ () 5. m 3 = e =0 r = dx dφ = x 4 ρ, dy dφ = x4 + x 6 ρ 2 + m 2 g (x, r, φ) + O(m 2 x 4 2 ρ ), dj dφ =0, dψ dφ = x 4 ρ. 2 x3 y ρ = ρ 0 m 2x 2 ( ) x 2 cos φ + + O(m 2 x 4 ρ 0 ( 2x 2 cos φ + x 4 ) 2), 3/2 ρ 0 = ± x 4 (y 2 2x 2 2J), x 4 (2) (3) ± m 3 J x y ψ m 2 =0 ρ = J 9

10 dx dφ = 2 x3 y +x 4 J, dy dφ = x4 + x 6 ρ 2 +x 4 J, dj dφ =0, dψ dφ = +x 4 J. (4) H(x, y, J) = 2 y2 + 2 x4 J 2 x 2. H(x, y, J) ( ) Figure J H(x, y, J) x y Figure (4) x y (0, 0) x 3 x 3 (0, 0) φ x = ξ(φ) y = η(φ) η(0) = 0 ξ(φ(t)) η(φ(t)) ξ(t, ρ) = 2 (3t ) 2/3 ( ), 2/3 + 9t2 + ρ 6 + 3t 9t2 + ρ 6 ρ 2 ± 2ξ η(t, ρ) = 2 (t) ξ 4 (t)ρ 2 for x 0, 2ξ 2 (t) ξ 4 (t)ρ 2 for x 0. (5) 0

11 ± ρ x 2 J ξ(0) = 2 J J = ± 2 x x = O x y (4) (2) m 2 J ψ φ 0 [0, 2π) Σ φ 0 φ = φ 0 Π Σ φ 0 ( ) O Π McGehee [] Robinson 2.[6] W s (O) W u (O) x>0 Xia[2] J W s (O) W u (O) m 2 e m 3 2 Σ φ 0=0 x 2 Σ φ 0=π x m 3 m m 2 m 3 ω m 3 α ( ) (Fig.2 ) p p γ γ p γ (ξ(φ),η(φ)), <φ< p γ (ξ(φ + π),η(φ + π)) <φ< Xia[2] J m 2 γ γ m 2 J Figure 2 R p 2 W s (O) W u (O) (Fig.3 ) Π q R k = k(q) Π k (q) R ( ) k>0 q R D Π(q) =Π k (q) for all q D.

12 Π R Π D (shift) D (Moser[3] ) Figure Π I D S = N Z ( ) I Π S τ τ : S I σ S Πτ = τσ. φ 0 =0 Σ φ 0 p x x ε>0 k p ε p 2k t φ n p n I 5. ( nnnn, nnn ) γ n n p n p n p n Π p n W s (p n ) W u (p n ) 5. I n p n p 0 O W s (O) W u (O) (C ) W s (p n ) W u (O) W u (O) W s (O) W s (O) W u (p n ) λ (inclination lemma) n W u (p n ) W s (p n ) n p n Π p n n Π (Π n = Π ) (2) ψ 2

