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, Massachusetts 0238 Abstract 2 2 [0] McGehee[7]. Sitnikov[3] Alekseev[] Sitnikov Moser[9] [3,8,2] 2 Smale Sitnikov [0] 2 McGehee[7] Easton[2] Robinson[]
2 Llibre and Simó[5,6] J.Llibre [5] Llibre and Simó C ± 2 [5,6] K.Meyer 2. R 2 3 P,P 2,P 3 P P 2 μ μ 0 μ P 3 P 3 P P 2 P P 2 P P 2 q =(q,q 2 ) P 3 p =(p,p 2 )=(q,q 2) P 3 P 3 q = p, () p = U q. U U q q U U = μ (q μx 2 ) 2 +(q 2 μy 2 ) + μ (q 2 +( μ)x 2 ) 2 +(q 2 +( μ)y 2 ). 2 x 2,y 2 P P 2 x y x 2 =cost, y 2 =sint. μ =0 μ μ U 2
U = ( + μ q 2 + q2 2 (q 2 + q2) + q cos t + q 2 sin t 2 /2 (q 2 + q2) 2 3/2 ) + + O(μ 2 ). ((q +cost) 2 +(q 2 +sint) 2 ) /2 μ =0 t t μ q = x 2 S s q = x 2 s p = ys + x 2 ρis. is s ρ P 3 U = x 2 + μx 2 ( +x 2 cos(t θ)+ (2) ) + O(μ 2 ), (3) ( + 2x 2 cos(t θ)+x 4 ) 3/2 x = 2 x3 y, y = x 4 + x 6 ρ 2 + μg (x, t θ)+o(μ 2 ), θ = x 4 ρ, ρ = μg 2 (x, t θ)+o(μ 2 ). (4) θ S s =(cosθ, sin θ) g (x, t θ),g 2 (x, t θ) g (x, t θ) =x ( 4 2x 2 +x 2 ) cos(t θ) cos(t θ), ( ( + 2x 2 cos(t θ)+x ) 4 ) 3/2 (5) g 2 (x, t θ) =x 4 sin(t θ). ( + 2x 2 cos(t θ)+x 4 ) 3/2 5 2 y2 + 2 x4 ρ 2 U ρ = C, 3
C ρ ρ μx 2 ( ) ρ = ρ 0 +x 2 cos(t θ)+ + O(μ 2 ), ( x 4 ρ 0 ) ( + 2x 2 cos(t θ)+x 4 ) 3/2 ρ 0 = ± (6) x 4 (y 2 2x 2 2C). x 4 ± P 3 t t θ s s = t θ, s S. x = 2 x3 y, y = x 4 + x 6 ρ 2 + μg (x, s)+o(μ 2 ), s = x 4 ρ. ρ x, y, s C (6) μ =0 x = 2 x3 y, y = x 4 + x 6 ρ 2, (8) s = x 4 ρ. ρ (ρ P 3 ) x, y s H x y H(x, y, ρ) = 2 y2 + 2 x4 ρ 2 x 2. (9) (7) Figure H(x, y, ρ) (H P 3 ) Figure 4
x, y (x >0 ) H (0, 0) x 3 x(t) =ξ(t, C) = 2 (3t + 9t 2 + C 6 ) 2/3 +(3t 9t 2 + C 6 ) 2/3 C 2, ± 2ξ 2 (t) ξ 4 (t)c 2 for x 0, y(t) =η(t, C) = 2ξ 2 (t) ξ 4 (t)c 2 for x 0. (0) ± C ξ(t, C) t η(t, C) t t 0 R (ξ(t t 0,C),η(t t 0,C)) ξ(t, C) η(t, C) s s x 0 (s) y 0 (s) s x 0 (s) =ξ[t(s),c] y 0 (s) =η[t(s),c] θ 0 (?) x 0 (s) s y 0 (s) 0 <μ x y s dx ds = 2 x3 y x 4 ρ, dy ds = x4 + x 6 ρ 2 + μg (x, s) x 4 ρ x 4 ρ + O(μ2 ). ρ x, y, s C (6) s x 4 ρ 0x ρ<0 μ 0 s 2π t ω t α x 3 ω α McGehee[7] ω α. () {x 0} γ : x =0,y =0,s S x>0 W s (γ) {x>0} W u (γ) {x>0} C ω α () 5
