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3 Stability of periodic solutions of Lagrange equations and planar Hamiltonian systems Dissertation Submitted to Tsinghua University in partial fulfillment of the requirement for the degree of Doctor of Sciences by Jifeng Chu ( Mathematics ) Dissertation Supervisor : Professor Meirong Zhang May, 28

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7 ... Hill Ermakov-Pinney r(t) Hill ρ. Ermakov-Pinney x + e x = σ + h(t) x + x 2 = σ + h(t) x + a(t)x = 1/x γ, x >.. Ortega. Ermakov-Pinney.... Birkhoff Moser. I

8 Abstract Abstract We are mainly interested in the Lyapunov stability of periodic solutions of Lagrange equations and planar nonlinear Hamiltonian systems. This paper is divided into four parts. In Chapter 1, we introduce the historical background and some recent results obtained in the literature. We also state some important basic results. In Chapter 2, we study the Lyapunov stability of elliptic periodic solutions of Lagrange equations. First we give some reasonable estimates of the periodic solutions of Ermakov-Pinney equations when the linearized equation is in the first stability zone. These estimates can also give the estimates of the rotation numbers of the Hill equations. The results concerning the lower bounds of the rotation numbers are completely new in the literature. By using these estimates, we prove that two classes of nonlinear, scalar, time-periodic, Lagrange equations will have twist periodic solutions, one class bing regular, including x + e x = σ + h(t) and x + x 2 = σ + h(t), another class being singular, x + a(t)x = 1 x + e(t), γ >. γ In Chapter 3, we try to extend the analytical method in studying the stability of periodic solutions of Lagrange equations to the nonlinear planar Hamiltonian systems. First, we establish two important facts on linear planar Hamiltonian systems. One is the reduction from ellipticity to R-ellipticity. Another is the relation between the stability of linear systems and the existence of periodic solutions of the generalized Ermakov-Pinney equations. Based on these two basic facts and the Birkhoff normal forms of area-preserving mappings, we compute the twist coefficients of planar nonlinear Hamiltonian systems. Such twist coefficients play an important role in studying the Lyapunov stability of periodic solutions. For some special nonlinear systems, we state and prove the stability results. As an example, the stability of the equilibrium of the one-dimensional Φ-Laplacian is given. As can be seen in Chapter 2 and 3, the existence and the estimates of periodic so- II

9 Abstract lutions of singular equations play important roles in the stability theory. Therefore, we develop some existence results for second order non-autonomous dynamical systems in Chapter 4. The first one is based on a nonlinear alternative principle of Leray Schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity. The second one is based on Schauder s fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions. The third existence result is concerned with the nontrivial periodic solutions and the proof is based on a well-known fixed point theorem in cones. Key words: Lagrange equations planar Hamiltonian systems Birkhoff normal form Moser twist theorem Lyapunov stability. III

10 Sobolev Hill Ermakov-Pinney R Ermakov-Pinney IV

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12 Lyapunov. Lyapunov.. [14 16,18,31,39,48,55,56,58,59,62,7,71,79,95,113] [5,43,47,73,82,94]. Lyapunov.. KAM Moser [8,12,32,94] 3/2 2. Birkhoff 3/2 2 Birkhoff Moser,. Birkhoff. Birkhoff Moser Ortega. x + f (t, x) =, (1-1) 1

13 1 f (t, x) = f (t + 2π, x) (1-1) 2π x = ϕ(t) (1-1) ( ) Hill x + a(t)x =, (1-2) x + a(t)x + b(t)x 2 + c(t)x 3 + =, (1-3) a(t) = f x (t, ϕ(t)), b(t) = 1 2 f xx(t, ϕ(t)), c(t) = 1 6 f xxx(t, ϕ(t)). ϕ(t) (1-1) 3 Ortega (1-1) (1-2) (1-3) Birkhoff β β (1-1) ϕ(t). Ortega β ( swing) x + a(t) sin x =, a(t) = a(t + 2π), (1-4). (1-4) (1-4). Ortega Núñez [75,84] (1-4) (1-4) x = x + a(t)x a(t) 6 x3 =.. 23 [5,119] Ortega β β. β β = Q a (b) + l a (c), (1-5) Q a (b) b L 2 (S 2π ) l a (c) c L 2 (S 2π ) Q a (b) (1-2). 2

14 1 Hill (1-2) ρ(a) Floquet r(t) ϕ(t). [52,121] (1-2) r(t). r(t) Ermakov-Pinney. r + a(t)r = 1, r >, (1-6) r3 [121] (1-2) (1-6) r = r a (t). [5,52] (1-2) (1-6) Riccati Ż = Z 2 + a(t), Z C, (1-7) x + ω 2 sin x = p(t), p(t) = p(t + 2π), (1-8) x ω (t) P(ω) p L 1 < P(ω) x ω (t) P(ω) 4 ω + O(ω 1/2 ). O(ω 1/2 ).. (1-1) (1-2) Sobolev Ermakov-Pinney r(t) (1-2) ρ A p = a + p [34] a(t) L 1. Ermakov-Pinney 3

15 1 x + e x = σ + h(t), (1-9) x + x 2 = σ + h(t), (1-1) Ermakov-Pinney x + a(t)x = 1, γ >, (1-11) xγ σ R h(t) L 1 (R/TZ) = {h L 1 (R/TZ) : h = }, T >. (1-9) Σ 2 (T) > 1/(2T 2 ) σ (, Σ 2 (T)] h L 1 (R/TZ) (1-9) T- ψ(t) = ψ σ,h (t) Lyapunov. (1-1). (1-11) L(γ) > a(t) L 1 (R/2πZ), a < A 1 = a 1 < L(γ), (1-11) T-. h(t), a(t) swing. Ermakov-Pinney r(t) β.... 4

16 1 ẋ = a(t)y, ẏ = b(t)x, (1-12) a(t), b(t) 2π a(t), t, (1-13) (1-12). Ortega (1-12) Ermakov-Pinney r + a(t)b(t)r = a2 (t) r 3 + ȧ(t) a(t)ṙ (1-14). (1-13) ẋ + J H x (t, x) = (1-15) β.. β.. ( Φ(x ) ) = f (t, x), f (t, x) = f (t + 2π, x) (1-16) Φ(s) = s, f (t, ) =. 1 s 2 (1-16) ẋ = y + ky 3 +, ẏ = f x (t, )x f xx(t, )x f xxx(t, )x 3 +, (1-17) 5

17 1. (1-16). ẋ = a(t)y + c(t)y 2n 1 + G (t, x, y), x ẏ = b(t)x d(t)x 2n 1 G (1-18) y (t, x, y), n 2 G : R B ɛ () R t R G(t, x, y) = O((x 2 + y 2 ) n+1/2 ), (x, y) (, ). (1) Ermakov-Pinney. (2).... ẍ + a(t)x = f (t, x) + e(t). [1,1,35,38,91,93,125].. [4,97,99,1]. Lazer Solimini [49] 6

18 1 Schauder [9,13,17,3,38,44,57,14]. Poincaré [89]. [24,37,9,17] Sobolev I I p [1, ] K(p, I ) Sobolev C φ 2 L p (I) φ 2 L 2 (I), φ H1 (I)., K(p, I ) = K(p)/ I 1+2/p, K(p) = K(p, 1), K(p) = ( ) 2π 2 1 2/p ( Γ(1/p) 2 p 2+p Γ(1/2+1/p)), 1 p <, 4, p =, Γ( ) Γ-, [123] (8) Talenti [98]. Sobolev K(p, I ) φ 2 L p (I) φ 2 L 2 (I), φ H1 (I). (1-19) p [1, ] K(p) K(1) = 12, K(2) = π 2, K( ) = 4. Sobolev [15,18,12]. 7

19 a(t) L 1 (R/2πZ), Hill x + a(t)x = (2-1) ρ = ρ(a) [, )., (2-1) Lyapunov n N ρ ((n 1)/2, n/2) (2-1) n. (2-1). [34] a(t) L p (R/2πZ) 1 p ( ) 1/2 ρ(a) ζ p a+ p, (2-2) ζ p 1.2 Sobolev a + (t) = max(a(t), ) a + p = a + L p (,2π) L p. ζ p a(t) L p (R/2πZ).. x + f (t, x) =, f (t, x) f (t + 2π, x) (2-3) Lyapunov (2-3) 2π- ψ(t) (2-3) (2-1) ψ(t) a(t) = f x (t, ψ(t)). (2-4) (2-3) 3/2 (2-1) (2-3). 8

20 2 Birkhoff Moser [94] Ortega [8,81,83,84]. [27,5,52,119,124]. x = y + ψ(t) a(t) (2-4) y + a(t)y + b(t)y 2 + c(t)y 3 + o(y 3 ) =, (2-5) b(t) = f xx(t, ψ(t)) 2, c(t) = f xxx(t, ψ(t)). 6 P (2-5) Poincaré ψ(t) P O ψ(t) P O D Birkhoff N(z, z) = µ(z + i β z 2 z + o( z 3 )), z = x + ix D C, µ ψ Floquet β β ψ [83]. β ψ. Moser. β [5] [83] β = b(t)b(s)r 3 (t)r 3 (s)χ θ ( ϕ(t) ϕ(s) )dtds 3 [,2π] 2 8 χ θ χ θ (x) = 3 cos(x θ/2) 16 sin(θ/2) + 2π cos 3(x θ/2), x [, θ]. 16 sin(3θ/2) c(t)r 4 (t)dt, (2-6) θ = 2πρ ρ (2-1) r(t) Ermakov-Pinney 2π- ϕ(t) r + a(t)r = 1, r >. (2-7) r3 ϕ(t) = t ds r 2 (s), t R (2-8) 9

21 2. χ θ (2-1). (2-6) (2-5) a(t) b(t) c(t) β = β(a, b, c) (a, b, c) (C(R/2πZ)) 3 B = {(a, b, c) : β(a, b, c) = } (C(R/2πZ)) 3, Moser (a, b, c) B x(t) =.. Núñez Ortega [75,81,84] x + l(t) sin x =, l(t) >, l(t) C(R/2πZ), x(t) = x + l(t)x l(t) 6 x3 = [5,52]. Ortega [51,53,74,76,77]. (2-6) Hill Ermakov-Pinney. a(t) L 1 (R/2πZ) [5] [121] Hill (2-1) Ermakov-Pinney (2-7) 2π- r(t) (2-1) ρ r(t) (2-2) ρ = 1 2π 2π a(t) L p (R/2πZ) 1 ϕ(2π) ds = r 2 (s) 2π. (2-9) ā > = ρ(a) >, ā > A p = a + p < L (p) = 4ζ 2 p, (2-1) 1

22 2 (2-1) Hill (2-1) (2-1) a(t) (2-1) ρ(a) (2-2). a(t) (2-1) ρ(a) ā A p. a(t) ρ = ā 1/2 (1 + O(A p )), A p +. p = 1 1 < p. Ermakov-Pinney (2-7) r(t). ρ r(t) (2-3) x + e x = σ + h(t) (2-11) σ R h(t) L 1 (R/TZ) = {h L 1 (R/TZ) : h = }, T >. (2-11) Landesman-Lazer. Σ 2 (T) > 1/(2T 2 ) σ (, Σ 2 (T)] h L 1 (R/TZ) (2-11) T- ψ(t) = ψ σ,h (t) Lyapunov x + x 2 = σ + h(t), (2-12) x + a(t)x = 1, x >, (2-13) xγ γ > γ 3 a(t) L 1 (R/2πZ), a(t). Ermakov-Pinney (2-7) r(t) 11

