Swift-Hohenberg

2014 E-mail: sakamoto@meiji.ac.jp 1 1 1.1......................... 2 1.2.............................. 6 1.3 R.......... 12 2 18 2.1........................ 20 2.2.................... 23 3 24 3.1.............. 24 3.2............................. 33 3.2.1.................... 33 3.2.2 Swift-Hohenberg............... 36 1

4 38 4.1 Swift-Hohenberg................... 38 4.2....................... 43 4.2.1.................... 46 4.2.2 1............ 49 5 53 5.1....................... 54 5.2....................... 59 5.3 Hopf........................... 63 5.3.1......................... 63 5.3.2 Hopf.............. 67 6 68 6.1....................... 68 6.2............................. 73 6.2.1................. 73 6.2.2................ 75 7 77 8 83 8.1.................. 83 8.2....................... 86

-Lecture notes for Ouyousugaku Benkyoukai 2014- Takashi Okuda Sakamoto (Meiji University) 1 [1, 3, 5, 6, 8, 9, 18, 19, 20] [5] 2nd ed. [9] 1st ed. [6] [18] 4.1 Swift-Hohenberg [18] 4.2 [16, 17] 1

1.1 a, b, c x ax 2 + bx + c = 0 a 0. (1.1) (1.1) D = b 2 4ac D = 0 D > 0 a = 1, b = 0 D = 4c c 2

1: a=1,b=0 (1.1) x = ± c {(x, c) ; c = x 2 } x 2 + c = 0 c = 0 dx dt = x2 + c, x(t) R, c R (1.2) t (1.2) t dx dt = 0 x 2 + c = 0 3

(1.2) x(t) ± c. (1.2) ± c t t 0 (± c) 2 + c = c + c = 0 (1.2) ± c c < 0 x y(t) x (1.2) dy dt y(t) = x(t) x, y(t) 1 = (y + x ) 2 + c = 2x y + y 2 + x 2 + c = ±2 c y + y 2 x x 2 + c = 0 dx /dt 0 y(t) y 2 dy dt = ±2 c y. 4

x = c y(t) = y(0)e 2t c ; x = c y(t) = y(0)e 2t c ; y(0) x(t) c x(t) c (1.2) 5

2: (1.2) 1.2 ẋ dx/dt [3], Theorem 1.0.1) U R n M f : U R n C 1 - x 0 U T > 0 ( T, T ) R x(t) ẋ = f(x) x(0) = x 0 6

x(t) ẋ = f(x) f f K x, y f(y) f(x) K x y, x, y U O δ M x a < δ f(x) < M g(x) f(x) = O(g(x)) as x a o lim f(x) x a g(x) = 0 f(x) = o(g(x)) as x a 7

dx dt = f(x), x Rn, (1.3) (Flow) I R (1.3) x(0) = x 0 (1.3) φ(t, x 0 ) φ : (t, x 0 ) I R n x(t, x 0 ) R n (1.3) I R (1.3) x 0 R n x(t, t 0, x 0 ) x(t 0 ) = x 0 (1.3) t = t 0 x 0 (1.3) {x R n ; x = x(t, t 0, x 0 ), t I} (1.3) f(x ) = 0 8

x R n x(t) (1.3) x(t) ε > 0 δ (1.3) x(t 0 ) y(t 0 ) < δ t > t 0 t R x(t) y(t) < ε x(t) x(t) γ x(t 0 ) y(t 0 ) < γ lim x(t) y(t) = 0 t 9

(1.3) x x f Df(x ) 0 t R ẋ = f(x; α), x R n, α R m (1.4) x = x ẇ = g(w; β), w R n, β R m (1.5) w = w (1.4) x = x (1.5) w = w x U h α (x ) = w h α : R n R n p : R m R m 10

U (1.4) V = h α (U) β = p(α) (1.5) (1.4) φ(t, x(0); α), (1.5) ψ(t, w(0); β) h α, p τ : R R t R τ (t) > 0 U (1.4) φ(t, x(0); α) ( ) ( h α φ(t, x(0); α) = ψ τ(t), hα (w(0)); p(α) ) - x u = Df(x )u [20] 5 [15] - 0 ẋ = f(x; λ), x R n, λ R p (1.6) (1.6) φ(t, x 0 ; λ) 11

x λ λ (1.6) x λ λ = λ 0 λ λ 0 λ x λ (1.3) φ(t, x 0 ; λ), λ λ 0 1 λ = λ 0 x λ (1.3) φ(t, x 0 ; λ 0 ) (1.6) x λ λ = λ 0 x λ0 f x λ0 Df(x λ0 ) 0 0 R 1.3 R ẋ = f(x, µ), x(t) R, µ R (1.7) (1.7) (1.7) (1.7) µ = 0 x(t) 0 f(0, 0) = 0 12

f (0, 0) = 0 x (transcritical bifurcation) (1.7) f(0, 0) = 0, f x (0, 0) = 0 f µ (0, 0) = 0, 2 f (0, 0) 0, x µ 2 f (0, 0) 0 x2 (x, µ) = (0, 0) (1.7) ẋ = µx ± x 2 (1.8) [9] f (1.8) (x, µ) = (0, 0) x = 0 µ = ±x f (0, 0) = 0 µ (0, 0) (1.8) x = 0 x = 0 (1.7) (1.7) ẋ = xf (x, µ) (1.9) f(x, µ), x 0, x F (x, µ) f (0, µ) x = 0. x 13

F (x, µ) = 0 (x, µ) = (0, 0) x = 0 x F (0, 0) = 0, (0, 0) F x (0, 0) = 2 f (0, 0), x2 2 F x (0, 0) = 3 f (0, 0), x2 F µ (0, 0) = 2 f (0, 0), x µ F (0, 0) 0 µ F (x, µ(x)) = 0 x µ(x) µ(x) µ = 0 0 < dµ dx (0) < F (x, µ(x)) = 0 x 0 = d F F [F (x, µ(x))] = (x, µ(x)) + (x, µ(x))dµ dx x µ dx (x). (x, µ) = (0, 0) F dµ (0, 0) 2 f (0, 0) dx (x) = x = x2 F µ (0, 0) 2. f (0, 0) x µ 14