McGehee[7] Robinson[] McGehee Robinson[] 2 () C μ, μ 0 (μ =0 ) 3. μ =0 W s (γ) W u (γ) μ 0 P 3 t =0 x ω α γ Σ s 0 s = s 0 (x u μ (s, s 0),yμ u(s, s 0)) W u (γ) (x u μ(0,s 0 ),yμ(0,s u 0 )) Σ s 0 (x s μ(s, s 0 ),yμ(s, s s 0 )) W s (γ) (x s μ (0,s 0),yμ s(0,s 0)) Σ s 0. C > 2 C x s μ (s, s 0)=x 0 (s s 0 )+μx s (s, s 0)+O(μ 2 ), s [s 0, ), yμ s(s, s 0)=y 0 (s s 0 )+μy s(s, s 0)+O(μ 2 ), s [s 0, ), x u μ (s, s 0)=x 0 (s s 0 )+μx u (s, s 0)+O(μ 2 ), s (,s 0 ], yμ u(s, s 0)=y 0 (s s 0 )+μy u(s, s 0)+O(μ 2 ), s (,s 0 ]. (2) x s (s, s 0 ),y s (s, s 0 ) x u (s, s 0 ),y u (s, s 0 ) (x 0,y 0 ) () 2 () Gronwall (x 0 (0),y 0 (0)) O(μ) (x 0 (s),y 0 (s)) O(μ) (2) W u (γ),w s (γ) C C μ>0 C 2 C > 2 6
H(x, y) = 2 y2 + 2 x4 ρ 2 x 2 (9) ρ (6) μ 0 H dh ds = μ(yg (x, s)+x 4 ρg 2 (x, s)) x 4 + O(μ 2 ). (3) ρ d(s 0 ) d(s 0 )=(x s μ (s 0,s 0 ),y s μ (s 0,s 0 )) (x u μ (s 0,s 0 ),y u μ (s 0,s 0 )). W s (γ) W u (γ) Σ s 0 s = s 0 d(s 0 ) W s (γ) W u (γ) H N (x 0 (0),y 0 (0)) (x 0 (0),y 0 (0)) H(x, y) H N (x 0 (0),y 0 (0)) = (H x (x 0 (0),y 0 (0)),H y (x 0 (0),y 0 (0))). d(s 0 ) (x 0 (0),y 0 (0)) H(x, y) d(s 0 ) d(s 0 ) = HN (x 0 (0),y 0 (0)) H N (x 0 (0),y 0 (0)) (xs μ(s 0,s 0 ) x u μ(s 0,s 0 ),y s μ(s 0,s 0 ) y u μ(s 0,s 0 )) = μ HN (x 0 (0),y 0 (0)) H N (x 0 (0),y 0 (0)) (xs (s 0,s 0 ) x u (s 0,s 0 ),y s (s 0,s 0 ) y u (s 0,s 0 )) + O(μ 2 ). x s (s, s 0),y s (s, s 0),x u (s, s 0),y u (s, s 0) d(s 0 ) Σ s 0 (x 0 (0),y 0 (0)) W u (γ),w s (γ) H(x s μ (s 0,s 0 ), y s μ (s 0,s 0 )) = H(x 0 (0),y 0 (0)) + μh N (x 0 (0),y 0 (0)) (x s (s 0,s 0 ),y s (s 0,s 0 )) + O(μ 2 ), H(x u μ (s 0,s 0 ), y u μ (s 0,s 0 )) = H(x 0 (0),y 0 (0)) + μh N (x 0 (0),y 0 (0)) (x u (s 0,s 0 ),y u (s 0,s 0 )) + O(μ 2 ). d(s 0 ) H N (x 0 (0),y 0 (0)) = H(x s μ (s 0,s 0 ),yμ s(s 0,s 0 )) H(x u μ (s 0,s 0 ),yμ u(s 0,s 0 )) + O(μ 2 ) dh(x, y) = ds + O(μ 2 ) ds = μm(s 0 )+O(μ 2 ), M(s 0 ) dh M(s 0 ) = μ ds (x 0(s s 0 ),y 0 (s s 0 ))ds yg (x 0 (s s 0 ),s)+x 4 = 0(s s 0 )ρg 2 (x 0 (s s 0 ),s) ds. x 4 0(s s 0 )ρ (4) (5) 7
ρ = C d(s 0 )= M(s 0 ) μ H N (x 0 (0),y 0 (0)) + O(μ2 ) = M(s 0 ) μ 2x 3 0(0)C 2 2x 0 (0) + O(μ2 ) (6) = μ C 2 2 M(s 0)+O(μ 2 ). g (x, s)+x 4 ρg 2 (x, s) x 4 ρ = g 2(x, s) x 4 ρ + d ( ( x 2 +x 2 cos(t θ) ds )) + + O(μ 2 ). ( + 2x 2 cos(t θ)+x 4 ) 3/2 ρ ρ 0 x(s),y(s) s M(s 0 )= = = = g 2 (x(s s 0 ),s) ds x 4 0(s s 0 )ρ 0 g 2 (x(s),s+ s 0 ) ds