23 2 L 4 (γ) > a(t) L 1 (R/2πZ), a < A 1 = a 1 < L 4 (γ), (2-13) 2π-. < γ < 3 β < γ > 3 β >. L p (R/2πZ) 1 p 2π- p. f i L 1 (R/2πZ) f 1 f 2 t f 1 (t) f 2 (t) f 1 > f 2. f (t) f min = min f (t), f max = max f (t). t t 2.2 Hill a(t) L 1 (R/2πZ) Hill (2-1). Hill (2-1) [64] [114,118]. M(x, y ) = (x(2π; x, y ), x (2π; x, y )) (2-1) Poincaré x(t; x, y ) (2-1) x() = x, x () = y. M µ i i = 1, 2 (2-1) Floquet. det M = 1 µ 1 µ 2 = 1. (2-1) : µ 1 = µ 2, µ 1 = 1, µ 1 ±1. : < µ 1 < 1 < µ 2. : µ 1 = µ 2 = ±1. µ i 1 (2-1) (2-1) 2π-. ψ(t) (2-3) 2π- (2-1) ψ(t). 12

24 2 (2-1) ϑ(t) ρ = lim, t t ϑ(t) ϑ = cos 2 ϑ + a(t) sin 2 ϑ ρ ϑ(t) [114,118] a L p (R/2πZ) 1 p (i) ρ = ρ(a) (ii) a 1 a 2 = ρ(a 1 ) ρ(a 2 ) (iii) (2-1) n N ρ ((n 1)/2, n/2). (2-1) Floquet µ 1,2 = exp(±iθ) θ = 2πρ (iv) (2-1) n N ρ (n 1, n). ρ ((n 1)/2, n/2) θ ((n 1)π, nπ) (2-1) n. (2-1). (2-1) (2-1) Floquet µ i µ k i 1, 1 k 4. n N, 1 n 4 m Z + ρ m/n. < ρ < 1/4 (2-1). Hill (2-1) Ermakov-Pinney (2-7) [5,121] Hill (2-1) Ermakov- Pinney (2-7). 2.2 [121] (2-7) 2π- r(t) (2-1) r(t). Hill. 2π- x() x(2π) = x () x (2π) =, (2-14) 13

25 2 2π- x() + x(2π) = x () + x (2π) = (2-15) x + (λ + a(t))x =. (2-16) [64] {λ n (a) : n N} {λ n (a) : n Z + }, Z + = {, 1, 2, }. 2.3 [64,118] (i) λ (a) < λ 1 (a) λ 1 (a) < λ 2 (a) λ 2 (a) < < λ n (a) λ n (a) <. (ii) λ (2-16)-(2-14) n λ = λ n (a) λ n (a). λ (2-16)-(2-15) n λ = λ n (a) λ n (a). (iii) (2-1) n N λ n 1 (a) < < λ n (a). [34]. 2.4 a L p (R/2πZ) 1 p ρ ζ p a + 1/2 p, ζ p = 1/(2(K(2p, 2π)) 1/2 ), p = p/(p 1). (2-17) p [1, ] 2.4. a min > (a min ) 1/2 ρ (a max ) 1/2.. 14

26 2 2.3 Ermakov-Pinney (2-1) (2-7) r(t). (2-1) r(t) (2-7) 2π-. (2-9) t r(t ) = r(t + 2π) = ρ 1/2 t =. r(t) = ρ 1/2 (1 + v(t)) v(t) = ρ 1/2 r(t) 1 v H 1 (, 2π). r(t) (2-7) 2π r 2 2 = r 2 dt = 2π a(t)r 2 dt 2π (1/r 2 )dt = (2-9). (2-18) v(t) 2π v 2 = 2π 2π a(t)r 2 2πρ, (2-18) a(t)(v + 1) 2 2πρ 2. (2-19) r(t) (2-19) 2πρ 2 ρ. p = Sobolev (1-19) u u 2 /(K(, 2π)) 1/2 = (π/2) 1/2 u 2, u H 1 (, 2π). Z = (π/2) 1/2 v p a(t) L p (R/2πZ) ā >, A p = a + p < L (p) = K(2p, 2π). (2-2) Z = (π/2) 1/2 v 2 R 1 (A p, a + ρ 2 ), (2-21) R 1 (A p, a + ρ 2 ) (2-28). M p = K(2p, 2π), N p = (K(p, 2π)) 1/2. 15

27 2 (2-19) Hölder Sobolev (1-19) 2π v 2 2 a + (t)(v(t) + 1) 2 dt 2πρ 2 = 2π a + (t)(v 2 (t) + 2v(t))dt + 2π a + (t)dt 2πρ 2 A p v 2 + 2v p + ˆB A p ( v 2 2p + 2 v p ) + ˆB A p ( v 2 2 /M p + 2 v 2 /N p ) + ˆB, (2-22) ˆB = 2π(a + ρ 2 ). Z = (π/2) 1/2 v 2 (2-22) (M p A p )Z 2 e 1 A p Z B, (2-23) T = 2π e 1 = K(2p )/(K(p )) 1/2, e 2 = K(2p )/4, B = e 2 T 1/p (a + ρ 2 ). (2-24) A p < M p (2-23) B e 2 1 A2 p + 4(M p A p )B. (2-25) R(A p, B) Q(z) = (M p A p )z 2 e 1 A p z B = (2-26) R(A p, B) = e 1A p + ( e 2 1 A2 p + 4(M p A p )B ) 1/2 2(M p A p ). (2-27) (2-23) Z R 1 (A p, a + ρ 2 ) = R ( A p, e 2 T 1/p (a + ρ 2 ) ). (2-28) (2-27) R(A p, B) B. R 1 (A p, a + ρ 2 ) ρ = ρ(a) a + T 1/p a + p = T 1/p A p, (2-29) 16

28 2 B < e 2 T 1/p a + e 2 A p, R 1 (A p, a + ρ 2 ) < R 2 (A p, a + ) = e 1A p + ( e 2 1 A2 p + 4e 2 (M p A p )T 1/p a + ) 1/2 2(M p A p ) R 3 (A p ) = e 1A p + ( e 2 1 A2 p + 4e 2 (M p A p )A p ) 1/2 2(M p A p ) (2-3). (2-31) R 1 (A p, ā ρ 2 ) R 2 (A p, a + ) R 3 (A p ) = O(A 1/2 p ), A p +. (2-32) (2-2) (2-1). 2.5 (2-21) (2-25) p a(t) L p (R/2πZ) (2-2). ρ a + + K(2p ) A 2 p T 1/p K(p ) M p A p 1/2 = R 4 (A p, a + ), (2-33) ρ ( Ap T 1/p ) 1/2 ( 1 + K(2p ) K(p ) A p M p A p ) 1/2 = R 5 (A p ). (2-34) (2-33) (2-25) (2-34) (2-29) (2-33). (2-34) (2-17). (2-17) ρ (A p /T 1/p ) 1/2 (π/(k(2p )) 1/2 ) = R 6 (A p ), (2-35) a(t) L p (R/2πZ) p = (2-35) π/(k(2p )) 1/2 = 1, (2-34) (2-35) 1 p < π 2 /K(2p ) > 1, 17

29 2 A p (2-34) (2-35). 2.5 (2-7) r(t). (2-2) r max ρ 1/2 (1 + R 1 (A p, a + ρ 2 )) < ρ 1/2 (1 + R 2 (A p, a + )) ρ 1/2 (1 + R 3 (A p )). (2-36) r min > 2.5 r min a(t) (2-31) R 3 (A p ) < 1 A p < (2-37) R 1 (A p, a + ρ 2 ) < 1. (2-37) M p 1 + e 1 + e 2 = L 1 (p), (2-38) 2.2 Ermakov-Pinney (2-7) r(t) (i) a(t) (2-2) r(t) (2-36) (ii) a(t) (2-38) r(t) r min ρ 1/2 (1 R 1 (A p, a + ρ 2 )) ρ 1/2 (1 R 2 (A p, a + )) ρ 1/2 (1 R 3 (A p )) >. (2-39) 2.2 Ermakov-Pinney. 2.1 < L 2 (p) < L 1 (p) = M p 1 + e 1 + e 2, ā >, A p = a + p < L 2 (p), (2-4) (2-7) 2π- r(t). (2-7) r(t) x + â(t)x =, (2-41) 18

30 2 â > ā > â + p A p + 3 1/r 4 p. â(t) = a(t) + 3 r 4 (t). (2-42) â + (t) a + (t) + 3 r 4 (t), (2-38) (2-39) R 1 = R 1 (A p, a + ρ 2 ) 2.4 a(t) 1/r 4 p T 1/p (r min ) 4 T 1/p ρ 2 (1 R 1 ) 4, â + p A p + 3T 1/p ρ 2 (1 R 1 ) 4. â + p A p + 3T 1/p ρ 2 (1 R 1 ) 4 < M p, (2-43) (2-41). (2-17) A p + ρ = O(A 1/2 p ). (2-32) A p + 3T 1/p ρ 2 (1 R 1 ) 4 = O(A p ), A p +. (2-43) A p. p [1, ] ˆ L 2 (p) = sup { C (, L (p)) : ā > a + p < C (2-7) r(t) }. 2.1 L 2 ˆ(p). ˆL 2 (p, T) ˆL 2 (p, T) = ˆL 2 (p)/t 2 1/p, ˆL 2 (p) 2.1. L 3 (p) = (2π) 2 1/p L 2 (p). 19

31 L 3 (p) p 2.1 ˆL 2 (p) L 3(p) 2.4 a(t) (2-2) ρ Ermakov-Pinney r(t). a(t) L p (R/2πZ) p [1, ] a, A p = a p < L (p) = M p = K(2p, 2π). (2-44) 2.6 a(t) L p (R/2πZ) (2-44) t r(t) (2-7) 2π-. R r(t ) = ā 1/4 a (2-7) 2π r min a 2π a(t)r = 2π (1/r 3 ) 2π/(r min ) 3. r min ā 1/4. r max ā 1/4. t r(t ) = ā 1/4. 2

32 2 r() = ā 1/4. r(t) = ā 1/4 (u(t) + 1) u(t) = ā 1/4 r(t) 1 H 1 (, 2π) (2-18) 2π u 2 = 2π a(t)(u + 1) 2 ā 2π (1/(u + 1) 2 ). (2-45) p a(t) (2-44). Y = (π/2) 1/2 u 2 S 1 (A p, ā), (2-46) S 1 (A p, ā) A p, ā p [1, ]. Sobolev (1-19) u Y 2π (1/(u + 1) 2 ) 2π/(Y + 1) 2. Hölder Sobolev (1-19) (2-45) 2π u 2 2 = a(t)(u 2 + 2u) + 2πā ā 2π (1/(u + 1) 2 ) A p ( u 2 2 /M p + 2 u 2 /N p ) + 2πā(1 1/(Y + 1) 2 ). (2-47) (2-24) (2-47) (M p A p )Y 2 e 1 A p Y + e 2 T 1/p ā(1 1/(Y + 1) 2 ). (2-48) P(y) = (y + 1) 2 ((M p A p )y e 1 A p ) e 2 T 1/p ā(y + 2). P() = e 1 A p 2e 2 T 1/p ā < (2-44) P(y) S 1 (A p, ā) (2-48) P(Y) Y S 1 (A p, ā).. S 1 (A p, ā) P(y) = (y + 1) 2 ((M p A p )y e 1 A p ) e 2 T 1/p ā(y + 2) = (2-49) S 1 (A p, ā). (2-29) = P(S 1 (A p, ā)) P 1 (S 1 (A p, ā)), (2-5) 21

33 2 P 1 (y) = (y + 1) 2 ((M p A p )y e 1 A p ) e 2 A p (y + 2) A p p. (2-44) P 1 (y) = (y + 1) 2 ((M p A p )y e 1 A p ) e 2 A p (y + 2) = (2-51) S 2 (A p ). (2-49) (2-51) (2-5) S 1 (A p, ā) S 2 (A p ). (2-52) (2-51) S 2 (A p ) A p (, L (p)). A p + A p L (p) S 1 (A p, ā) S 2 (A p ) = (e 1 + 2e 2 )A p /M p + o(a p ). (2-53) lim S 1(A p, ā) = lim S 2(A p ) = +. (2-54) A p L (p) A p L (p) 2.7 Ermakov-Pinney (2-7) r(t) Hill (2-1) 2.3 a(t) (2-44) (i) (2-7) 2π- r(t) r max ā 1/4 (1 + S 1 (A p, ā)) ā 1/4 (1 + S 2 (A p )). (2-55) (ii) (2-1) ρ ρ ā 1/2 /(1 + S 1 (A p, ā)) 2 ā 1/2 /(1 + S 2 (A p )) 2. (2-56) (i) 2.7 (2-9) (2-55) (2-56). ρ = 1 2π 2π dt r 2 (t) ā 1/2 (1 + S 1 (A p, ā)). 2 22