2 f (0, 0) 0 x µ 2 f (0, 0) 0 x2 f(x, µ) (x, µ) = (0, 0) (1.8) (1.8) - (saddle-node bifurcation) (1.7) f(0, 0) = 0, f x (0, 0) = 0 f µ (0, 0) 0, 2 f (0, 0) 0 x2 (x, µ) = (0, 0) (1.7) ẋ = µ ± x 2 (1.10) (fold bifurcation) [9] f [8] 3.3 (1.10), µ > 0 µ < 0 (x, µ) = (0, 0) x (1.7) 15

(0, 0) f µ 0 µ = µ(x), µ(0) = 0 f(x, µ(x)) = 0 µ (0) = 0 (1.11) x 2 µ (0) 0 (1.12) x2 f(x, µ(x)) = 0 (1.11), (1.12) f x 0 = d dx (x, µ) = (0, 0) f f [f(x, µ(x))] = (x, µ(x)) + (x, µ(x))dµ x µ dx (x). f dµ (0, 0) dx (x) = x. f µ (0, 0) f (0, 0) 0 µ f (0, 0) = 0 x 16

dµ dx (0) = 0 µ(x) x (0, 0) f(x, µ(x)) = 0 x 0 = 2 x [f(x, µ(x))] = 2 f 2 x (x, µ(x)) + 2 f 2 + 2 f (x, µ(x)) µ 2 ( dµ dx (x) ) 2 + f x µ (x, µ(x))dµ dx (x) µ (x, µ(x))d2 µ dx 2 (x). (x, µ) = (0, 0) (dµ/dx) x=0 = 0 2 f f (0, 0) + x2 µ (0, µ 0)d2 (0) = 0. dx2 2 f d 2 µ (0, 0) dx (0) = x2 2 f (0, 0) µ f (0, 0) 0 µ 2 f (0, 0) 0 x2 d 2 µ dx 2 (0) 0 (pitchfork bifurcation) (1.7) f(0, 0) = 0, f x (0, 0) = 0 f µ (0, 0) = 0, 2 f (0, 0) = 0, x2 2 f (0, 0) 0, x µ 3 f (0, 0) 0 x3 (x, µ) = (0, 0) (1.7) ẋ = µx + ςx 3, ς = ±1 (1.13) 17

ς = 1 (subcritical pitchfork bifurcation), ς = 1 (supercritical pitchfork bifurcation), (1.7) (1.13) f x = 0 F f(x, µ) = xf (x, µ) F (0, 0) = 0, F µ 0 F (x, µ) = 0 µ(x) dµ dx (0) = 0, d 2 µ dx 2 (0) 0 F (x, µ(x)) = 0 f f ± 0 2 18

{ ẋ = xy x 5 ẏ = y + x 2 (2.1) (2.1) (0, 0) (0, 0) 1.1 (2.1) x(t), y(t) { ẋ = xy ẏ = y. y y(t) = y(0)e t (2.1) y = 0 lim y(t) = lim t t y(0)e t = 0 ẋ ẋ = x 5 { < 0 x > 0 > 0 x < 0 {(x, y) R 2 ; (x, y) (0, 0)} 19

x (0, 0) (0, 0) (2.1) 2.1 { ẋ = Ax + f(x, y), (x, y) R n R m. (2.2) ẏ = Bx + g(x, y), n n A, m m B, f, g A 0 B f, g C r - r 2 f(0, 0) = 0, D x f(0, 0) = 0, D y f(0, 0) = 0, g(0, 0) = 0, D x g(0, 0) = 0, D y g(0, 0) = 0 R n R m M (2.2) 0 R n R m U (x, y) (x(0), y(0)) M U 20

t [0, T ] u(t) U (2.2) t [0, T ] (x(t), y(t)) M (2.2) W c δ x < δ h(0) = 0, Dh(0) = 0 h(x) W c = {(x, y) R n R m ; y = h(x)} (2.2) [1] [1] [1], Theorem 1 (2.2) (2.2) u = Au + f(u, h(u)), u < δ, u R n (2.3) [1] [1], Theorem 2 (2.2) (x, y) (0, 0) (2.3) u 0. (2.2) (x(t), y(t)) 21

(x(0), y(0)), x(0), y(0) 1 (2.2) (2.3) u(t) t x(t) = u(t) + O(e γt ), y(t) = h(u(t)) + O(e γt ), (γ > 0) [1] h(x) y = h(x) t ẏ = Dh(x)ẋ = Dh(x){ Ax + f(x, h(x)) } ẏ = By + g(x, y) = Bh(x) + g(x, h(x)) Bh(x) + g(x, h(x)) = Dh(x){ Ax + f(x, h(x)) } (2.4) h(x) [1], Theorem 3 C 1 - ϕ : R n R m ϕ(0) = Dϕ(0) = 0 q Dϕ(x){ Ax + f(x, ϕ(x)) } Bϕ(x) + g(x, ϕ(x)) = O( x q ) as x 0 22

h(x) ϕ(x) = O( x q ) [1] (2.4) O( x p ) O( x p ) (2.1) 2.2 (2.1) { ẋ = xy x 5 ẏ = y + x 2 (2.1) h(x) (2.1) (2.4) A = 0, B = 1, f(x, y) = xy x 5, g(x, y) = x 2 h(x) + x 2 = dh dx (x){ xh(x) x5 } (2.5) h(x) = a 2 x 2 +a 3 x 3 + a 2, a 3,... h(0) = h (0) = 0 23

h(x) O(x 2 ) (2.5) h (x) = O(x), xh(x) = O(x 3 ), x 5 = O(x 5 ) h(x) + x 2 = O(x){O(x 3 ) + O(x 5 )}} h(x) = x 2 + O(x 4 ) (2.1) ẋ = x 3 + O(x 5 ) x 0 y = 0 3 3.1 24

Haragus Iooss [6] [6] X, Y X Y L(X, Y ) L L(X,Y ) := sup Lu Y u X =1 X = Y L(X, X) L(X) X Y k C k (X, Y ) F : X Y, F C k (X, Y ) ( ) F C k = max sup D j F (x) L(X j=0,...k j,y ) x X C k (X, Y ) η C η (R, X) { } ( ) C η (R, X) := u C 0 (R, X) ; u Cη = sup e η t u(t) X < t R C η (R, X) Cη F η { F η (R, X) := u C 0 (R, X) ; u Fη 25 } ( ) = sup e ηt u(t) X < t R