x 4 0(s)ρ 0 g 2 (x 0 (s),s+ s 0 )dt ( x 4 0 (s)sin(s + s 0) ) dt. ( + 2x 2 0(s)cos(s + s 0 )+x 4 0) 3/2 (7) s 0 = π M(s 0 )=0 s 0 = π M(s 0 ) M (s 0 ) s0 =π 0 M (π) = x 4 0 (s)cos(s + π)dt x 4 0 (s)cos(s + π) ( + 2x 2 0(s)cos(s + π)+x 4 0(s)) 3/2 dt 3x 6 0 sin 2 (s + π) dt. ( + 2x 2 0(s)cos(s + π)+x 4 5/2 0(s)) (8) x 0 (s) =ξ(t, C) (0) s s = t θ = t t 0 x 4 ρdt = t + t 0 ξ 4 (t, C)Cdt. C = ± 2 M(s 0 ) M (π) t =0 t =0 C = ± 2 +2ξ 2 cos(s + π) +ξ 4 =0 +2ξ 2 cos(s + π) +ξ 4 t =0 C > 2 C ± 2 8
M (0) C > 2 C 2 M (π) 0 C μ γ W s (γ) W u (γ) μ W s (γ) W u (γ) μ s 0 = π Σ s 0 W s (γ) x W u (γ) C μ W u (γ) W s (γ) ( C ) ([4] ) W u (γ) W s (γ) μ C x p(μ, C) k(μ, C) p(μ, C) W s (γ) p(μ, C) W u (γ) k(μ, C) k(μ, C) 0 p(μ, C) W u (γ) W s (γ) W u (γ) W s (γ) C μ k(μ, C) k(μ, C) 0 ( W u (γ) x ) C μ k(μ, C) 0 C μ k(μ, C) C μ μ C > 2 W u (γ) W s (γ) k(μ, C) 0 k(μ, C) C μ μ C k(μ, C) 0 k(μ, C) C μ W u (γ) W s (γ) C = C C μ k(μ, C ) k(μ, C ) 0 k(μ, C) μ [0, ] k(μ, C ) k(μ, C ) 0 2. μ γ W s (γ) W u (γ) 4. Smale-Birkhoff ( s 9
) t ± lim sup r = lim inf r< r P 3 r = q 2 + q2 2 = x 2 t ± r t lim sup r< Moser[9] s 0 Γ=Σ s 0 C Γ p R 2 W s (p) W u (p) (Fig.2 ) φ q R k = k(q) φ k (q) R ( ) k>0 q R D φ(q) =φ k (q) for all q D. Moser φ R φ D (shift) D Fig.2 3. φ I D S = N Z ( ) I φ S τ τ : S I σ S φτ = τσ. [9] Smale-Birkhoff 0
q I k(q) 4. I 3 q I S I s = {...,s 3,s 2,s,s 0,s,s 2,s 3,...) sup{s i,i Z} = Fig.2 R R 5. (989)[0] 2 μ 5. μ [0, ] P P 2 μ μ References. V.M. Alekseev, Quasirandom dynamical systems, I, II, III, Math. USSR-Sb. 5 (968), 73-28; 6 (968), 505-560; 7 (969), -43. 2. R. Easton, Parabolic orbits for the planar three-body problem, J. Differential Equations 52 (984), 6-34. 3. R. Easton and R. McGehee, Homoclinic phenomena for orbits doubly asymptotic to an invariant three-sphere, Indiana Univ. Math. J. 28 (979), 2-240. 4. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 983. 5. J. Llibre and C. Simó, Some homoclinic phenomena in the three-body problem, J. Differential Equations 37 (980), 444-465. 6. J. Llibre and C. Simó, Oscillatory solutions in the planar restricted three-body problem, Math. Ann. 248 (980), 53-84.
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