34 2 2.3 r min a(t). S 1 (A p, ā) < 1 S 2 (A p ) < 1. S 2 (A p ) (2-51) P 1 (1) > S 2 (A p ) < 1. A p < M p 1 + e 1 + 3e 2 /4 = L 3(p), (2-57) 2.4 a(t) (2-57) (i) (2-7) 2π- r(t) r min ā 1/4 (1 S 1 (A p, ā)) ā 1/4 (1 S 2 (A p )) (> ). (2-58) (ii) (2-1) ρ ρ ā 1/2 /(1 S 1 (A p, ā)) 2 ā 1/2 /(1 S 2 (A p )) 2. (2-59) 2.1 (2-55) (2-58) A p + r(t) (2-53) r max /r min = 1 + O(A p ) r(t) ρ. 2.1 a(t) (2-44) A p + r(t) = ā 1/4 (1 + O(A p )), t, (2-6) ρ(a) = ā 1/2 (1 + O(A p )). (2-61) p = 1 ā = A 1 /(2π), M 1 = 2/π, e 1 = 2, e 2 = 1. (2-44) < A 1 < M 1 = 2/π. (2-62) 23

35 2 S 1 (A 1, ā) = S 2 (A 1 ). (2-56) ρ (2/π)y(y + 1) 2 = A 1 (y + 2)((y + 1) 2 + 1) (2-63) ρ (A 1 /(2π)) 1/2 /(1 + S 2 (A 1 )) 2 = S 3 (A 1 ) >, (2-64) (2-62) a(t). lim S 2(A 1 ) = +, A 1 2/π lim S 3(A 1 ) =. A 1 2/π (2-64). 2.5 a(t) L 1 (R/2πZ) a a(t) L 1 A 1 S 4 (A 1 ) ρ S 4 (A 1 ), A 1 (, ). (2-65) ξ 1 (, 2/π) ξ 2 = S 3 (ξ 1 ) = max {S 3 (y) : y (, 2/π)}. S 4 (A 1 ) = S 3 (A 1 ), < A 1 ξ 1, ξ 2, ξ 1 < A 1 <. (2-65) A 1 (, ). < A 1 ξ 1 (2-65) (2-64). ξ 1 < A 1 < a (t) = (ξ 1 /A 1 )a(t), a 1 = ξ 1. a(t) = (A 1 /ξ 1 )a (t) A 1 /ξ 1 > 1 2.1(ii) ρ(a) = ρ((a 1 /ξ 1 )a ) ρ(a ) ξ 2. (2-65). 24

36 2 ρ (2-34) (2-35). ( A1 /(2π π 2 A 1 ) ) 1/2, A1 (, ξ 3 ], S 5 (A 1 ) = (πa 1 /8) 1/2, A 1 (ξ 3, ), (2-66) ξ 3 = (2π 2 8)/π (2-57) ρ S 5 (A 1 ), A 1 (, ). (2-67) < A 1 < M e 1 + 3e 2 /4 = 8 15π. ρ (A 1 /2π) 1/2 /(1 S 2 (A 1 )) 2, A 1 (, 8/15π). 2.6 a(t) L 1 (R/2πZ) a (2-1) (2-67). 2.2 A a (2-57) (i) (2-7) T- r(t) < ā 1/4 (1 S 2 (A p )) r min r(t) r max ā 1/4 (1 + S 2 (A p )). (2-68) (ii) (2-1) ρ ā 1/2 /(1 + S 2 (A p )) 2 ρ ā 1/2 /(1 S 2 (A p )) 2. (2-69) 2.7 (2-1) 4- a (2-57) b(t) c(t). µ(ā, A p ) b(t) c(t) c 1 < µ(ā, A p ) b 2 1, (2-7) 25

37 ρ A A 1 (2-5) x = µ(ā, A p ) (2-73). β >. b(t) I = b(t)b(s)r 3 (t)r 3 (s)χ θ ( ϕ(t) ϕ(s) )dsdt r 6 min min χ θ(u) b 2 1. [,2π] 2 x θ [119] < θ < arccos( 1/4) χ θ (u) > u [, θ] min χ θ(u) = χ θ () = u [,θ] 5 (1 + 4 cos θ) cos(θ/2) 8 sin(3θ/2) 5 = χ (θ). a(t) (2-57) θ = 2πρ (, arccos( 1/4)). θ (, π/2) χ (θ). 2.8 ( I (ā) S 1 (A p ) ) 6 χ (2πρ) b

38 c(t) 2 ( (ā) S 1 (A p ) ) 6 χ 2π(ā) 1 2 b 2 (1 S 1 (A p )) 2 1. (2-71) II = 3 8 2π c(t)r 4 (t)dt 3 8 r4 max c (ā) 1 (1 + S 1 (A p )) 4 c 1. (2-72) µ(ā, A p ) = 8(1 S 1(A p )) 6 2π(ā) 1 2 3(ā) 1 2 (1 + S 1 (A p )) 4 χ, (2-73) (1 S 1 (A p )) 2 (2-71) (2-72) (2-7) β >. 2.5 x + e x = σ + h(t) (2-74) h L 1 (R/TZ, R), T >. (2-74) e x (2-74) Landesman-Lazer [33]. [111] (2-74) T- σ >. h L 1 (R/TZ) E (h) = { σ > < σ < σ (2-74) T- }. E (h) [85] inf h E (h) = Σ (T) = 16/T 2. σ (, E (h)) ψ σ,h (t) (2-74) T- E 1 (h) = inf { < σ < E (h) < σ < σ (2-74) ψ σ,h (t) }. [85] inf h E 1(h) = Σ 1 (T) = 4/T 2. 27

39 2 E 2 (h) = inf { < σ < E 1 (h) < σ < σ (2-74) ψ σ,h (t) }. E 2 (h) Σ 2 (T) 1/(2T 2 ). inf h E 2(h) = Σ 2 (T) >, (2-75) T > Σ i (T) = Σ i /T 2, i =, 1, 2. T 2π. (2-74) 2π- ψ(t) ψ(t) 2π e ψ(t) dt = 2πσ. (2-76) σ < σ < Σ 1 (2π) = 1/π 2 h L 1 (R/TZ) (2-74) ψ σ,h (t). (2-74) ψ σ,h (t) x + a(t)x =, a(t) = e ψ σ,h(t) >. (2-77) (2-76) a 1 = 2πσ. 2.4 (2-77) ψ σ,h (t). ρ (1/(2(2/π) 1/2 )(2πσ) 1/2 = (π/2)σ 1/2. (2-78) (2-74) < σ < 1/(2π) 2.253, (2-78) y + a(t)y a(t)y a(t)y3 + =, a(t) (2-77). ψ σ,h (t) β = 1 4 [,2π] 2 χ θ ( ϕ(t) ϕ(s) )r 3 (t)r 3 (s)a(t)a(s)dtds π r 4 (t)a(t)dt. 28

40 2 2.8 h L 1 (R/2πZ) Σ 2 > < σ Σ 2 h L 1 (R/2πZ) (2-74) ψ σ,h (t) Lyapunov. σ (2-78) (2-76) a(t) = e ψ σ,h(t) A 1 = a 1 = Tσ a(t) L 1 σ. p = 1 M 1 = 2/π, e 1 = 2, e 2 = 1. (2-78) (2-57) < 2πσ < 4/(15π 2 ). [118] < θ < arccos( 1/4) u [, θ] χ θ (u) > min χ θ(u) = χ θ () u [,θ] = 3 16 cot θ θ cot (1 + 4 cos θ) cos(θ/2) = 8 sin(3θ/2) 5 = : χ (θ) >. (2-79) σ (2-78) θ = 2πρ (, π/2) (, arccos( 1/4)). χ (θ) θ (, π/2). (2-66) ρ (2-77) ρ S 6 (σ) := S 5 (2πσ). (2-8) Ermakov-Pinney (2-7) r(t) r max σ 1/4 (1 + S 7 (σ)), r min σ 1/4 (1 S 7 (σ)), (2-81) S 7 (σ) = S 1 (2πσ, σ) = S 2 (2πσ) 2.3 y(y + 1) 2 = π 2 σ(y + 2)((y + 1) 2 + 1). β := 1 4 [,2π] 2 χ θ ( ϕ(t) ϕ(s) )r 3 (t)r 3 (s)a(t)a(s)dtds π r 4 (t)a(t)dt 29

41 2 4χ (θ)r 6 min rmax 4 2π [,2π] 2 a(t)a(s)dtds a(t)dt = 8πσχ (2πρ)r 6 min /r4 max, a 1 = 2πσ. (2-81) (2-8) χ (θ) B(σ) σ. σ + r 6 min /r4 max σ 1/2 (1 S 7 (σ)) 6 /(1 + S 7 (σ)) 4. β 8πσ 1/2 χ (2πρ) (1 S 7(σ)) 6 (1 + S 7 (σ)) 4. β 8πσ 1/2 χ (2πS 6 (σ)) (1 S 7(σ)) 6 =: B(σ). (2-82) (1 + S 7 (σ)) 4 S 6 (σ) = σ 1/2 (1 + O(σ)), S 7 (σ) = O(σ). (2-79) χ (θ) = 5 12θ (1 + O(θ2 )), θ +. B(σ) χ (2πS 6 (σ)) = 5 (1 + O(σ)). 24πσ1/2 B(σ) = 5/3 + O(σ), σ +. Σ 2 > β B(2πσ) > 1 σ (, Σ 2 ] β > ψ σ,h (t) σ (, Σ 2 ]. 3

42 B * (A 1 ) A B (A 1 ) := B(A 1 /2π) A 1 = 2πσ πσ 1/2 χ (2πS 6 (σ)) > (1 + S 7(σ)) 4 (1 S 7 (σ)) 6. (2-83) (2-83) ρ S 6 (σ) r(t) S 7 (σ) S 6 (σ) S 7 (σ). < σ Σ 2 =.128. Σ 2 = (2π)2 Σ 2 = , Σ 2 (T) 1/(2T 2 )

43 2 2.6 x + x 2 = σ + h(t), (2-84) σ R h L 1 := { h L 1 (R/2πZ, R) : h }. (2-84) Ambrosetti-Prodi [3]. [92]. [36] h L 1 E(h) R σ < E(h) σ = E(h) σ > E(h) (2-84). E(h) (2-84). h L 1 E(h) L 1 - E() = E(h) > h. [85] Σ > E(h) < σ < Σ (2-85) (2-84) 2π- ψ ± (t) Σ {σ n } {h n } σ n Σ σ = σ n h = h n n = 1, 2, (2-84). E(h) < σ < Σ 1 = Σ /16 (2-86) 2π- ψ (t) ψ + (t). [34] [119] ψ + (t) E(h) < σ Σ 3 = (π 1 arccos( 1/4)) 4 Σ 1 (, Σ 1 ), (2-87) ψ (t) ψ + (t). (2-87) Σ 2 32

44 2 σ σ=e(h) Σ two solutions Σ 1 elliptic and unstable Σ 2 stable and unstable O h 2.4. (Σ 3, Σ 1 ) E(h) < σ < Σ 2 (2-88) (2-84) ψ + (t) (2-84) x = y + ψ + (t) y y + a(t)y + y 2 =, (2-89) a(t) = 2ψ + (t) a(t) σ h a 2 = 2 ψ + 2 = (8πσ) 1/2. (2-9) (2-89) Hill 33