L : X Y im L im L := {Lu Y ; u X} Y. L ker L ker L := {u X ; Lu = 0} X. X Y L L(X, Y ) ρ(l) ρ ; ρ := {λ C ; λi L : X Y is bijective}. I L σ(l) σ σ := C \ ρ. X, Y, Z X Y Y Z Z du dt = Lu + N(u) (3.1) L N L L(X, Z) k 2 0 X V N C k (V, Y ) N(0) = 0, DN(0) = 0 D 26

N(0) = 0 u(t) 0 (3.1) L σ σ = σ + σ 0 σ σ + = {λ σ; Re λ > 0}, σ 0 = {λ σ; Re λ = 0}, σ = {λ σ; Re λ < 0}, Re λ λ 2 γ sup Re λ < γ, λ σ γ < inf λ σ + Re λ σ 0 (3.1) L 0 σ 0 Γ {λ ; Re λ < γ} σ 0 Dunford ([7], Section III. 4, [21], 1.3 ) P 0 = 1 2πi Γ (λi L) 1 dλ L(Z, X) 27

P 0 P 2 0 = P 0, P 0 Lu = LP 0 u for all u X dim(im P 0 ) P h P h = I P 0 P h P 2 h = P 0, P h Lu = LP h u for all u X P 0 L(X, Y ) X Y Y Z P h L(X) L(Y ) L(Z) E 0 = im P 0 = ker P h X, Z h = im P h = ker P 0 Z Z Z = E 0 X h X h = P h X X, Y h = P h Y Y L 0, L h L E 0, X h L 0 L(E 0 ), L h L(Z h, X h ) L 0 σ 0 L h σ h = σ + σ (3.1) L 28

3 η [0, γ] f C η (R, Y h ) u h du h dt = L hu h + f(t) u h = K h f C η (R, Z h ) K h L(C η (R, Y h ), C η (R, Z h )) C : [0, γ] R K h L(Cη(R,Y h ),C η(r,z h )) C(η). 2.1 ([6], Theorem 2.9) 1, 2, 3 Ψ C k (E 0, X h ) Ψ(0) = 0, DΨ(0) = 0, 0 X U M 0 = {u 0 + Ψ(u 0 ) ; u 0 E 0 } X (i) M 0 u u(0) M 0 U t [0, T ] u(t) U (3.1) t [0, T ] u(t) M 0 (ii) M 0 t R U (3.1) u t R u(t) U (3.1) u(0) M 0. [6] 29

([6], Collorary 2.12) (3.1) u(t) t I I R u(t) M 0 u = u 0 + Ψ(u 0 ) u 0 Ψ du 0 dt = L 0u 0 + P 0 N(u 0 + Ψ(u 0 )). DΨ(u 0 ){ L 0 u 0 + P 0 N(u 0 + Ψ(u 0 )) } = L h Ψ(u 0 ) + P h N(u 0 + Ψ(u 0 )), for all u 0 E 0. [6] u = u 0 + Ψ(u 0 ) (3.1) P 0, P h 3 Y Z with Y Z 3 3.1 3.1 ω 0, c α [0, 1) ω R, ω ω 0 iω L ρ(l) (iωi L) 1 L(Z) c ω, (3.2) (iωi L) 1 L(Y,X) c. (3.3) ω1 α 30

L L 3.1 [4, 21] [2] [1] 6 A L(X, Z) A D(A) X φ (0, π/2) M 1 a S a,φ = {λ ; φ < arg(λ a) < π, λ a} (3.4) A ρ(a) (λi A) 1 L(Z) M λ a, for all λ S a,φ ( Z {T (t)} t 0 {T (t)} t 0 (i) T (0) = I s, t > 0 T (t)t (s) = T (t + s) (ii) u Z t +0 T (t)u u; 31

(iii) u Z t T (t)u 0 < t < {T (t)} t 0 1 lim (T (t)u u) = Lu, t +0 t u Z L T (t) ( [4], Theorem 1.3.4) A A {e At } t 0 : e At = 1 (λi + A) 1 e λt dλ 2πi Γ Γ λ argλ θ A ρ( A) A A [4] X, Y, Z Z = Y 2.1 [6], Theorem 2.20 X, Y, Z 1, 2, L h (3.2) 2.1 [6] 32

([6], Theorem 3.22) σ + = (3.1) M 0 u(t) U, t > 0 (3.1) u(t) M 0 (3.1) u(t; u(0)) u(0) U u(t; u(0)) U, t > 0 (3.1) ũ M 0 U γ u(t; u(0)) = u(t; ũ) + O(e γ t ) as t. [6] 3.2 3.2.1 u t = 2 u x + u + g(u), 2 u(0, t) = u(π, t). (3.5) u(x, t) (x, t) (0, π) R g g C k (R, R), k 2 g(0) = 0, g (0) = 0 33

X = {u H 2 (0, π) ; u(0) = u(π) = 0}, Y = Z = {u L 2 (0, π) ; u(0) = u(π) = 0} X, Y (= Z) (3.5) u(t, x) 0 Lu := u + u λ j = 1 j 2, sin jx, j N j = 1 λ 0 = 0, λ j < 0, j > 1 σ 0 = {0}, σ = {λ j ; j > 1, j N}, σ + = λ 0 = 0 sin x 0 P 0 u Z sin x [0, π] L 2 P 0 u = 2 ( π ) u sin x dx sin x π 0 (3.2) v Z w (I P 0 )X (iω L)w = v (w = (iω L) 1 v (iω 1)w w = v. w (0, π) (iω 1) π w 2 dx π w w dx = π 0 0 0 34 vw dx

π 0 w w dx = [w w] π 0 π 0 w w dx = w 2 L 2 π (iω 1) w 2 L + 2 w 2 L = vw dx. 2 π ω w 2 L = Im vw dx. 2 ω w 2 L v 2 L 2 w L 2. 0 0 w L 2 1 ω v L 2 w L 2 = (iωi L) 1 v L 2 1 ω v L 2 (iωi L) 1 L(Z) = sup (iω L) 1 v L 2 1 v L 2 =1 ω. (3.5) L L [21] 35