13 Ω n Ω n = {(x, y, J, ψ, φ) (x, y)=p n (J); φ =0;ψ S ; J T R}. T T =( 2+δ, 2+δ 2 ) δ 2 >δ > 0 δ δ 2 Ω n Π Π n Π Ω n δ 2 >δ > 0 Π n J T ψ S Ω n 5.. n δ 2 >δ > 0 ε>0 C J = ( 2+δ, 2+δ 2 ) 0 <m 2 ε 0 Ω n Π f n (C) Π n (J) =J, Πn (ψ) =ψ + f n (J); J T df n (J)/dJ 0. (4) Π n (J) =J Π n (ψ) = 2nπ 0 +x 4 dφ + ψ, J x γ n n m 2 γ n v =( x 2 ρ 2 )s yρis. (6) v m 2 =0 dv dφ = 2m 2x 2 ρg 2 (x, r, φ)s + m 2 (ρg (x, r, φ)+yg 2 (x, r, φ))is x 4 + O(m 2 ρ 2). (7) α S v x v = v e αi dα dφ = 2m 2x 2 ρ 2 yg 2 (x, r, φ)+m 2 ( x 2 ρ 2 )(ρg (x, r, φ)+yg 2 (x, r, φ)) (y 2 ρ 2 +( x 2 ρ 2 ) 2 )( x 4 ρ) = m 2( + x 2 )x 4 y sin φ + m 2 ( x 2 ρ 2 )ρx 4 ( + 2x 2 cos φ) (y 2 ρ 2 +( x 2 ρ 2 ) 2 )( x 4 ρ) m 2x 4 [( + x 2 ρ 2 )y sin φ +( x 2 ρ 2 )p( x 2 cos φ)] ( 2x 2 cos φ + x 4 ) 3/2 (y 2 ρ 2 +( x 2 ρ 2 ) 2 )( x 4 ρ) + O(m2 2). + O(m 2 2) (8) 3

14 m 2 =0 α m 2 Δα(γ) γ α Δα(γ) γ m 2 α Δα(γ) ( 2m2 x 2 ρ 2 yg 2 (x, r, φ)+m 2 ( x 2 ρ 2 ) ) (ρg (x, r, φ)+yg 2 (x, r, φ)) Δα(γ) = (y 2 ρ 2 +( x 2 ρ 2 ) 2 )( x 4 dφ + O(m 2 ρ) 2). p = J x y γ m 2 Fig. x y x = ξ(t, ρ),y = η(t, ρ) ξ(t, ρ) =ξ( t, ρ) η(t, ρ) = η( t, ρ) Δα(γ) = θ m 2 ( + ξ 2 ρ 2 )ξ 4 η sin(t θ) +m 2 ( ξ 2 ρ 2 )ρξ 4 ( + 2ξ 2 cos(t θ))dt m 2 ξ 4 ( + ξ 2 ρ 2 )η sin(t θ) + dt ( 2ξ 2 cos(t θ)+ξ 4 ) 3/2 m 2 ξ 4 ( ξ 2 ρ 2 )ρ( ξ 2 cos(t θ)) dt + O(m 2 ( 2ξ 2 cos(t θ)+ξ 4 ) 3/2 2 ). θ(t, ρ) = t 0 (9) ξ 4 (t, ρ)ρdt. (20) ξ = 2 J 2 2t 2 / J 7 (9) J = 2 δ 2 >δ > 0 J T =( 2+δ, 2+δ 2 ) Δα(γ) 0 δ 2 >δ > 0 Δα(γ) J J T c 0 Δα(γ) c log(j 2) Δψ(γ n ) γ n ψ Π n (ψ) = ψ +Δψ(γ n ) Δψ(γ n )= 2nπ 0 2nπ x 4 ρ dφ = 0 x 4 ρ dφ mod(2π), x 4 ρ x γ n Δθ(γ n ) γ n θ 2nπ x 4 ρ Δθ(γ) = 0 x 4 ρ dφ. x γ n Δψ(γ n )=Δθ(γ n ) γ n γ x 4 ρ Δψ(γ n )=Δθ(γ n )= x 4 ρ dφ + ε n, 4

15 n ε n 0 x γ α ( (6)) x = y =0 α = θ Δψ(γ n )=Δθ(γ n )=Δα(γ)+ε n + O(m 2 2 ). δ 2 >δ > 0 m 2 > 0 n Δψ(γ n )(J) J J ( 2+δ, 2+δ 2 ) m 3 e 3 x = y =0 ( ψ φ : ) {x = y =0} {x = y =0} J T J J = J T J 6. m 3 e 0 <m 3 e Ω n m 3 e 6.. e>0 Ω n Ω e n Ω e n Π n. Ω n Ω n Ω n Ω e n Π n Π n 2- ω Ω e n ω n ω n Ω n Π n 2- ω n Π n Ω e n Π n Ω e n KAM(Kolmogorov-Arnold-Moser) Ω e n Ωe n J ψ Π n Π n (J) =J + O(m 3,e), Πn (ψ) =ψ + f n (J)+O(m 3,e). J T Ω n T J e >0 m 3 > 0 T e J T e J 5