45 2 (2-89) β = χ θ ( ϕ(t) ϕ(s) )r 3 (t)r 3 (s)dtds, (2-91) [,2π] 2 θ, r ϕ χ θ χ θ (y) = 3 cos(y θ/2) 16 sin(θ/2) + cos 3(y θ/2), y [, θ]. 16 sin(3θ/2) (2-91) ξ = ϕ(t) η = ϕ(s) (2-8) β = χ θ ( ξ η )R 5 (ξ)r 5 (η)dξdη, (2-92) [,θ] 2 R(ξ) = r(ϕ 1 (ξ)), ξ [, θ]. (2-93) (2-92) R 5 (ξ) Hilbert L 2 (, θ), L : L 2 (, θ) L 2 (, θ) (Ly)(ξ) = θ L χ θ [,θ] χ θ ( ξ η )y(η)dη, β = LR 5, R 5. χ θ ( ξ η )dη = 5, ξ [, θ]. 12 (2-85) (2-86) (2-9) [,θ] 2 χ θ ( ξ η )dξdη = 5θ 12. (2-94) Σ 1 = K2 (4) 64π 4, Σ = 16Σ 1. θ ζ 1 a 2 = ζ 2 σ 1/4, (2-95) 34

46 2 ζ 2 = (8π) 1/4 ζ 1. Θ = 2π, Θ 1 = π Σ i ζ 2 Σ 1/4 i = Θ i, i =, 1. (2-96) Θ 3 = arccos( 1/4) (π/2, 2π/3). (2-97) [119] ξ [, θ] χ θ (ξ) < θ Θ 3 θ π/2 β > θ = π/2 (2-92) [83] (2-89) 2.2 [119] θ (2-84) ψ (2-98) ψ +. < θ Θ 3, (2-98) E(h) < σ Σ 3 := (Θ 3 /ζ 2 ) 4 = (Θ 3 /π) 4 Σ 1, (2-99) θ (Θ 3, Θ 4 ) Θ 4 = 2π/3. θ = θ (θ) = arccos ( (3/2) cos θ ) = 1 arccos( 1 3 cos θ), 2 < θ (θ) < θ/2. θ 1 = θ/2 θ θ 2 = θ/2 + θ. 2.9 χ θ (ξ), θ 1 ξ θ 2, χ θ (ξ), ξ θ 1 θ 2 ξ θ. (ξ, η) [, θ] 2 χ + θ (ξ, η) = max(χ θ(ξ, η), ), χ + θ (ξ, η) = min(χ θ(ξ, η), ) 35

47 2 C + (θ) = χ + θ (ξ, η)dξdη, C (θ) = [,θ] 2 (2-94) C ± (θ) R (θ) = C + (θ) C (θ) + 5θ 12. [,θ] 2 χ θ C + (θ) C (θ) = 2 1/2 (2 + 3 cos θ) 3/2 2 1/2 (2 + 3 cos θ) 3/2 5 sin(3θ/2). 2.4 ψ + +, < θ Θ 3, (C + (θ)/c (θ)) 1/1, Θ 3 < θ < Θ 4. (ξ, η)dξdη. (2-1) (2-11) θ (, Θ 4 ), (2-12) ψ +. r max r min < R (θ), (2-13) (2-92) β = χ + θ (ξ, η)r5 (ξ)r 5 (η)dξdη χ θ (ξ, η)r5 (ξ)r 5 (η)dξdη [,θ] 2 [,θ] 2 R 1 min (ξ, η)dξdη R1 max (ξ, η)dξdη [,θ] 2 χ + θ = R 1 min C+ (θ) R 1 maxc (θ), [,θ] 2 χ θ C (θ) =, θ (, Θ 3 ]. (2-93) R max = r max, R min = r min R (θ) (2-13) β > ψ +. E(h) < σ < Σ 4 := (Θ 4 /ζ 2 ) 4 = Σ 1, (2-14) 36

48 2 (2-12). u(t) = (r(t) r )/r u 2π u 2 2 = a(t)(u + 1) 2 θ 2 /(2π). (2-15) R 1 (θ) = 2.1 (2-12) u(t) ψ +. ( ) 1/2 2 R (θ) 1 π R (θ) + 1, θ (, Θ 4). (2-16) u 2 < R 1 (θ), (2-17) Sobolev (1-19) u u 2 /(K(, 2π)) 1/2 = (2/π) 1/2 u 2. (2-17) (2-16) u < (R (θ) 1)/(R (θ) + 1) 1. r max < r (1 + (R (θ) 1)/(R (θ) + 1)) = 2r R /(R + 1), r min > r (1 (R (θ) 1)/(R (θ) + 1)) = 2r /(R + 1). r max /r min < R (2-12). 2.4 ψ a 2 < K 4 := K(4, 2π), σ < Σ 1, (2-18) R 2 ( a 2, θ) u(t) u 2 R 2 ( a 2, θ). (2-19) 37

49 2 r(t) u(t) H 1 (, 2π) W4,1 (R/2πZ). (2-15) Hölder Sobolev u 2 2 a 2 (u + 1) 2 2 θ 2 /(2π) ( 2π = a 2 u π u π u π u + 2π 1) 1/2 θ 2 /(2π) ( a 2 u u u u 1 + 2π ) 1/2 θ 2 /(2π) a 2 S 1 ( u 2 ) θ 2 /(2π), (2-11) S 1 (x) = x4 K x3 + 6x2 K 3/2 3 K 2 + 4x K 1/ π 1/2, x [, ), (2-111) K i = K(i, 2π), i = 1, 2, 3, 4. (2-11) a 2 u 2 2 /K 4. (2-18) S 2 (x) := x 2 a 2 S 1 (x) θ 2 /(2π) = (2-112) R 2 ( a 2, θ). (2-11) (2-112) (2-19). 2.9 Σ 2 (Σ 3, Σ 1 ) h L 1 E(h) < σ < Σ 2 (2-84) ψ +. σ (2-14) (2-12) ψ +. R 2 ( a 2, θ) < R 1 (θ), (2-113) (2-14) S 2 (x). S 2 (x) >, x [, + ). 38

50 2 (2-112) R 2 (2-113) R 1 (θ) (2-112) S 2 (R 1(θ)) > S 2 (R 1 (θ)) >. a 2 < 2R 1(θ) S 1 (R 1(θ)) =: R 3(θ), (2-114) a 2 < 2π(R 1(θ)) 2 + θ 2 2πS 1 (R 1 (θ)) =: R 4 (θ). (2-115) R 5 (θ) = min (R 3 (θ), R 4 (θ)), θ (, Θ 4 ). (2-114) (2-115) a 2 < R 5 (θ). (2-116) lim R 3 (θ) = θ Θ 3 2 3/2 π 1/2 S 1 ((2/π)1/2 ).2291, lim R 4 (θ) = 4 + (arccos( 1/4))2 θ Θ 3 2πS 1 ((2/π) 1/2 ) R 5 (Θ 3 +) = R 4 (Θ 3 +).1728 > (Θ 3 /ζ 1 ) , (2-117) Θ 21 (Θ 3, Θ 4 ) R 5 (θ) = R 4 (θ), θ (Θ 3, Θ 21 ) R 5 (θ) θ (Θ 3, Θ 21 ). R 5 (θ) (θ Θ 3 ) 1/5, θ Θ 3 +, lim R 5 (θ) =. θ Θ 3 39

51 θ Θ 3 R 3 (θ) R 4 (θ) R 6 (σ) = R 5 (ζ 2 σ 1/4 ), σ (Σ 3, Σ 4 ). Σ 21 := (Θ 21 /ζ 2 ) 4 (Σ 3, Σ 4 ), R 6 (σ) σ (Σ 3, Σ 21 ). (2-95) (2-99) (2-117) R 7 (σ) = (8πσ) 1/2, σ (Σ 3, Σ 4 ), lim(r 7 (σ) R 6 (σ)) = (8πΣ 3 ) 1/2 R 5 (Θ 3 +) = (Θ 3 /ζ 1 ) 2 R 5 (Θ 3 +) <. σ Σ 3 lim(r 7 (σ) R 6 (σ)) = (8πΣ 4 ) 1/2 >. σ Σ 4 R 6 (σ) = R 7 (σ) (2-118) (Σ 3, Σ 4 ) Σ 22 (Σ 3, Σ 4 ). Σ 2 = min(σ 21, Σ 22 ) (Σ 3, Σ 4 ). 4

52 2 Σ 2. (σ, h) E(h) < σ < Σ 2 (2-14) (2-116). σ Σ ψ + σ (Σ 3, Σ 2 ). 1 θ Θ ψ +. 2 θ > Θ 3. (2-95) θ ζ 2 σ 1/4 < ζ 2 Σ 1/4 2 ζ 2 Σ 1/4 21 = Θ 21. R 5 (θ) θ (Θ 3, Θ 21 ) R 5 (θ) R 5 (ζ 2 σ 1/4 ) = R 6 (σ). (2-116) σ (8πσ) 1/2 < R 6 (σ) R 7 (σ) < R 6 (σ). σ (Σ 3, Σ 2 ) (Σ 3, Σ 22 ) Σ γ > a(t) L 1 (R/2πZ) a Ermakov-Pinney x + a(t)x = 1/x γ, γ >. (2-119) γ = 3 (2-119) Ermakov-Pinney (2-7) a(t) a > r (t) a 1/4. Lazer Solimini [49] [2,11,12,16,19]. [19] Brillouin x + λ(1 + δ cos t)x = 1 x (2-12) 2π- λ >, < δ < 1 γ, δ λ(1 + δ) < 1 32, (2-121) 41

53 2 8(1 + δ) 9 < 81e 16λδ (1 δ + e 4λδ ) 5. (2-122) 2.1 γ > γ 3 L 4 (γ) > a(t) L 1 (R/2πZ), a A 1 = a 1 L 4 (γ), (2-119) 2π-. γ (, 3) β < γ (3, ) β > a(t) L 1 (R/2πZ) a [19,25] A 1 = a 1 < L (1) = 2/π, (2-123) (2-119) 2π- ψ(t). (2-123) 2.6 t R ψ(t ) = ā 1/(γ+1). t = ψ(t) = ā 1/(γ+1) (1 + v(t)), v(t) = ā 1/(γ+1) ψ(t) 1 H 1 (, 2π). (2-119) ψ(t) 2π v 2 dt = 2π 2π 2.5 < a(t)(v + 1) 2 dt ā a(t)(v + 1) 2 dt. 2π (v + 1) γ dt 42

54 (2-123) (2-119) ψ(t) v (π/2) 1/2 v 2 < R 3 (A 1 ), R 3 (A 1 ) (2-31). A 1 (2-57) ψ min ā 1/(γ+1) (1 R 3 (A 1 )) (> ), ψ max ā 1/(γ+1) (1 + R 3 (A 1 )), (2-124) ā = A 1 /(2π). (2-124) A ψ(t) = ā 1/(γ+1) (1 + O(A 1/2 1 )), t. (2-125) 2.14 < L 5 (γ) < L (1) = 2/π (2-119) 2π- ψ(t). A 1 < L 5 (γ), (2-126) 2.15 (2-126) (2-119) 2π- ψ(t). (2-124) (2-119) D := { x : x S 8 (A 1 ) := ā 1/(γ+1) (1 R 3 (A 1 )) }. (2-119) D ψ i (t) x(t) := ψ 1 (t) ψ 2 (t) ( ) x + α(t)x =, (2-127) 1 dτ α(t) = a(t) + γ (ψ 2 (t) + τ(ψ 1 (t) ψ 2 (t))) γ+1 a(t) + γ/(s 8 (A 1 )) γ+1. 43

55 2 α(t) > α 1 A 1 + γ/(s 8 (A 1 )) γ+1 = A 1 + γa 1 /(2π(1 R 3 (A 1 )) γ+1 ), A 1 +. Lyapunov α 1 2/π (2-127) 2π- x(t) = ψ 1 (t) ψ 2 (t) < L 6 (γ) < L 5 (γ) A 1 < L 6 (γ), (2-128) (2-119) 2π- ψ(t) ρ (, 1/4). x + â(t)x =, â(t) = a(t) + γ/(ψ(t)) γ+1, (2-129) 2.1 (2-128) (2-119) ψ(t) â(t) (2-129) y + â(t)y + ˆb(t)y 2 + ĉ(t)y 3 + =, γ(γ + 1) γ(γ + 1)(γ + 2) ˆb(t) =, ĉ(t) =. 2(ψ(t)) γ+2 6(ψ(t)) γ+3 ˆβ = γ2 (γ + 1) 2 4 γ(γ + 1)(γ + 2) 16 [,2π] 2 2π ˆr 3 (t)ˆr 3 (s) ψ γ+2 (t)ψ γ+2 χˆθ ( ˆϕ(t) ˆϕ(s) )dtds (s) ˆr 4 (t) dt, (2-13) ψ γ+3 (t) ˆθ = 2πˆρ ˆρ (2-129) ˆr(t) (2-7) a(t) = â(t) 2π- ˆϕ(t) (2-8) r(t) ˆr(t). A 1 + ψ(t) (2-125) â(t) = a(t) + γā(1 + O(A 1/2 1 )), t, 44