3.2.2 Swift-Hohenberg Swift-Hohenberg (SHE) u t = u xxxx 2u xx (1 ν)u u 3, x R l X = Hper 4 := {u Hloc(R) 4 ; u(x) = u(x + l)}, Y = Z = L 2 per := {u L 2 loc(r) ; u(x) = u(x + l)} u(t, x) u = u m (t)e imk0x, k 0 = 2π/l. m Z u(t, x) R u m (t) = u m (t) u(t, x) 0 u t = L u, Lu = u (4) 2u (1 ν)u λ j = ν (1 j 2 k0) 2 2, e ±ijk0x, j Z (ν, k 0 ) C j := {(ν, k 0 ) R R + ; ν = (1 j 2 k 2 0) 2 }, j Z (ν, k 0 ) C j λ ±j = 0 j Z (ν, k 0 ) = (0, 1 j ) 36

λ ±j = 0, λ m < 0, m Z \ { j, j} L w Hper 4 (iωi L)w = v (iωi L)w = w (4) + 2w + (iω + 1 ν)w w (0, l) w 2 L 2 per + 2 w 2 L 2 per + (iω + 1 ν) w 2 L 2 per w 2 L 2 per + 2 w 2 L 2 per + (iω + 1 ν) w 2 L 2 per l = wv dx. 0 ω w 2 L 2 per = Im ( π 0 ) wv dx w L 2 per v L 2 per (iωi L) 1 L(Z) = sup (iω L) 1 v L 2 per 1 v L 2 =1 ω. per (ν, k 0 ) = (0, 1/ j ) (SHE) 37

4 Swift-Hohenberg 4.1 Swift-Hohenberg Swift-Hohenberg (SHE) u t = u xxxx 2u xx (1 ν)u u 3, x (0, l) l X = Hper 4 := {u Hloc 4 ; u(x) = u(x + l)}, Y = Z = L 2 per := {u L 2 loc ; u(x) = u(x + l)} u(t, x) u = u m (t)e imk0x, k 0 = 2π/l m Z (SHE) u m e imk 0x m Z = m Z + m 1 Z λ m u m e imk 0x u m1 u m2 u m3 e i(m 1+m 2 +m 3 )k 0 x (4.1) m 2 Z m 3 Z u u m (t) = u m (t) λ m = ν (1 m 2 k0) 2 2 38

u(t, x) 0 e ±imk 0x P n P n u = 1 ( l ) ue ink0x dx e ink0x, l 0 n Z (4.1) e ink 0x (0, l) m Z l = m Z m n m Z 0 l l 0 u m e i(m n)nk 0x dx 0 u m e i(m n)nk 0x dx + l λ m u m e i(m n)nk 0x dx = l λ m u m, l m 1 Z m 2 Z m 3 Z 0 l = + = l m 1,m 2,m 3 Z 0 m 1 +m 2 +m 3 n l m 1,m 2,m 3 Z m 1 +m 2 +m 3 =n m 1,m 2,m 3 Z m 1 +m 2 +m 3 =n 0 0 u n dx = l u n, u m1 u m2 u m3 e i(m 1+m 2 +m 3 n)k 0 x u m1 u m2 u m3 e i(m 1+m 2 +m 3 n)k 0 x u m1 u m2 u m3 e i(m 1+m 2 +m 3 n)k 0 x u m1 u m2 u m3 (SHE) F u n = λ n u n m 1,m 2,m 3 Z m 1 +m 2 +m 3 =n u m1 u m2 u m3, n Z (4.2) {u m } m Z X F := {{u m } m Z ; u m = u m, {u m } m Z 2 X F := m Z(1+m 2 ) 4 u m 2 < } 39

X F {u m } m Z P n : L 2 per L 2 per, n Z H 4 per u(t, x) = m Z u m(t)e imk 0x H 4 per : {u m } m Z = { 1 l l 0 } (P m u) e imk0x dx m Z j Z (ν, k 0 ) = (0, 1/ j ) λ ±j = 0, λ m < 0, m Z \ { j, j} λ ±j = 0, (4.3) u j = 0 u j + λu j u m1 u m2 u m3, (4.4) u j = 0 u j + λū j u m = λ m u m m 1,m 2,m 3 Z m 1 +m 2 +m 3 =j m 1,m 2,m 3 Z m 1 +m 2 +m 3 = j m 1,m 2,m 3 Z m 1 +m 2 +m 3 =m u m1 u m2 u m3, (4.5) u m1 u m2 u m3, ( m = j ). (4.6) λ ±j = 0 M 0 X F R s λ j < s ν, k 0 (ν, k 0 )- (0, 1/ j ) U p (ν, k 0 ) U p u j = u j u j m = j u m = h m (u j, u j, λ j ), m Z \ { j, j} (4.7) h m (0, 0, 0) = h m u j (0, 0, 0) = h m u j (0, 0, 0) = h m λ j (0, 0, 0) = 0 40

(4.7) t λ m h m (u j, u j, λ j ) m 1,m 2,m 3 Z m 1 +m 2 +m 3 =m = h u j + h u j + h λj. u j u j λ j u m1 u m2 u m3 {u m } m R, m j XF + λ j = O(δ) h m (u j, u j, λ j ) = O(δ 2 ) (4.4) Σ m 1, m 2, m 3 j m j = j j + j j = j j, j, j (m 1, m 2, m 3 ) = (j, j, j), (j, j, j), ( j, j, j) u j = λ j u j 3u 2 ju j + h.o.t.. u j = λ j u j 3u j u j 2 + h.o.t (4.8) u j (t) R z = u j R, λ j = C ż = Cz 3z 3 (4.9) z(t) 0 C < 0 C > 0 C = 0 z(t) ± C/3 41

C > 0 Swift-Hohenberg u(x) = ±2 λ j /3 cos(2πjx/l) + O( u 4 H ) per 2 λ j > 0 λ j 0 (4.3) (SHE) u(t, x) u(t, x + θ) for all θ R u(t, x) u(t, x + θ) u(x) = ±2 λ j /3 cos(2πjx/l + θ) + O( u 4 H ), per 2 θ [0, 2π) λ k = 0, (ν, k 0 ) = (ν, 2π/L) = (0, 1/ j ) (SHE) (ν, k 0 ) C m n, j Z, n = j C n C j C n λ n = 0, C j λ j = 0 λ n = λ j = 0 Swift-Hohenberg [18] 3.5 42

u 3 h m [18] 3.5 [12] [11] 4.2 - Integro-Reaction-Diffusion system : u t = D 2 u 1 x + au + bv + F (u, v) + s l u(t, x) dx, x (0, l), t > 0, 2 l 0 (IRD) v t = D 2 v 2 + cu + dv + G(u, v), x (0, l), t > 0, x2 (NBC) u x = v x = 0 at x = 0, l D 1, D 2 l a, b, c, d, s (IRD) (A1) F, G F (ξ, η) = O( (ξ, η) 3 ), G(ξ, η) = O( (ξ, η) 3 ) (A2) F (u, v) F ( u, v), G(u, v) G( u, v) (A3) a, c > 0, b, d < 0, a + d < 0, := ad bc > 0 43