16 T j Ω n ( ) (stochasitic layer) e T J 2 W s (p n ) T J 2 W u (p n ) T J e T e J 2 W s (T e J ) 2 W u (T e J ) W s (p n ) W u (p n ) Σ s 0 W s (T J ) W u (T J ) W s (T J ) W u (T J2 ) J J 2 J J 2 W s (T e J ) W u (T e J 2 ) 6.. n m 2 δ 2 >δ > 0 e 0 > 0 ε > 0 J T =( 2+δ, 2+δ 2 ) T J TJ e 0 <e<e 0 0 <m 3 <ε TJ e W s (TJ e ) W u (TJ e ) ε >0 J J 2 ε J,J 2 T W u (T J ) W s (T J2 ) S = {x = y =0,J R,ψ S } p n p n 6.2. [5] 7. S = {x = y =0;J R; ψ S ; φ =0}. 6

17 S m 3 J T S Π 2. x>0 q S q W u (q) x >0 Σ φ 0 φ0 =0 S W s (S) W u (q) W s (S) m 3 = e =0 J T W u (q) W s (S) q W u (q) W s (S) q S q W s (q ) D S S d d(q) =q W u (q) W s (S) d q q q q q d W u (q) W s (S) ( ) p 7.. D 0 S D 0 = {x = y =0;J T ( 2+δ, 2+δ 2 ); ψ S ; φ =0}.. δ 2 >δ,ε,ε 2, ε 3 J T,0 <m 2 ε, 0 m 3 ε 2 0 e ε 3 d D 0 D 0 D 2. m 3 = e =0 d D 0 3. d D. Xia[2] m 3 = e =0 q S J m 2 x y δ 2 >δ > 0 ε > 0 J ( 2+δ, 2+δ 2 ) 0 <m 2 ε W s (q) W u (q) x y W u (q) W s (S) m 3 e J T W u (q) W s (S) m 3 = e =0 d d(j) =J d(ψ) =ψ + +x 4 J dφ. x γ 5. +x 4 J dφ =Δα(γ)+O(m2 2), Δα(γ) J δ δ 2 J T m 2 d D 0 7

18 d ω ω Σ 0 Π ω S ω ω d ω = ω D d q D D d D D d n Π n D d(d) d 2 : D d(d) n Π n Π d d 2 ω d =(d ) d 2 ω d ω = ω d (exact) KAM D S d m 3 = e =0 D J = J e m 3 D 0 2. C J 2 W s (C J ) 2 W u (C J ) Σ 0 Π d 2 W s (C J ) W u (C J ) d (C J ) 2 Σ 0 D 0 C J 7.. δ 2 >δ,ε,ε 2 ε 3 7. C J D 0 ε 2 ε 3 0 m 3 ε 2 0 <e ε 3 W s (C J ) W u (C J ) ε >0 J,J 2 T J J 2 ε J W s (C J ) J 2 W u (C J2 ) (pseudo Arnold diffusion) D 0 ( 8. ) n t lim sup max r ij = lim inf max r ij < r ij i j 7.2. t lim sup x > 0 lim inf x =0, ω 7. C J,C J2,C J3,... t 8