56 2 â = (γ + 1)ā(1 + O(A 1/2 1 )), Â 1 = â 1 = O(A 1 ), ˆρ = (γ + 1) 1/2 ā 1/2 (1 + O(A 1/2 1 )), ˆθ = 2πˆρ = 2π(γ + 1) 1/2 ā 1/2 (1 + O(A 1/2 1 )), ˆr(t) = (γ + 1) 1/4 ā 1/4 (1 + O(A 1/2 1 )), t, χˆθ (x) = 5 12ˆθ (1 + O(ˆθ 2 )), x [, ˆθ], 5 χˆθ ( ˆϕ(t) ˆϕ(s) ) = 24π (γ + 1) 1/2 ā 1/2 (1 + O(A 1/2 1 )), (t, s). (2-13) ˆβ := ˆβ 1 ˆβ 2 = ˆβ 1 = 5π 24 γ2 ā 2/(γ+1) (1 + O(A 1/2 1 )), ˆβ 2 = π 8 γ(γ + 2)ā2/(γ+1) (1 + O(A 1/2 1 )). 5γ (1 + O(A1/2 3(γ + 2) 1 )), A 1 +. (2-131) < γ < 3 5γ/(3(γ + 2)) < 1 L 4 (γ) > A 1 L 4 (γ) ˆβ < 1 ˆβ <. 3 < γ < 5γ/(3(γ + 2)) > 1 L 4 (γ) > A 1 L 4 (γ) ˆβ > 1 ˆβ >. 2.1 Brillouin (2-12) 2.2 λ > < δ < 1 λ (2-12) 2π δ < δ < 1 [19] (2-121) (2-122). 45

57 Birkhoff Moser. ẏ = J y H(t, y), (3-1) J = 1 1 H = H(t, y) : R R 2 R. 3.1 (3-1) ẋ = a(t)y, ẏ = b(t)x. (3-2) (3-1) (3-2) (3-2). (3-2) 3.1. Ermakov-Pinney r + a(t)b(t)r = a2 (t) r 3 r + a(t)b(t) r = b2 (t) r 3 + ȧ(t) a(t)ṙ (3-3) + ḃ(t) b(t) r, (3-4) 46

58 3 (3-2) 3.2. a(t) 1 (3-2) Hill Ermakov-Pinney ẍ + b(t)x =, (3-5) ẍ + b(t)r = 1 r 3. (3-6) (3-6) (3-5) [121]. 3.4 ẋ = a(t)y + c(t)y 2n 1 + G (t, x, y), x ẏ = b(t)x d(t)x 2n 1 G (3-7) y (t, x, y), a, b, c, d T- c(t) dt, d(t) dt n 2 G : R B ɛ() R T- G(t, x, y) = O ( (x 2 + y 2 ) n+1/2), (x, y) (, ), (3-8) t R. (3-2) (3-7). c(t), d(t), c(t), d(t), 3.5 Φ ( Φ(x ) ) = f (t, x), (3-9) 47

59 3 Φ Φ() =. Φ(s) = s 1 s 2. Φ- [6,7,65,66] Leray- Schauder, Mawhin.. Φ 1 (s) = s 1 + s 2. [11,42]. 3.7 ẋ = a(t)y + a 1 (t)y 2 + a 2 (t)y 3 +, ẏ = b(t)x b 1 (t)x 2 b 2 (t)x 3, (3-1) a, b, a i, b i, i = 1, 2 T- T (3-1). 3.8 ẋ = y + a 2 (t)y 3 +, ẏ = b(t)x b 1 (t)x 2 b 2 (t)x 3,. (3-11). (3-11) 3.2 R (3-1). H(t, y) t T- y = 48

60 3 ϕ(t) (3-1) T- y = ϕ(t) + ỹ (3-1) ỹ = JB(t)ỹ + ỹ H(t, ỹ), (3-12) B(t) = y H(t, ϕ(t)), H(t, y) H(t, ) =, ỹ H(t, ) =. ỹ = (ỹ 1, ỹ 2 ) T (3-12) ỹ = JB(t)ỹ, (3-13) B(t) = α(t) β(t) β(t) γ(t), α(t), β(t), γ(t) T [54] t ψ(t) ỹ = R ψ(t) x (3-14) (3-13) ẋ 1 = a(t)x 2, ẋ 2 = b(t)x 1. (3-15) θ R, R θ = cos θ sin θ sin θ cos θ. ψ(t) [54] B(t) T- R ψ(t) T-. (3-14) (3-15). 49

61 3 (3-15) Poincaré φ 1 (T) M = ψ 1 (T) φ 2 (T) ψ 2 (T), (φ 1 (t), ψ 1 (t)) T (φ 2 (t), ψ 2 (t)) T (3-15), φ 1 () = 1, ψ 1 () =, φ 2 () =, ψ 2 () = 1. M λ 1,2 (3-15) Floquet λ 1 λ 2 = 1 (3-15) : λ 1 = λ 2, λ 1 = 1, λ 1 ±1; < λ 1 < 1 < λ 2 ; λ 1 = λ 2 = ±1. [43] 7.2 (3-15) (3-15) λ 1 = λ 2 = 1 (3-15) T- λ 1 = λ 2 = 1 2T-. (3-15) Ẋ = A(t)X, (3-16) X = x 1 x 2, A(t) = a(t) b(t). M A T Poincaré M = M(A, T). (3-16) Floquet λ λ λ = e iθ, θ >, θ nπ, n = 1, 2,. M(A, T) Symp(R 2 ) = {M : R 2 R 2, det M = 1}, 5

62 3 R θ. Symp(R 2 ). 3.1 [83] (3-16) R- (3-16) Poincaré. - (3-16). Φ(t) = Φ(t, A) (3-16) (3-16) Φ() = I 2. M(A, T) = Φ(T, A). T α,t (t) := α(t t ), t R, (3-17) α > t R. s = T α,t (t) Φ() = I 2, Y(s) = X(t + s/α), (3-16) Y (s) = A α,t (s)y(s), (3-18) A α,t (s) = α 1 A(t + s/α) s T α := αt = d ds. (3-18) Φ(s, A α,t ) Φ(t + s/α, A)Φ(t, A) 1. (3-18) Poincaré M(A α,t, T α ) M(A α,t, T α ) = Φ(t + T, A)Φ(t, A) 1 = Φ(t, A)Φ(T, A)Φ(t, A) 1 = Φ(t, A)M(A, T)Φ(t, A) 1, (3-19) Φ(t + T, A) = Φ(t, A)Φ(T, A) A(t) T-. 51

63 3 3.2 (3-16) Floquet λ λ λ = exp(iθ), θ >, θ nπ, n = 1, 2,. a(t) a(t), t R. (3-2) t R α > D α = diag(α, α 1 ) (i) M(A α,t, T α ) = D α R θ D 1 α, (ii) M(A α,t, T α ) = D α R θ D 1 α. λ = exp(iθ) (3-16) Floquet v C 2 M(A, T) λ (3-16) X() = v (x 1 (t), x 2 (t)) T x 1 (t + T) = λx 1 (t), x 2 (t + T) = λx 2 (t), t R. (3-21) λ = 1 t x 1 (t) 2 T- t R d dt x 1(t) 2 =, t = t. x 1 (t) t x 1 (t ). φ(t) = x 1 (t)/x 1 (t ), ψ(t) = x 2 (t)/x 1 (t ). φ(t ) = 1, d dt φ(t) 2 =, t = t. (3-22) φ(t) = φ 1 (t) + iφ 2 (t), ψ(t) = ψ 1 (t) + iψ 2 (t). (3-22) φ 1 (t ) = 1, φ 2 (t ) =, φ 1 (t ) =. (3-23) (3-15) (3-23) ψ(t ) = φ(t )/a(t ) = i φ 2 (t )/a(t ) ψ 1 (t ) =, ψ 2 (t ). (3-24) 52

64 3 (φ(t), ψ(t)) T (3-16) (φ k (t), ψ k (t)) T, k = 1, 2 (3-16) X 1 (t) := (φ 1 (t), ψ 1 (t)) T, X 2 (t) := (φ 2 (t)/ψ 2 (t ), ψ 2 (t)/ψ 2 (t )) T (3-16). (3-23) (3-24) X 1 (t ) = (1, ) T, X 2 (t ) = (, 1) T, (3-16) (3-16) Φ(t, A)Φ(t, A) 1 = (X 1 (t), X 2 (t))., M(A α,t, T α ) = Φ(t + T, A)Φ(t, A) 1 φ 1 (t + T) φ 2 (t + T)/ψ 2 (t ) = ψ 1 (t + T) ψ 2 (t + T)/ψ 2 (t ) = : M. λ = ν 1 + iν 2 ν k R, k = 1, 2 (3-21) (3-23) (3-24) φ 1 (t + T) = ν 1, φ 2 (t + T) = ν 2, ψ 1 (t + T) = ν 2 ψ 2 (t ), ψ 2 (t + T) = ν 1 ψ 2 (t )., M = ν 1 ν 2 /ψ 2 (t ) ν 2 ψ 2 (t ) ν 1. (1, iψ 2 (t )) T M λ 3.3 M = PR θ P 1, P = diag(1, ψ 2 (t )). (3-25) α = ψ 2 (t ) 1/2 >. ψ 2 (t ) < (3-25) M = D α R θ D 1 α. ψ 2 (t ) > (3-25) M = D α R θ D 1 α.. 53

65 3 3.3 [81] M 2 2 λ λ ω = (a + ib, c + id) T C 2 λ a b P = c d. M = PR θ P 1. s = α(t t ), x (s) = x(t + s/α), y (s) = α 1 y(t + s/α), (3-26) t R α > (3-16) ẋ = a (s)y, ẏ = b (s)x, (3-27) a (s) = a(t + s/α), b (s) = α 2 b(t + s/α) s T = αt. 3.1 (3-16) 3.2 α > t R (3-26) (3-27) Poincaré R θ. t, α 3.2 Φ(t + T, t ) = D α R θ D 1 α, (3-28) exp(iθ) θ R (3-16) Floquet. (3-26) (3-27) T = αt. (φ 1 (t, t ), ψ 1 (t, t )) T (φ 2 (t, t ), ψ 2 (t, t )) T (3-15) φ 1 (t ) = ψ 2 (t ) = 1, φ 2 (t ) = ψ 1 (t ) =. (φ 1 (τ, ), ψ 1 (τ, ))T (φ 2 (τ, ), ψ 2 (τ, ))T (3-27) φ 1 () = ψ 2 () = 1, φ 2 () = ψ 1 () =. (3-28) (3-27) Φ (T, ) = R θ. 54

66 3 3.3 Ermakov-Pinney (3-15) Ermakov-Pinney r + a(t)b(t)r = a2 (t) r 3 (3-2). + ȧ(t) a(t)ṙ (3-29) (φ 1 (t), ψ 1 (t)) T (φ 2 (t), ψ 2 (t)) T (3-15) φ(t) = φ 1 (t) + iφ 2 (t), ψ(t) = ψ 1 (t) + iψ 2 (t). φ(t) = R(t)e iϕ(t), (3-3) R(t) ϕ(t) t R R(t) >. ψ(t) = φ(t) a(t), ψ(t) = b(t)φ(t), R(t) ϕ(t) a R + a 2 br ar ϕ 2 ȧṙ =, 2aṘ ϕ + ar ϕ ȧr ϕ =. (3-31) ϕ = c a(t) R 2, (3-32) c = R 2 () ϕ()/a() (φ 1 (t), ψ 1 (t)) T (φ 2 (t), ψ 2 (t)) T. (3-3) ψ i () = φ i ()/a(), i = 1, 2 R 2 () = φ 2 1 () + φ2 2 (), ϕ() = φ 1() φ 2 () φ 2 () φ 1 () φ 2 1 () + φ2 2 (). c = ϕ()r 2 ()/a() = φ 1 () ψ 2 () φ 2 () ψ 1 (). (3-33) 55