(A4) bc d + d < 0 [17] (u(t, x), v(t, x)) 0 {(u, v) H 2 (0, l) H 2 (0, l) ; u, v satisfiy (NBC)} L ( ) ( u D1 u + au + bv + (s/l) l L = u dx ) 0 v D 2 v + cu + dv (u (t, x), v (t, x)) t > 0, x [0, l] (IRD) t > 0, x [0, 2l] (u (t, x), v (t, x)) (u (t, x), v (t, x)) = { (u (t, x), v (t, x)), x [0, l) (u (t, 2l x), v (t, 2l x)), x [l, 2l] (u, v ) x = l (u, v ) x = l l =..., l, 2l, l, 0, 4l,... x R (u per, v per ) u per, v per x R 2l u t = D 2 u 1 x + au + bv + F (u, v) + s 2l u(t, x) dx, x (0, 2l), t > 0, 2 2l (IRD 0 ) v t = D 2 v 2 + cu + dv + G(u, v), x (0, 2l), t > 0, x2 2l (IRD ) (u(t, x), v(t, x)) = (u(t, x), v(t, x)) (4.10) 44

(0, l) (IRD)-(NBC) (IRD ) X = {(u, v) H 2 per H 2 per ; u, v satisfiy (4.10)} Y = Z = {(u, v) L 2 per L 2 per ; u, v satisfiy (4.10)} F, G F (u, v) = F jk u j v k + o( (u, v) 3 X), j+k=3 G(u, v) = G jk u j v k + o (u, v) 3 X), j+k=3 F jk = 1 jk F j!k! u j v (0, 0), G k jk = 1 jk G (0, 0), j + k = 3, j, k N. j!k! u j vk u, v u(t, x) = u m (t)e imk0x, m Z v(t, x) = v m (t)e imk0x, (k 0 = π/l) m Z (IRD )-(PBC) P m : P n (u, v) = 1 2l ( l 0 ) (u, v)e ink0x dx e ink0x, n Z ( ) ( d um um = M m dt v m v m ) + ( Fm G m ), m 0, (4.11) 45

( a + s b ) M m = c d ( a D1 m 2 k0 2 b ) (m = 0), (4.12) c d D 2 m 2 k 2 0 (m 0), F m = G m = m 1 +m 2 +m 3 =m m 1,m 2,m 3 Z m 1 +m 2 +m 3 =m m 1,m 2,m 3 Z (F 30 u m1 u m2 u m3 + F 21 u m1 u m2 v m3 +F 12 u m1 v m2 v m3 + F 03 v m1 v m2 v m3 ), (G 30 u m1 u m2 u m3 + G 21 u m1 u m2 v m3 +G 12 u m1 v m2 v m3 + G 03 v m1 v m2 v m3 ) (4.11) X F := X F X F X F := {{α m } m Z ; α m = α m R, {α m } m Z 2 X F := m Z(1+m 2 ) 2 α m 2 < } {(u m, v m )} m Z X F (u, v) X P m { 1 l } {(u m, v m )} m Z = (P m (u, v)) e imk0x dx 2l 0. m Z 4.2.1 (4.11) (IRD )-(PBC) {λ m } m Z {M m } m Z 0 det M m = 0 46

D 2, k 0, s det M 0 = det M 1 = det M 2 = 0 ([16], [17]) L [ k 0 = k0 1 { := 5 } ] 1/2 25 8dD 2 16ad, 1 D 2 = {dd 1(k 0) 2 } (k 0) 2 {D 1 (k 0) 2 a}, s = s := /d, 0 (3.2) (3.4) v = (u, v) X L : X Z ( ) ( u L D1 u + au + bv + (s/(2l)) 2l = u dx ) 0 v D 2 v + cu + dv T (t)v := e L t v = m Z e M mt P m v, v X M m, (m Z) 6 ( m = 0, ±1, ±2) 0 { 1 l } (P m (u, v)) e imk0x dx 2l 0 47 m Z = {(u m, v m )} m Z X F

e Mmt P m v P m v <. m Z m Z lim t +0 ( ) e L t v v /t = m Z lim t +0 = m Z M m P m v = L v ( e M m t P m v P m v ) /t L 0 (k 0, D 2, s) = (k0, D2, s ) M 0, M 1, M 2 ( ) ( ) d bc/d d + D2 m 2 k0 2 a D 1 m 2 k0 2 T 0 =, T m =, m = 1, 2, c c c c ( ũ m ) ( 0 0 ) ( ũm ) ( fm ) ṽ m = 0 µ m ṽ m + T 1 m g m, m = 0, 1, 2. µ 0 := d + bc/d, µ m := (a + d) m 2 (D 1 + D 1,2 2 )(k 1,2 0 ) 2, f m := f m t (u mj,v mj )=T m t (ũ mj,ṽ mj ), g m := g m t (u mj,v mj )=T m t (ũ mj,ṽ mj ) ṽ j = h (2) j (ũ 0, ũ 1, ũ 2 ), j = 0, 1, 2, (u m, v m ) = (h (1) m, h (2) m )(ũ 0, ũ 1, ũ 2 ), m N \ {1, 2} 48