19 C J D 0 m 3 = e =0 J J = J H (Xia[2]) Lyapunov-Schmit ruduction e m 3 W s (C J ) W u (C J ) J e m 3 J ΔJ dj/dφ = 2m 2ex 4 ( ) sin φ cos ψ O(m 2 x 4 ρ ( 2x 2 cos φ + x 4 ) 5/2 2 e, m 3,e 2 ). m 2 e J ΔJ γ D 0 p J φ =0 ψ ψ 0 0 2m 2 ex 4 ( ) sin φ cos ψ 3 ΔJ = + dφ x 4 ρ ( 2x 2 cos φ + x 4 ) 5/2 0 2m 2 ex 4 sin φ cos(ψ ψ 0 ) = cosψ 0 ( x 4 ρ ) 3 + dφ ( 2x 2 cos φ + x 4 ) 5/2 (2) 0 2m 2 ex 4 sin φ sin(ψ ψ 0 ) +sinψ 0 ( x 4 ρ ) 3 + dφ ( 2x 2 cos φ + x 4 ) 5/2 = A cos ψ 0 + B sin ψ 0. γ A B γ (5) 0 2m 2 eξ 4 ( ) sin φ cos(ψ ψ 0 ) 3 A = + dφ +Jx 4 ( 2ξ 2 cos φ + ξ 4 ) 5/2 +O(m 2 2e), 0 2m 2 eξ 4 ( ) (22) sin φ sin(ψ ψ 0 ) 3 B = + dφ +Jx 4 ( 2ξ 2 cos φ + ξ 4 ) 5/2 +O(m 2 2e). J = 2 2 δ 2 >δ > 0 J T =( 2+δ, 2+δ 2 ) A B J 2 c 2 0 B c 2 (J 2) 2 9

20 D 0 S 7. C J D 0 J = h(ψ) C J h(ψ) ψ 2π J = h (ψ) d J = h(ψ) 7. d S q d W u (q) W s (S) q m 2 e d d :(J, ψ) (J + A cos(ψ + b)+b sin(ψ + b),ψ+ b). b p 0 ψ b = 0 h(ψ) h (ψ) x 4 ρ dφ = 2 Δα(γ)+o(m2 2e) 0. h (ψ) =h(ψ b)+a cos ψ + B sin ψ + o(m 2 e). J = h 2 (ψ) d 2 J = h(ψ) 7. h 2 (ψ) =h(ψ + b)+a cos ψ B sin ψ + o(m 2 e). d h 2 (ψ) =h (ψ) h(ψ b) h(ψ + b)+2b sin ψ =0. (23) h(ψ) ψ 2π h(ψ) cosine (23) h(ψ) = a n cos(nψ). n=0 2a n sin(nb)sin(nψ) =0, for n 2a sin b sin ψ +2bsin ψ =0. m 2 n a n a = B/sin b = 0 h(ψ) =a 0 + B cos ψ sin b + h.o.t. γ J 5 O γ <φ< x = ξ(φ + π) y = η(φ + π) 20

21 ΔJ(γ) γ J π 2m 2 ex 4 ( ) sin φ cos ψ 3 ΔJ(γ) = + dφ x 4 ρ ( 2x 2 cos φ + x 4 ) 5/2 (24) = A cos ψ 0 + B sin ψ 0. x ψ γ ψ 0 φ =0 ψ γ A = B = 0 2m 2 eξ 4 ( sin φ cos(ψ ψ 0 ) + +Jx 4 +o(m 2 e), 0 2m 2 eξ 4 ( sin φ sin(ψ ψ 0 ) + +Jx 4 +o(m 2 e). ) 3 dφ ( + 2ξ 2 cos φ + ξ 4 ) 5/2 ) 3 dφ ( + 2ξ 2 cos φ + ξ 4 ) 5/2 J = 2 (5) A(J) =A( J) B(J) =B( J) A(J) B(J) A(J) B(J) C J d, d d 2 W s (q) W u (S) p p p d, d d 2 J = h (ψ) d C J h (ψ) h (ψ) = h(ψ b)+a cos ψ + B sin ψ + O(m 2 2 e) = a 0 + B sin b cos(ψ b)+a cos ψ + B sin ψ + O(m2 2e). b γ ψ b = π x 4 ρ dφ = 0 +Jξ 4 (φ) dφ + O(m2 2,e,m 3 ), J = h 2 (ψ) d 2 C J h 2 (ψ) =a 0 + B sin b cos(ψ + b)+acos ψ B sin ψ + O(m2 2 e). 2 J = h (ψ) J = h 2 (ψ) C J B B sin b/ sin b 2 h (ψ) h 2 (ψ) e m 3 J 2 B b B c 2 (J 2) 2 b c log(c 2) δ 2 >δ > 0 J T =( 2+δ, 2+δ 2 ) B B sin b/ sin b W s (C J ) W u (C J ) 2 (25)