67 3 (3-32) (3-31) R(t) R + abr = c2 a 2 R 3 + ȧ aṙ. r(t) Ermakov-Pinney (3-29). r(t) (3-29) ϕ(t) r(t) = c 1/2 R(t). (3-34) ϕ(t) = t a(s) r 2 (s) ds, (3-15) x 1 (t) = Ar(t) sin(ϕ(t) + B), x 2 (t) = A ṙ(t) 1 a(t) sin(ϕ(t) + B) + A r(t) cos(ϕ(t) + B), (3-35) A, B R. 3.2 : (i) Ermakov-Pinney (3-29) T-. (ii) (3-15) T- 2T-. (iii) (3-15). [43] (ii) (iii) (ii) (i). (3-15) λ S 1 \{±1} Floquet λ M v = (v 1, v 2 ) T C 2 λ : Mv = λv. (x 1 (t), x 2 (t)) T (3-15) x 1 () = v 1, x 2 () = v 2. x 1 (t) = x 1 1 (t) + ix2 1 (t), x 2(t) = x 1 2 (t) + ix2 2 (t), 56

68 3 t R x 1 (t), x 2 (t) x 1 (t + T) λx 1 (t), x 2 (t + T) λx 2 (t). (3-36) λ x1 1 x2 1. R(t) = x(t) R(t) >, t R. (3-36) R(t + T) R(t). (3-3)-(3-33)-(3-34) r(t) = cr(t) (3-29) T- c = 1 x 11 ()x22 () x21 ()x12 (). (3-15) Floquet λ = λ 1,2 = ±1 x 1 (t + T) λx 1 (t) (φ 1 (t), ψ 1 (t)) T (φ 2 (t), ψ 2 (t)) T (3-29) T-. r(t) = c ( φ 2 1 (t) + φ2 2 (t)) 1/2 c >. (3-29) T- (ii) (i). (i) (iii). (3-29) T- r(t) (3-35) (3-15). 3.3 (3-15) (3-29) T-.. r j (t), j = 1, 2 (3-29) T- ϕ j (t) = t a(s) j = 1, 2, r 2 j (s)ds, ϕ j (t + T) ϕ j (t) + θ j, j = 1, 2, θ j. (3-35) x j (t) = r j (t) exp(iϕ j (t)), y j (t) = ṙ a exp(iϕ j(t)) + 1 r exp(iϕ j(t)) (3-15) (3-37) x j (t + T) exp(iθ j )x j (t), y j (t + T) exp(iθ j )y j (t), j = 1, 2. 57

69 3 (x j (t), y j (t)) T (3-15) Floquet Floquet exp(iθ j ), j = 1, 2. exp(iθ 2 ) = exp(iθ 1 ) exp(iθ 2 ) = exp( iθ 1 ). Floquet γ C\{} x 2 (t) γx 1 (t) x 2 (t) γx 1 (t) x 1 (t) x 1 (t). t = r 2 () = γr 1 () γ (3-29) T- r 1 (t) = x 1 (t) r 2 (t) = x 2 (t) = γr 1 (t). r 1 + abr 1 = a2 r ȧ a r 1, γ r 1 + γabr 1 = γ = 1 r 1 (t) = r 2 (t). a2 γ 3 r γȧ a r t R b(t). ψ(t) = R(t)e i ϕ(t) Ermakov-Pinney r + a(t)b(t) r = b2 (t) r (3-15) (3-38). + ḃ(t) b(t) r. (3-38) 3.4 (3-7). [2,81,95]. F : Ω C C z = F F. F F = F(z, z). 3.4 [81] m 3 F(z, z) = λz + O( z m 1 ), z, (λ S 1 ). m H = H(z, z) F(z, z) = λ (z + 2i z H(z, z) + O( z m )), z. (3-39) 58

70 3 F 3.4 m = 2n, n 2 H β R α k C, k =,, n 1. n 1 ( H(z, z) = β z 2n + αk z k z 2n k + ᾱ k z k z 2n k), (3-4) k= 3.5 [81] λ S 1 F (3-39) m = 2n n 2 H (3-4)., (C 1 ) p = 1,, n λ 2p 1 β. (C 2 ) p = 1,, n λ 2p = 1 H # (z, z), z C {} F z =. H # (z, z) = 1 2p 1 H(λ r z, λ r z). 2p r= (3-15) φ(t + T) = λφ(t), ψ(t + T) = λψ(t), t R, (3-41) λ S 1 M. 3.2 (3-41). 3.2 (3-15) λ = 1, λ ±1 - (3-41). (3-15) (3-41) (3-15) T- 2T-. (3-41) (3-15) M. λ ±1 M ±I 2 λ = ±1 M ±I 2. ẋ = a(t)y + f (t), ẏ = b(t)x g(t), (3-42) f g (3-41). 59

71 3 3.6 (x(t), y(t)) T (3-42) x() = y() = (3-41) x(t) + iy(t) = iλ ( f (t)ψ(t) + g(t)φ(t)) dt. 3.4 (3-15) c(t) dt, d(t) dt (3-7) : (H 1 ) c(t), d(t). (H 2 ) c(t), d(t). (3-41) 3.2 [83,84]. (x(t), y(t)) T = (x(t, z, z), y(t, z, z)) T (3-7) x() = q, y() = p, z = q + ip., x(t, z, z) = φ(t)z + φ(t) z 2 + O( z 2 ), z, (3-43) y(t, z, z) = ψ(t)z + ψ(t) z 2 t [, T]. (3-7) (3-42) f (t) = c(t)y 2n 1 (t) + G (t, x(t), y(t)), x g(t) (3-8) (3-43) (3-44) + O( z 2 ), z. (3-44) = d(t)x2n 1 (t) + G (t, x(t), y(t)). (3-45) y G G (t, x(t), y(t)), x y (t, x(t), y(t)) = O( z 2n ), z. (3-41) (3-7) 3.6 P(z, z) = λz iλ g(t)φ(t) dt iλ f (t)ψ(t) dt. (3-46) 6

72 3 (3-43) (3-44) (3-46) ( ) 2n 1 φ(t)z + φ(t) z P(z, z) = λz iλ c(t)φ(t) dt 2 ( ) 2n 1 ψ(t)z + ψ(t) z iλ d(t)ψ(t) dt + O( z 2n ). 2 P (3-39) H (3-4) β β = 1 2 2n+1 n H(z, z) = 1 2n n 1 2 2n 2 2n c(t)( φ(t)z + φ(t) z) 2n dt 2n d(t)( ψ(t)z + ψ(t) z) 2n dt. 2n ( c(t) φ(t) 2n dt + ) d(t) ψ(t) 2n dt. (φ 1 (t), ψ 1 (t)) T (φ 2 (t), ψ 2 (t)) T (3-15) t R φ(t), ψ(t). (3-29) T- r(t) (3-38) r(t) β β = 1 2 2n+1 n γ γ. 2n n ( ) γ c(t)r 2n (t) dt + γ d(t) r 2n (t) dt, (H 1 ) p = 1,, n λ 2p 1 (C 1 ) β < z C {} H(z, z) <. p = 1,, n λ 2p = 1 H # H H # (z, z) < z C {} (C 2 ). (H 2 ). ẋ = h(t) sin y, ẏ = l(t) sin x. (3-47) 3.1 h(t), l(t) T- (3-47). 61

73 3 (3-47) (3-7) n = 2 a(t) = h(t), b(t) = l(t), c(t) = h(t), d(t) = l(t) 6 6. h(t), l(t) T [6] ẋ = c(t)y 2m+1 + f 1 (x, y, t), ẏ = d(t)x 2n+1 f 2 (x, y, t), (3-48) m, n Z +, m + n 1, c(t), d(t), f 1, f 2. (3-48) G : (x, y) ( x, y). (3-48) (x, y) = (, ) 3.4. m = n [6]. 3.3 (3-7) (x, y) = (, ) (3-15). (3-15). [85] a(t) > b(t) > ( ) ( ) a(t) dt b(t) dt 4, (3-15). [122]. 3.5 Φ Φ : ( a, a) ( b, b) Φ() = < a, b Φ 1 C 3 Φ 1 (y) = y + ky 3 +, k = (Φ 1 ) (). 6 62

74 3 f (t, ) x = (3-9). Φ a < +. Φ b < +. a = b = + ( ) p- Φ 1 (x) = x p 2 x, p > 1. a < +, b = + ( ) Φ 2 (x) = x 1 x 2. a = +, b < + ( ) Φ 3 (x) = x 1 + x 2.. Φ 2 (x ) = y (3-9) ẋ = Φ 1 2 (y), ẏ = f (t, x). (3-49) (3-9) x = (3-49). 3.1 (3-49) ẋ = y + ky 3 +, ẏ = f x (t, )x f xx(t, )x f xxx(t, )x 3 +. k = 1 2. (3-5) 3.5 f (t, x) f (t, ) = x f x (t, )x = (3-9) (H 3 ) t R, f xxx (t, ). (H 4 ) t R, f xx (t, ) f xx (t, ), f xx(t, ) dt >. 63

75 3 3.1 l(t) 2π-. (Φ 2 (x )) + l(t) sin x =, (3-51) 3.2 x + l(t)x = (3-51). 3.2 Sitnikov Sitnikov [63,69,96] (Φ 2 (x )) + 2x =, (3-52) (x 2 + r(t) 2 ) 3/2 r(t) 2π (3-52). x + 2 r(t) 3 x = 3.6. [67,68,72,94]. D C A p, p = 1,, F : D C C, F = F(z, z), (1) F() = ; (2) F C p (D, C); (3) D, z F 2 z F 2 = [83] F A 3, λ C, λ = 1, λ ±1, df() S p (R 2 ) R λ : R λ (z, z) = λz. Ψ A N = Ψ 1 F Ψ 64

76 3 (1) λ n 1, n = 1, 2, 3, 4, N(z, z) = λ[z + iβ z 2 z + ]; (2) λ = ±i, N(z, z) = λ[z + iβ z 2 z + γ z 3 + ]; z z 3 β R, γ C. 3.7 [83] F A z = Lyapunov (1) λ n 1, n = 1, 2, 3, 4, β ; (2) λ = ±i, β > γ. (3) λ = ±i, β < γ z = Lyapunov. 3.7 [29,83] 3.6 F Taylor F(z, z) = λ z + F 2 (z, z) + F 3 (z, z) +, λ = e iθ, F 2 (z, z) = A z 2 + B z z + C z 2, λ n 1, n = 1, 2, 3, 4, F 3 (z, z) = M z 3 + N z 2 z + P z z 2 + Q z 3. λ = ±i, β = I( λ N) + 3 sin θ sin 3 θ 1 cos θ A cos 3 θ C 2. (3-53) β = I(i N) 3 A 2 ± C 2, γ = Q 2 Ā C. (3-54) 3.7 (3-2) φ(t + T) = λφ(t), ψ(t + T) = λψ(t), t R, (3-55) 65