h (j) m ũ j (0, 0, 0) = h (j) m (0, 0, 0) = 0, j = 1, 2 h ( j) m (IRD ) h (j) m = O( ũ 0, ũ 1, ũ 2 3 ) ([16], [17]) a, b, c, d, D 1 (k 0, D 2, s) = (k 0, D 2, s ) (IRD) W c loc (IRD) ż 0 = (µ 0 + a 1 z0 2 + a 2 z1 2 + a 3 z2)z 2 0 + a 4 z1z 2 2 + o( (z 0, z 1, z 2 ) 3 ), ż 1 = (µ 1 + b 1 z0 2 + b 2 z1 2 + b 3 z2)z 2 1 + b 4 z 0 z 1 z 2 + o( (z 0, z 1, z 2 ) 3 ), ż 2 = (µ 2 + c 1 z0 2 + c 2 z1 2 + c 3 z2)z 2 2 + c 4 z 0 z1 2 + o( (z 0, z 1, z 2 ) 3 ). (4.13) z j (t) = ũ j (t) = cu j(t) (a D 1 j 2 k 2 0)v j (t) c[j 2 k 2 0(D 1 + D 2 ) (a + d)] R, j = 0, 1, 2 µ j, a j, b j, c j (IRD) [16] 4.2.2 1 (4.13) ż 0 = (µ 0 + a 1 z0 2 + a 2 z1 2 + a 3 z2)z 2 0 + a 4 z1z 2 2, ż 1 = (µ 1 + b 1 z0 2 + b 2 z1 2 + b 3 z2)z 2 1 + b 4 z 0 z 1 z 2, ż 2 = (µ 2 + c 1 z0 2 + c 2 z1 2 + c 3 z2)z 2 2 + c 4 z 0 z1. 2 (4.14) 49

(4.14) (z 0 (t), z 1 (t), z 2 (t)) = (0, ±z 1, 0), z 1 = µ 2 /b 2 e 1 := (0, ±z 1, 0) µ 1 e 1 µ 1 b 2 < 0 e 1 2µ 1 0 0 M e1 := 0 α β, 0 c 4 µ 1 /b 2 γ α = µ 0 a 2 µ 1 /b 2, β = a 4 µ 1 /b 2 and γ = µ 2 c 2 µ 1 /b 2 a 4 c 4 < 0 (µ 0, µ 2 ) {(µ 0, µ 2 ) ; tr M e1 } = {(µ 0, µ 1 ) ; µ 0 + µ 2 (a 2 + c 2 )µ 1 /b 2 = 0} SB = {(µ 0, µ 2 ) ; det M e1 = 0} = {(µ 0, µ 2 ) ; (µ 0 a 2 µ 1 /b 2 )(µ 2 c 2 µ 1 /b 2 ) a 4 c 4 µ 2 1/b 2 2 = 0} (µ 0, µ 2 ) = (µ ± 0, µ ± 2 ) M e1 0 (µ 0, µ 2 ) tr M e1 = det M e1 = 0 µ 0, µ 2 µ j = µ P (±) j, (j = 0, 2) M e1 2µ 1 0 0 T 1 M e1 T = 0 0 1, 0 0 0 50

1 0 0 T = 0 2β 0. 0 α γ 2 (4.14) ±e 1 b 2 < 0 z 1 = z 1 z, z = µ 1 /b 2 ( z 1 (t), z 0 (t), z 2 (t)) z 1 2µ 1 0 0 z 1 N 1 ( z 1, z 0, z 2 ) ż 0 = 0 α β z 0 + N 0 ( z 1, z 0, z 2 ), ż 2 0 c 4 µ 1 /b 2 γ z 2 N 2 ( z 1, z 0, z 2 ) (4.15) N 0 ( z 1, z 0, z 2 ) = 2a 4 z z 1 z 2 + 2a 2 z z 1 z 0 + F 0 ( z 1, z 0, z 2 ), N 1 ( z 1, z 0, z 2 ) = b 1 z z 2 0 + 3b 2 z z 2 1 + b 3 z z 2 2 + b 4 z z 0 z 2 + F 1 ( z 1, z 0, z 2 ), N 2 ( z 1, z 0, z 2 ) = 2c 2 z z 1 z 2 + 2c 4 z z 1 z 0 + F 2 ( z 1, z 0, z 2 ). (z, x, y) z z 1 x = T 1 z 0. y (µ 0, µ 2 ) p 1, p 2, p j 1 (4.15) ż 2µ 1 0 0 z Ñ 1 (z, x, y) ẋ = 0 p 1 1 x + Ñ 0 (z, x, y), (4.16) ẏ 0 0 p 2 y Ñ 2 (z, x, y) 51 z 1

Ñ 1 (z, x, y) = N 1 (z, 2βx, (α γ)x 2y), Ñ 0 (z, x, y) = N 0 (z, 2βx, (α γ)x 2y), Ñ 2 (z, x, y) = N 2 (z, 2βx, (α γ)x 2y). p j < 2µ 1, j = 1, 2, (4.14) M c M c (4.16) ( ẋ where ẏ ) = ( p1 1 0 p 2 ) ( x y ) + j,k N j+k=3 f 30 = 2z (a 2 a 4 α/β)h 20 + 4(a 1 β 2 + a 3 α 2 ), ( fjk x j y k f 21 = 2a 4 z H 20 /β + 2z (a 2 a 4 α/β)h 11 8a 3 α, f 12 = 2a 4 z H 11 /β + 2z (a 2 a 4 α/β)h 02 + 4a 3, f 03 = 2a 4 z H 02 /β, g jk x j y k ), (4.17) g 30 = 2z [(a 2 c 2 )αβ a 4 α 2 + c 4 β 2 ]H 20 /β + 4α[(a 1 c 1 )β 2 + (a 3 c 3 )α 2 ], g 21 = 2z (a 4 α/β + c 2 )H 20 + 2z [α(a 2 c 2 ) a 4 α 2 /β + c 4 β]h 11 +4[α 2 (3c 3 2a 3 ) + β 2 c 1 ], g 12 = 2z [(a 2 c 2 )α a 4 α 2 /β + c 4 β]h 02 + 2z (a 4 α/β + c 2 )H 11 + 4α(a 3 3c 3 ), g 03 = 2z (a 4 α/β c 2 )H 02 + 4c 3, H 20 = 2z µ 1 (β 2 b 1 + b 3 α 2 βb 4 α), H 11 = 2z µ 1 (βb 4 2αb 3 ) H 20 µ 1, H 02 = z [2b µ 2 3 (µ 1 + α) + b 4 β] H 11. 1 2µ 2 1 52

(4.17) (4.16) p 1 = p 2 = µ 1 = 0 0 0 0 0 0 1 0 0 0 (4.16), (z, x, y) (z, x, y) Dumorutier- Kokubu [13] µ 1 b 2 < 0 (4.16) 5 (4.17) [3] 3.3, [6] 3, [9] 19 [8] [14] [18] 3.3 [14] 53