22 Arnold[2] (obstructing set) M X Ω x M M x M N Ω M Ω 8.. () C J S d U Σ φ 0 3 q C J W s (C J ) Ω = φ 0 U(φ) U C J W u (C J ) q Ω q W u (C J ) (2) C J,C J2,C J3, i =, 2, 3,... C Ji C Ji+ C Ji C Ji+ Σ φ 0 U Σ φ 0 3 i C Ji Ω = φ 0 U(φ) U Ω j W u (C Jj ) W u (C Ji ). q W s (C J ) q C J S φ q(φ) q q W u (q ) Ω q W u (C J ) S 2. (Robinson[6]) (graph transform) U C S S S S q ε>0 W u (S ) q U n Π n (U) C U ε q W u (S ) n Π n (U) Ω Ω q W u (q ) W u (C J ) Ω q W u (C J ) W u (C J ) q,q 2,...,q k q W s (q ),d(q )=q 2,d(q 2 )=q 3,...,d(q k )=d(q k ) q W u (q k ) q = d (q ) q W u (q ) Ω q W u (C J ) d n Π n (U) W s (q 2 ) U W s (q 2 ) U i =, 2,...,k W s (q i ) q W u (q k ) Ω q W u (C J ) U q W u (C J ) C J d q U q W s (C J ) C J q,q 2,...,q k q W s (q ),d(q )=q 2,...,d(q k )=q k q W u (q k ) U W s (C J ) W s (C J ) 7. 22

23 7.2 d C J U W s (C J ) 3 C J,C J2,C J3,... q,q 2,q 3,... U q W u (q ) q U U q U q 2 q 2 U 2 U q 2 U 2 q 3 U 2 W s (q 3 ) q 3 U U U U 2 U 23, q q q φ lim sup x > 0 lim inf x =0. q,q 2,q 3, q ω q TJ e Ω n J T =( 2+δ, 2+δ 2 ) e = m 3 =0 TJ e S e>0 TJ e S s s S S s s 2 TJ e S s 2 α s s δ 2 >δ > 0,ε > 0,e 0 > 0 0 <e<e 0 0 m 3 <ε J T =( 2+δ, 2+δ 2 ) s s 2 Σ φ 0. J ψ S J = h(ψ) s h(ψ) s 2 J = h( ψ) e = m 3 =0 W u (s ) Σφ=0 {(x, y) =p; φ =0} Σ φ=π {(x, y) =p; φ = π} W s (s 2 ) J ψ e e J ψ γ γ s s 2 J C = h (ψ) d J = h(ψ) h (ψ) =h(ψ b)+a cos ψ + B sin ψ + h.o.t. A, B b d 2 s 2 J = h ( ψ) =h( ψ b)+a cos( ψ)+b sin( ψ)+h.o.t. 23