77 3 λ S 1 M. P(z, z) (3-1) Poincaré P P(z, z) = λz + P 2 (z, z) + P 3 (z, z) +, λ R- P 2, P (x(t, z, z), y(t, z, z)) T (3-1) x() = q, y() = p, z = q + ip. x(t, z, z) = φ(t)z + φ(t) z 2 + O( z 2 ), z, (3-56) y(t, z, z) = ψ(t)z + ψ(t) z 2 t [, T]. 3.8 X(t) Ż = A(t)Z + F(t). Z(t ) = Z(t) = X(t) t + O( z 2 ), z, (3-57) X 1 (s)f(s)ds. (3-42) (x(t), y(t)) T (3-42) x() = y() = x(t) = y(t) = t t G 1 (t, s) f (s)ds + G 3 (t, s) f (s)ds + t t G 2 (t, s)g(s)ds, G 4 (t, s)g(s)ds, t [, T], t [, T], G 1 (t, s) = φ 1 (t)ψ 2 (s) φ 2 (t)ψ 1 (s), G 2 (t, s) = φ 1 (t)φ 2 (s) φ 2 (t)φ 1 (s), 66

78 3 G 3 (t, s) = ψ 1 (t)ψ 2 (s) ψ 2 (t)ψ 1 (s), G 2 (t, s) = ψ 1 (t)φ 2 (s) ψ 2 (t)φ 1 (s). (3-2) (3-55) x(t) + iy(t) = iλ [ f (t)ψ(t) + g(t)φ(t)]dt. 3.8 (3-2) (3-55) P 2 (z, z) = Az 2 + Bz z + C z 2, P 3 (z, z) = Mz 3 + Nz 2 z + Pz z 2 + Q z 3. A = iλ 4 C = iλ 4 [b 1 (t)φ(t) φ 2 (t) + a 1 (t)ψ(t) ψ 2 (t)]dt, [b 1 (t)φ 3 (t) + a 1 (t)ψ 3 (t)]dt, N = 3iλ [b 2 (t) φ(t) 4 + a 2 (t) ψ(t) 4 ]dt 8 iλ G 1 (t, s)a 1 (s)b 1 (t)[2 φ(t) 2 ψ(s) 2 + φ 2 (t) ψ 2 (s)]dsdt 4 T iλ G 2 (t, s)b 1 (t)b 1 (s)[2 φ(t) 2 φ(s) 2 + φ 2 (t) φ 2 (s)]dsdt 4 T iλ G 3 (t, s)a 1 (t)a 1 (s)[2 ψ(t) 2 ψ(s) 2 + ψ 2 (t) ψ 2 (s)]dsdt 4 T iλ G 4 (t, s)a 1 (t)b 1 (s)[2 ψ(t) 2 φ(s) 2 + ψ 2 (t) φ 2 (s)]dsdt, 4 T Q = iλ 8 iλ 4 iλ 4 iλ 4 iλ 4 [b 2 (t)φ 4 (t) + a 2 (t)ψ 4 (t)]dt G 1 (t, s)a 1 (s)b 1 (t)φ 2 (t)ψ 2 (s)dsdt G 2 (t, s)b 1 (t)b 1 (s)φ 2 (t)φ 2 (s)dsdt G 3 (t, s)a 1 (t)a 1 (s)ψ 2 (t)ψ 2 (s)dsdt G 4 (t, s)a 1 (t)b 1 (s)ψ 2 (t)φ 2 (s)dsdt, T T = {(t, s) R 2 : < s < t, < t < T}. 67

79 3 (3-1) (3-42) f (t) = a 1 (t)y 2 (t) + a 2 (t)y 3 (t) + r 1 (t, y(t)), (3-58) g(t) = b 1 (t)x 2 (t) + b 2 (t)x 3 (t) + r 2 (t, x(t)). (3-59) x(t, z, z) = φ(t)z + φ(t) z 2 + t G 1 (t, s) f (s)ds + t G 2 (t, s)g(s)ds, (3-6) y(t, z, z) = ψ(t)z + ψ(t) z 2 + t G 3 (t, s) f (s)ds + (3-56) (3-57) t [, T] x(t, z, z) = φ(t)z + φ(t) z t y(t, z, z) = ψ(t)z + ψ(t) z t t t G 1 (t, s)a 1 (s)( ψ(t)z + ψ(t) z ) 2 ds 2 G 4 (t, s)g(s)ds. (3-61) G 2 (t, s)b 1 (s)( φ(t)z + φ(t) z ) 2 ds + O( z 3 ), z, (3-62) 2 t G 3 (t, s)a 1 (s)( ψ(t)z + ψ(t) z ) 2 ds 2 G 4 (t, s)b 1 (s)( φ(t)z + φ(t) z ) 2 ds + O( z 3 ), z. (3-63) 2 (3-55) 3.9 P(z, z) = λz iλ f, g (3-58) (3-59). g(t)φ(t)dt iλ (3-64) (3-62) (3-63) f (t)ψ(t)dt, (3-64) P 2 (z, z) = iλ P 3 (z, z) = iλ b 1 (t)φ(t)( φ(t)z + φ(t) z ) 2 dt iλ 2 b 2 (t)φ(t)( φ(t)z + φ(t) z T ) 3 dt iλ 2 a 1 (t)ψ(t)( ψ(t)z + ψ(t) z ) 2 dt, 2 a 2 (t)ψ(t)( ψ(t)z + ψ(t) z ) 3 dt 2 68

80 3 2iλ 2iλ 2iλ 2iλ φ(t)b 1 (t) φ(t)z + φ(t) z 2 φ(t)b 1 (t) φ(t)z + φ(t) z 2 ψ(t)a 1 (t) ψ(t)z + ψ(t) z 2 ψ(t)a 1 (t) ψ(t)z + ψ(t) z 2 t t t t. G 1 (t, s)a 1 (s)( ψ(t)z + ψ(t) z ) 2 dsdt 2 G 2 (t, s)b 1 (s)( φ(t)z + φ(t) z ) 2 dsdt 2 G 3 (t, s)a 1 (s)( ψ(t)z + ψ(t) z ) 2 dsdt 2 G 4 (t, s)b 1 (s)( φ(t)z + φ(t) z ) 2 dsdt ẋ = a ε (t)y + a 1ε (t)y 2 + a 2ε (t)y 3 +, ẏ = b ε (t)x b 1ε (t)x 2 b 2ε (t)x 3, (3-65) ε a ε, b ε, a iε, b iε, i = 1, 2 ε ε =. ε = (3-65) ε (3-65) β = β(ε) ε. ε = a 1 =. 3.4 (3-2) (3-55) a 1 (t). A = iλ b 1 (t)φ(t) φ 2 (t)dt, C = iλ b 1 (t)φ 3 (t)dt, 4 4 N = 3iλ [b 2 (t) φ(t) 4 + a 2 (t) ψ(t) 4 ]dt 8 iλ G 2 (t, s)b 1 (t)b 1 (s)[2 φ(t) 2 φ(s) 2 + φ 2 (t) φ 2 (s)]dsdt, 4 T Q = iλ 8 iλ 4 [b 2 (t)φ 4 (t) + a 2 (t)ψ 4 (t)]dt T G 2 (t, s)b 1 (t)b 1 (s)φ 2 (t)φ 2 (s)dsdt. 69

81 (H 5 ) (3-5) T; (H 6 ) t R, a 2 (t), b 2 (t) ; (H 7 ) t R, b 1 (t) b 1 (t) b 1(t) dt > ; (H 8 ) (3-2) Floquet exp(±iθ) < θ arccos( 1 4 ) 2π 3 < θ < π. (3-66) (3-11). θ π 2 β. β = Im( λn) + 3 sin θ sin 3θ 1 cos θ A cos 3θ C 2 = 3 b 2 (t) φ(t) 4 dt 3 a 2 (t) ψ(t) 4 dt + 3 sin θ cos θ A G 2 (t, s)b 1 (t)b 1 (s)[2 φ(t) 2 φ(s) 2 + Re(φ 2 (t) φ 2 (s))]dsdt 4 T = β a 2 (t) ψ(t) 4 dt. [83] 3.2 β 1 a 2 (t) a 2(t) ψ(t) 4 dt. β >. θ = π 2. sin 3θ 1 cos 3θ C 2 >. Q 1 [b 2 (t) φ(t) 4 + a 2 (t) ψ(t) 4 ]dt 8 1 G 2 (t, s)b 1 (t)b 1 (s) φ(t) 2 φ(s) 2 dsdt, 4 T 2 φ(t) 2 φ(s) 2 + Re(φ 2 (t) φ 2 (s)) > φ(t) 2 φ(s) 2. Im( λn) > Q. (3-67) 7

82 3, β > Q + 3 A 2 C 2 Q + A 2 + C 2 Q + 2 A C γ, C A. a(t) 1, a 1 (t) Hill (3-15) R- 4. r(t) r(t) β. (3-53) β = Im( λn) + A, C, N 3.4. φ(t) = r(t)e iϕ(t) r(t) ϕ(t) A, C, A = iλ 4 3 sin θ sin 3θ 1 cos θ A cos 3θ C 2, (3-68) b 1 (t)r 3 (t)e iϕ(t) dt, C = iλ 4 b 1 (t)r 3 (t)e 3iϕ(t). Im( λn) = 3 [b 2 (t) φ(t) 4 + a 2 (t) ψ(t) 4 ]dt 8 1 G 2 (t, s)b 1 (t)b 1 (s)[2 φ(t) 2 φ(s) 2 + Re(φ 2 (t) φ 2 (s))]dsdt 4 T = 3 [b 2 (t) r(t) 4 + a 2 (t)c r(t) 4 ]dt 8 + b 1 (t)b 1 (s)r 3 (t)r 3 (s)χ 1 ( ϕ(t) ϕ(s) )dtds, [,2π] 2 χ 1 (x) = 1 8 (2 + cos 2x) sin x = 3 sin x 2 sin3 x, x [, θ], (3-69) 8 c. β β = 3 [b 2 (t)r 4 (t) + ca 2 (t) r 4 (t)dt]dt 8 [,2π] + b 1 (t)b 1 (s)r 3 (t)r 3 (s)χ 1 ( ϕ(t) ϕ(s) )dtds [,2π] cot θ 2 b(t)r 2 3 (t)e iϕ(t) dt + 1 3θ cot 16 2 [,2π] [,2π] [119] β. b(t)r 3 (t)e 3iϕ(t) dt 2. 71

83 3 3.1 (3-5) R- 4 (3-11) β β = 3 b 2 (t)r 4 (t)dt 3 ca 2 (t) r 4 (t)dt 8 [,2π] 8 [,2π] + b 1 (t)b 1 (s)r 3 (t)r 3 (s)χ 2 ( ϕ(t) ϕ(s) )dtds, [,2π] 2 c χ 2 ( ) χ 2 (x) = 3 cos(x θ/2) sin(θ/2) 16 cos 3(x θ/2), x [, θ]. sin(3θ/2) 3.5 (3-5) R- 3.2 t R α > (3-26) (3-5) R-, ẍ + b (s)x =. (3-7) (3-26) (3-11) ẋ = y + a 2 (s)y 3 +, ẏ = b (s)x b 1 (s)x 2 b 2 (s)x 3, (3-71) a (s) = a(t + s/α), b (s) = α 2 b(t + s/α), b 1 (s) = α 2 b 1 (t + s/α), b 2 (s) = α 2 b 2 (t + s/α) s T = αt. β (3-71), sign β = sign β. R (3-2) 4 (H 6 ) (H 7 ) (3-11). 72

84 T ẍ + a(t)x = f (t, x) + e(t) (4-1) ẍ + a(t)x = f (t, x) + e(t) (4-2) a(t), e(t) C(R/TZ, R N ), f (t, x) C((R/TZ) R N \{}, R N ). x(t) = (x 1 (t),, x N (t)) C 2 (R/TZ, R N ) (4-1)( (4-2)) T- x (4-1)( (4-2)) t i = 1, 2,, N x i (t) >. x C(R/TZ, R N ) (4-1)( (4-2)) t x(t) x (4-1)( (4-2)) T-.. i = 1, 2,, N, lim f i(t, x) = +, x + t. (4-1) (4-2).. [1,1,35,38,91,93,125]. 73

85 4. [4,97,99,1]. Poincaré [89] Gordon [4]. [4]. ẍ + x V(t, x) = f (t), (4-3) V(t, x) = 1 x α, α 2.. Lazer Solimini [49]. Schauder [9,13,17,3,38,44,57,14]. [21,36,41,87,88,11,112,115,116]. [49]. [24,37,9,17]. [45,17] Leray- Schauder Schauder [17].. [117] Leray-Schauder 74