5.1 R n dx = Lx + F (x), dt x(t) R, (5.1) F (x) = O(x 2 ). (5.1) F (x) F (x) = p N p 2 F p [x (p) ] H k R n k H k k H k R n x = (x, y) R 2 H k = H k R n. H 2 = span{x 2, xy, y 2 } H 2 = H 2 R 2 = span{x 2, xy, y 2 } span{ t (1, 0), t (0, 1)} {( ) ( ) ( ) ( ) x 2 xy y 2 0 ( 0 ) ( 0 )} = span 0, 0, 0, x 2, xy, y 2 54

([14], Theorem 2) (5.1) N(z) dz dt = F (z), z Rn D z N(z) L z L N(z) = 0 (5.2) L L L (L ) ad L : P (x) D x P (x) L x L P (x) (5.3) (5.1) F (x) ad L [F ] := D x F (x) L x L F (x) = 0 [14] [14] 2.4 ( 0 1 ) L = 0 0 (5.4) ) ) ) ( ẋ ẏ = ( 0 1 0 0 ) ( x y + ( F1 (x, y) F 2 (x, y), t (x, y) R 2 (5.5) 55

(5.3) ( ) ( ) ( ) ( (F1 ) x (F 1 ) y 0 0 x 0 0 (F 2 ) x (F 2 ) y 1 0 y 1 0 ( ) ( ) ( ) (F1 ) x (F 1 ) y 0 0 = (F 2 ) x (F 2 ) y x F 1 ( ) x(f1 ) y = = 0 x(f 2 ) y F 1 x F 1 y = 0, x F 2 y = F 1 ) ( F1 (x, y) F 2 (x, y) ) F 1 = f(x) F 2 = yf(x)/x+g(x) F 1 (x) = f(x) = xϕ 1 (x) F 2 = yϕ 1 (x) + ϕ 2 (x) ϕ 2 = β(x) + α(x) ( ) ( ) ( F1 (x, y) x 0 = ϕ 1 (x) + F 2 (x, y) y 1 t (F 1, F 2 ) H 2 ) ( 0 β ( x) + x ) α ( x). (F 1, F 2 ) = (ax 2, axy + bx 2 ) t (F 1, F 2 ) H 3 (F 1, F 2 ) = (ax 3, ax 2 y + bx 3 ) t (F 1, F 2 ) H k (F 1, F 2 ) = (ax k, ax k 1 y + bx k ). (5.6) 56

(P 1, P 2 ) = ( ax k, ax k 1 y) H k (5.7) ad L [(P 1, P 2 )] = 0 ad L [(P 1, P 2 ) + (ax k, ax k 1 y + bx k )] = ad L [(0, ax k 1 y + b x k )] = 0. (5.5) ( ) ( ẋ = ẏ y ax k 1 y + bx k ) F (x) F = F (x; µ) [14], Theorem 5 ẋ = Lx + F (x; µ) ż = N(z; µ) N(e L t z; µ) = e L t N(z; µ) F (0; µ) ker L D x F (0; µ) L [14] 57

(5.4): L = ( 0 1 0 0 ) t (0, µ) kerl L L E ( ) 0 F (0; µ) = DF (0; µ) = µ 0 ( µ1 0 0 µ 2 ) ( ) 0 0 + µ 3 0 ( µ1 0 ) L DF (0; µ) = ( µ1 µ 2 0 0 = ( 0 0 µ 3 µ 2 0 µ 1 ) ( ) ( 0 0 µ1 0 + + µ 3 0 0 µ 1 ), (µ 2 = µ 3 µ 1 ) ) (5.4) ( ) ( ẋ 0 = ẏ µ 0 ) ( ) ( ) ( 0 0 x + + p 1 p 2 y (4.17) ) y +O( (x, y) k ) ax k 1 y + bx k 58

5.2 [8] 8.4 (4.17) (u, w) u = x, w = y + p 1 x + f 30 x 3 + f 21 x 2 y + f 12 xy 2 + f 03 y 3, (4.17) u = w, ẇ = p 1 u + p 2 w + σ 1 u 3 + σ 2 u 2 w + (g 12 + 2f 21 )uw 2 +(g 03 + f 12 )w 3 + O( (x, y) 5 ) + ( p 1 + p 2 (x, y) 3 ), (5.8) p 1 = p 1 p 2, p 2 = p 1 + p 2, σ 1 = g 30, σ 2 = (g 21 + 3f 30 ) ɛ, ɛ 1 : σ1 u = ɛ U, w = ɛ 2 σ 1 3/2 W, T = ɛ σ 1 σ 2 t p 1 = ɛ 2 σ2 1 σ 2 2 σ 2 2 P 1, p 2 = ɛ 2 σ 1 P 2. σ 2 u = ɛũ, w = ɛ 2 w, t = ɛt σ 2 p 1 = ɛ 2 p 1, p 2 = ɛ 2 p 2, du dt = W, (5.9) dw dt = P 1 U + ɛ P 2 W + (sign σ 1 )U 3 + ɛ (sign σ 2 ) U 2 W + O(ɛ 2 ). 59

sign σ 2 = +1 t W T T, W W, [17]) µ 1 b 2 µ 1 < 0. (4.17) ( (4.15) M c ) { u = w, ẇ = p 1 u + p 2 w + ςu 3 ɛ u 2 w, (5.10) p 1 = P 1 p 2 = ɛp 2, ɛ, ɛ 1 ς = sign σ 1 = sign g 30. [17] (IRD) (4.14) (4.14) (4.17) (4.17) (5.10) (5.10) 3,4 chapter 4 of [1] [3] 7.3 60

(IRD) 3,4 L1 3: ς > 0 (5.10) {(p 1, p 2 ) ; p 1 < 0, p 2 = 0} {(p 1, p 2 ) ; p 1 = 0} L 1 := {(p 1, p 2 ) ; p 2 = ɛ p 1 /5 + O((ɛp 1 ) 2 )} 61

L2 L3 L2 L3 L4 4: ς < 0 (5.10) 3 L 2 := {(p 1, p 2 ) ; p 2 = ɛ p 1 } L 3 := {(p 1, p 2 ) ; p 2 = 4ɛ p 1 /5+O(p 2 1)} p 1 L 3 L 4 := {(p 1, p 2 ) ; p 2 0.752ɛ p 1 } L 3 L 4 62