24 2 h ( b)+b 0 C J h ( b)+b 0 W u (s ) W s (s 2 ) p Σ 0 p p 2 (Fig.3 ) ε>0 p p p ε d d 2 d,p d 2,p W u (q) W s (S) p p J = h 2(ψ) d,p s J = h 2 (ψ) =h(ψ 2b)+2B sin(ψ b)+o(ε)+h.o.t. d 2,p s 2 J = h 2 ( ψ) =h( ψ 2b)+2B sin( ψ b)+o(ε)+h.o.t. 2 h 2(0) 0 h ( 2b)+2B cos( b)+o(ε) 0, ε h ( 2b)+2Bcos( b) 0 J T =( 2+δ, 2+δ 2 ) h ( 2b)+ 2B cos( b) 0 p 2 p 2 p 4 p 2 p ε h ( 4b)+2B cos( 3b)+2B cos( b) 0. h ( 4b)+2B cos( 3b)+2B cos( b) 0. k =, 2, 3,... h ( 2kb) h ((2k 2)b)+2B cos((2k )b) 0. b/2π h (ψ) h (ψ + b) h (ψ b)+2b cos ψ =0, for all ψ S. (26) s s 2 b h(ψ) h(ψ) = a n cos nψ + b n sin nψ. k=0 24

25 (26) a n b n n 0, a n = b n =0 b =0,a = B/sin b ( s m 3.) s J = h(ψ) =a 0 + B cos ψ. sin b 7. s = s 2 S d s s 2 Σφ 0 6. TJ e S s 2. W u (TJ) e C s W s (TJ e) C s 2 s s 2 TJ e Σφ 0 References. V.M. Alekseev, Quasirandom dynamical systems, I, II, III, Math. USSR-Sb. 5 (968), 73-28; 6 (968), ; 7 (969), V.I. Arnold, Instability of dynamical systems with several degrees of freedom, Dokl. Akad. Nauk SSSR 56 (964), V.I. Arnold(Ed), Dynamical Systems III, Encyclopaedia of Mathematical Sciences, Vol.3, Springer-Verlag, New York, V.I. Arnold, Sur une propriété topologique des applications globalement canonique de la mécanique classique, C. R. Acad. Sci. Paris 26 (965), B.V. Chirikov, A universal instability of many dimensional oscillator systems, Phys. Rep. 52 (979), R. Easton, Parabolic orbits for the planar three-body problem, J. Differential Equations 52 (984), R. Easton, Capture orbits and Melnikov integrals in the planar 3-body problem, Celestial Mech., toappear. 8. R. Easton and R. McGehee, Homoclinic phenomena for orbits doubly asymptotic to an invariant three-sphere, Indiana Univ. Math. J. 28 (979), P. Holmes and J. Marsden, Melnikov method and Arnold diffusion for the perturbation of integrable Hamiltonian systems, J. Math. Phys. 23 (982), C. Lim, A combinatorial perturbation method and Arnold s whiskered tori in vortex dynamics, preprint. 25

26 . R. McGehee, A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, J. Differential Equations 4 (973), V.K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc. 2 (963),. 3. J. Moser, Stable and Random Motions in Dynamical Systems, Annals of Mathematics Studies, Vol.77, Princeton Univ. Press, Princeton, NJ, N.N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Russian Math. Surveys 32 (977), H. Poincaré, Les méthodes nouvelles de la mécanique céleste III, Gauthier-Villars, Paris, C. Robinson, Homoclinic orbits and oscillation for the planar three-body problem, J. Differential Equations 52 (984), C. Robinson, Horseshoes for autonomous Hamiltonian systems using Melnikov integrals, Ergodic Theory & Dynamical Systems 8 (988), D. Saari and Z. Xia, The existence of oscillatory and super-hyperbolic motions in Newtonian systems, J. Differential Equations 82 (989), K. Sitnikov, The existence of oscillatory motion in the three-body problem, Dokl. Akad. Nauk USSR 33 (960), A. Weinstein, Lagrangian submanifolds and Hamiltonian systems, Ann. of Math. 98 (973), Z. Xia, Melnikov method and transversal homoclinic orbits in the restricted three-body problem, J. Differential Equations 96 (992), Z. Xia, Arnold duffusion in the elliptic restricted three-body problem, J. Dynamics and Diff. Equations 5 (993),

xia2.dvi

xia2.dvi Journal of Differential Equations 96 (992), 70-84 Melnikov method and transversal homoclinic points in the restricted three-body problem Zhihong Xia Department of Mathematics, Harvard University Cambridge,