86 4 [19,22,23,25,27] Leray-Schauder. 4.4 Schauder ẍ + a 1 (t)x = (x 2 + y 2 ) α + µ (x 2 + y 2 ) β + e 1 (t), (4-4) ÿ + a 2 (t)y = (x 2 + y 2 ) α + µ (x 2 + y 2 ) β + e 2 (t), a 1, a 2, e 1, e 2 C[, T] α, β > µ R. e 1, e 2 [21,37,45,57].. R N + = {x R N : x i, i = 1, 2,, N} x = max i x i. x = (x 1,, x N ), y = (y 1,, y N ), x y x y = (x 1 y 1,, x N y N ) R N +. ϕ : R N R x y x, y R N ϕ(x) ϕ(y). ψ L 1 [, T] ψ t [, T] ψ(t). p L 1 [, T] p p. p L p p p 1 p + 1 p = a 1, a 2,, a N e 1, e 2,, e N a(t), e(t) C(R/TZ, R N ). i = 1, 2,, N, x() = x(t), x () = x (T) (4-5) x + a i (t)x = e i (t). (4-6) (A) (t, s) [, T] [, T] i = 1, 2,, N (4-6)- (4-5) G i (t, s)

87 4 (4-5) G i (t, s). (B) (t, s) [, T] [, T] i = 1, 2,, N (4-6)- (4-6)-(4-5) ( ) (4-6)- (4-5) x(t) = G i (t, s)e i (s)ds. a i (t) = k 2 (A) < k 2 < λ 1 = ( π T )2 (B) < k 2 λ 1 λ 1 Dirichlet x() = x(t) =. sin k(t s) + sin k(t t + s), s t T, G i (t, s) = 2k(1 cos kt) sin k(s t) + sin k(t s + t), t s T, 2k(1 cos kt) 1 2k cot kt 2 G i(t, s) 1 2k sin kt 2 [38,13]. a(t) [13] L p i = 1, 2,, N a i (t) 1 p a i L p [, T]. (A). (B). (A) a i p < K(2 p), a i p K(2 p),. m i = min s,t T G i(t, s), M i > m i > < σ i < 1. M i = max s,t T G i(t, s), σ i = m i /M i. (4-7) 76

88 4 γ : R R N γ i (t) = G i (t, s)e i (s)ds, i = 1, 2,, N, γ = min γ i (t), i,t γ = max γ i (t). i,t γ ẍ + a(t)x = e(t) T Leray-Schauder [19,28]. 4.2 [78] X K X Ω K T : Ω K Ω (I) T Ω. (II) x Ω < λ < 1 x = λt x. X = C[, T] C[, T] (N ) C[, T]. T : X X T x = (T 1 x, T 2 x,, T N x) T, (4-8) (T i x)(t) = T (4-1). 4.1 a(t) (A) G i (t, s) f i (s, x(s) + γ(s))ds, i = 1, 2,, N. (4-9) 77

89 4 (A 1 ) L > φ L t [, T] x [ L, L] f f i f i (t, x) φ L (t) (A 2 ) f f i g i (x) h i (x) k i (t) f i (t, x) k i (t){g i (x) + h i (x)}, (t, x) [, T] R N +\{}, g i (x) > h i (x)/g i (x) (A 3 ) r > i = 1, 2,, N g i (γ,, γ, σ i r + γ, γ,, γ ) r { 1 + h i(r + γ,, r + γ } > Ki ). g i (r + γ,, r + γ ) K i (t) = G i(t, s)k i (s)ds. γ (4-1) T- x t x(t) > γ(t) < x γ < r. ẍ + a(t)x = f (t, x(t) + γ(t)) (4-1) T- x t [, T] x(t) + γ(t) > < x < r. u(t) = x(t) + γ(t) (4-1) T- < u γ < r ü + a(t)u = ẍ + γ + a(t)x + a(t)γ = f (t, x + γ) + e(t) = f (t, u) + e(t). (A 3 ) n {1, 2, } 1 n < σr + γ { Ki g i(γ,, γ, σ i r + γ, γ,, γ ) 1 + h i(r + γ,, r + γ } ) g i (r + γ,, r + γ + 1 < r ) n i = 1, 2,, N σ = min{σ 1, σ 2,, σ N }. N = {n, n + 1, } n N. ẍ + a(t)x = λ f n (t, x(t) + γ(t)) + a(t) n, (4-11) λ [, 1] i = 1, 2,, N fi n f i (t, x), if x i 1 n (t, x) =, f i (t, x 1,, x i 1, 1 n, x i+1,, x N ), if x i 1 n. 78

90 4 (4-11) i = 1, 2,, N x i (t) = λ G i (t, s) f n i (s, x(s) + γ(s))ds + 1 n = λ(t n i x)(t) + 1 n. (4-12) λ [, 1] (4-12) x x r. λ [, 1] (4-12) x x = r. j {1, 2,, N} x j = r. t x j (t) 1 n = λ G j (t, s) f j n (s, x(s) + γ(s))ds λm j = σ j M j λ σ j max t f j n f j n { λ = σ j x j 1 n. (s, x(s) + γ(s))ds (s, x(s) + γ(s))ds G j (t, s) f n j (s, x(s) + γ(s))ds } x j (t) σ j x j 1 n + 1 n σ j( x j 1 n ) + 1 n σ jr. 1 n 1 n < σr + γ (A 2 ) t x j (t) = λ = λ x j (t) + γ j (t) σ j r + γ > 1 n. G j (t, s) f n j (s, x(s) + γ(s))ds + 1 n G j (t, s) f j (s, x(s) + γ(s))ds + 1 n G j (t, s) f j (s, x(s) + γ(s))ds + 1 n G j (t, s)k j (s)g j (x(s) + γ(s)) g j (γ,, γ, σ j r + γ, γ,, γ ) { 1 + h j(x(s) + γ(s)) g j (x(s) + γ(s)) } ds + 1 n { 1 + h j(r + γ,, r + γ ) g j (r + γ,, r + γ ) } K j + 1 n, 79

91 4 i {1,, N} \ { j} x i (t) 1 n γ. r = x j g j (γ,, γ, σ j r + γ, γ,, γ ) n x r. 4.2 { 1 + h j(r + γ,, r + γ } ) g j (r + γ,, r + γ K j ) + 1, n x(t) = (T n x)(t) + 1 n (4-13) B r = {x X : x < r} x n ẍ + a(t)x = f n (t, x(t) + γ(t)) + a(t) n (4-14) T- x n x n < r. i = 1,, N t [, T] x n i (t) 1 n > xn (4-14) T-. x n (t) + γ(t) n N δ > n N { min x n i (t) + γ i (t) } δ. (4-15) i, t (A 1 ) φ r+γ (t) f f i : t x r + γ x r+γ (t) f i (t, x) φ r+γ (t). ẍ + a(t)x = Φ(t) T- Φ(t) = (φ r+γ (t),, φ r+γ (t)) T i = 1,, N x r+γ i (t) + γ i (t) = G i (t, s)φ r+γ (s)ds + γ i (t) Φ + γ >, Φ = min Φ i (t), Φ i (t) = G i (t, s)φ r+γ (s)ds. i, t 8

92 4 δ = Φ + γ >, (4-15). i = 1,, N, x n i (t) + γ i(t) r + γ x n i (t) + γ 1 n, xi n (t) + γ i(t) = = G i (t, s) f n i (s, xn (s) + γ(s))ds + γ i (t) + 1 n G i (t, s)φ r+γ (s)ds + γ i (t) G i (t, s)φ r+γ (s)ds + γ i (t) Φ + γ = δ. (4-14) x n (4-1) : H > n n ẋ n H. (4-16) t [, T] ẋ n (t ) =. (4-14) T a(t)x n (t)dt = i = 1,, N, ẋi n = max t ẍ t i n (s)ds t t = max t = 2 t [ f n (t, x n (t) + γ(t)) + a(t) ] dt. n [ fi n (s, x n(s) + γ(s)) + a i(s) n [ fi n (s, x n(s) + γ(s)) + a i(s) n a i (s)x n i (s)ds < 2r a i 1 = H i, ] ds + ] a i(s)xi n (s) ds a i (s)x n i (s)ds a i 1 = max i a i(s)ds. H = max i {H i }, (4-16). x n < r (4-16) i = 1, 2,, N, { } xi n [, T] n N. Arzela Ascoli { } xi n { } x n k n N i [, T] x i C[, T]. x = (x 1,, x N ), x n < r (4-15) x t i = 1,, N δ x i (t) + γ i (t) r + γ. k N 81

93 4 x n k i k x n k i (t) = x i (t) = G i (t, s) f i (s, x n k (s) + γ(s))ds + 1 n k, i = 1,, N. G i (t, s) f i (s, x(s) + γ(s))ds, i = 1,, N, f i (t, x) [, T] [δ, r + γ ]. x (4-1) < x r. (A 3 ) x < r. 4.1 a 1 (t), a 2 (t) (A) α > β e 1 (t), e 2 (t) C(R/TZ, R) γ (i) β < 1, µ > (4-4) T-. (ii) β 1 µ 1, < µ < µ 1 (4-4) T φ L (t) = ( 2L) α (A 1 ). g 1 (x, y) = g 2 (x, y) = (x 2 + y 2 ) α, h 1 (x, y) = h 2 (x, y) = µ (x 2 + y 2 ) β k 1 (t) = k 2 (t) = 1, (A 2 ). ω 1 (t) = G 1 (t, s)ds, ω 2 (t) = (A 3 ) : r > G 2 (t, s)ds. µ < r[(σ ir + γ ) 2 + γ 2 ] α/2 ω i [ 2(r + γ )] α+β, i = 1, 2. < µ < µ 1 := min sup r[(σ i r + γ ) 2 + γ ] 2 α/2 ω i i=1,2 r> [, 2(r + γ )] α+β (4-4) T-. β < 1 µ 1 = β 1 µ 1 <.. 82

94 α >. e [37,38,45]. 4.2 a(t) (A) α >, β < 1. b, ˆb f f i (F) ˆb(t) x α f i(t, x) b(t) x α + b(t) x β, t. γ, (4-1) T φ L (t) = ˆb(t) L α, k i(t) = b(t), g i (x) = 1 x α, h i(x) = x β. (A 1 ) (A 2 ) (A 3 ) : r > r(σ i r + γ ) α 1 + (r + γ ) α+β > β i, i = 1,, N (4-17) β i (t) = G i (t, s)b(s)ds. α >, β < 1 γ, r > (4-17) a(t) = k 2 k = λ 1 λ Schauder a i (t), i = 1,..., N (A 1 )-(A 2 )-(A 3 ). γ (4-2) T- x x(t) > γ(t) < x γ < r. 83

95 4 4.4 Schauder (4-1). Schauder. 4.3 X Banach Ω X T : Ω Ω T Ω. 4.3 a(t) (B) f (t, x) (A 1 )-(A 2 ). (G 1 ) R > R > Φ, Φ + γ > i = 1,, N, { R g i (Φ + γ,, Φ + γ ) 1 + h i(r + γ,, R + γ } ) g i (R + γ,, R + γ Ki ), Φ = min Φ i (t), Φ i (t) = G i (t, s)φ R+γ (s)ds (4-1) T-. i, t (4-1) (4-1) T : X X T (4-9). T. R (G 1 ) r = Φ >, R > r >. Ω = {x X : r x i (t) R, t, i = 1,, N}. (4-18) Ω T(Ω) Ω. x Ω i = 1,, N G i (t, s) (A 1 ) (T i x)(t) (A 2 ) (G 1 ) (T i x)(t) G i (t, s)k i (s)g i (x(s) + γ(s)) g i (Φ + γ,, Φ + γ ) G i (t, s)φ R+γ (s)ds Φ = r >. { 1 + h } i(x(s) + γ(s)) g i (x(s) + γ(s)) { 1 + h i(r + γ,, R + γ ) g i (R + γ,, R + γ ) T(Ω) Ω. Schauder. ds } Ki R. 84