5.3 Hopf (5.10) p 2 = 0 (5.10) (0, 0) ± p 1 p 1 < 0 5.3.1 [8] 5.10) Ẋ = AX + F (X), X = t (x, y) R A 2 2 λ = α + iω, ω > 0. λ q = (q 1, q 2 ) C 2 λ A p = (p 1, p 2 ) C 2 λ A t A < p, q >:= p 1 q 1 + p 2 q 2 = 1 z(t) C z =< p, (x, y) >. (x, y) (x, y) = zq + z q. 63

(5.11) ż = λz+ < p, F (zq + z q) >, z C g(z, z) =< p, F (zq + z q) > g kl g g(z, z) = k+l 2 g 1 k!l! g klz k z l (5.11) g 20, g 11, g 02 z z = w + h 20 2 w2 + h 11 w w + h 02 2 w2 w(t) C w = z h 20 2 z2 h 11 z z h 02 2 z2 + O( z 3 ) ẇ = ż h 20 zż h 11 (ż z z z) h 02 z z + = λw + (g 20 λh 20 )w 2 + (g 11 2 λh 11 )w w + (g 02 (2 λ λ)h 02 ) w 2 + O( w 3 ) 2 h 20 = g 20 λ, h 11 = g 11 λ, h 02 = g 02 2 λ λ ż = λz + k+l 3 64 1 k!l! g klz k z l

g kl z = w + h 30 6 w3 + h 21 2 w2 w + h 12 2 w w2 + h 03 6 w3 w = z h 30 6 z3 h 21 2 z2 z h 12 2 z z2 h 03 6 z3 + O( z 4 ) t ẇ = ż h 30 2 z2 ż h 21 2 (2z zż + z2 z) h 12 2 (ż z2 + 2z z z) h 03 2 z2 z = λw + g 30 2λh 30 6 w 3 + g 21 (λ + λ)h 21 w 2 w 2 + g 12 2 λh 12 w w 2 + g 03 + (λ 3 λ)h 03 w 3 + O( w 4 ) 2 6 h 30 = g 30 2λ, h 12 = g 12 2 λ, h 03 = g 03 3 λ λ α = 0 λ + λ = 0 g 21 z 2 z (5.11) ż = λz + c 1 z z (5.12) c 1 (5.11) g kl c 1 = g 20g 11 (2λ + λ) + g 11 2 2 λ 2 λ + g 02 2 2(2λ λ) + g 21 2 65

(5.11) ( d y1 dt y 2 ) = d ( β 1 1 β +d(y 2 1 + y 2 2) ) ( y1 y 2 ( y1 y 2 ) ) sign (Re c 1 ) = 1 ω 2 Re (ig 20g 11 + ωg 21 ) + O( (y 1, y 2 ) 4 ) (5.13) [8] (5.13) y 1 (t) = r(t) cos θ(t), y 2 (t) = r(t) sin θ(t) ṙ = (β + dr 2 )r, θ = 1 r(t) = β/d, θ = t βd < 0 d d > 0 d < 0 Hopf bifurcation) - - Poincaré-Andronov-Hopf bifurcation 66

5.3.2 Hopf (5.10) : { u = w, ẇ = p 1 u + p 2 w + ςu 3 ɛ u 2 w, (5.14) p 2 = 0, p 1 < 0 p1 i 1 t ( i/ p 1, 1), 2 p 1 i t ( i p 1, 1) < t ( i p 1, 1), 1 t ( i/ p 1, 1) >= 1 2 z =< (u, w), ( i p 1, 1) >= i p 1 u + w, (u, w) = ( i 2 (z z), 1 ) (z + z) p 1 2 g(z, z) :=< ( i p 1, 1), (0, ςu 3 ɛ u 2 w) > 67

g := 1 8p 1 [( ) ( ) ] iς iς ɛ z + ɛ z (z z z) 2 p1 p1 z 2 z c 1 := p ( ) 1 iς ɛ 4 p1 p 1 < 0, ɛ > 0 Re c 1 = p 1 ɛ/4 < 0. p 1 = 0 (5.10) (0, 0) 6 (5.10) : { u = w, ẇ = p 1 u + p 2 w + ςu 3 ɛ u 2 w, (6.1) [3] 4.5 [9] 28, [19] 5.7 [1] 4 (5.10) 6.1 ẋ = f(x) + ɛg(x), x R 2 (6.2) g g(x, t) = g(x, t + T ) t (5.10) 68

ɛ = 0 (6.2) H(x) ( ) H f(x) = t (f 1 (x, y), f 2 (x, y)) = t (x, y), H (x, y). y x ε = 0 (6.2) p 0 = (0, 0) q 0 (t); lim t q0 (t) = p 0, lim t + q0 (t) = p 0 Γ 0 Γ 0 := {q 0 (t) ; t R} {p 0 } Γ 0 α 0 Γ 0 q α, α ( 1, 0) α 0 q α q 0 t p 0 t = t 0 x ɛ 0 ɛ = 0 q 0 (t t 0 ) q 0 (0) {(x, y) ; y = 0} (6.2) qε(t, s t 0 ), qε u (t, t 0 ) ɛ ɛ qε(t, s t 0 ) = q 0 (t t 0 ) + εq1(t, s t 0 ), t [t 0, ), qε u (t, t 0 ) = q 0 (t t 0 ) + εq1 u (t, t 0 ), t (, t 0 ] 69

(6.2) f(q 0 + εq s 1) = f(q 0 ) + εdf(q 0 )q s 1 + O(ε 2 ), f(q 0 + εq u 1 ) = f(q 0 ) + εdf(q 0 )q u 1 + O(ε 2 ) q s 1 = Df(q 0 (t t 0 ))q s 1(t, t 0 ) + g(q 0 (t t 0 )), q u 1 = Df(q 0 (t t 0 ))q u 1 (t, t 0 ) + g(q 0 (t t 0 )) (6.3) q s ε qu ε p 0 ɛ 0 (6.2) (ẋ, ẏ)(t 0 ) = f(q 0 (0)) f(q 0 ) (x(t), y(t)) = q 0 (t t 0 ) t = t 0 f (q 0 (0)) V = span{f (q 0 (0))} V q s ε(t 0 ) := q s ε(t 0, t 0 ) q s ε(t 0 ) := q s ε(t 0, t 0 ) d(t 0 ) d(t 0 ) = ε f (q 0 (0)) (q u 1 (t 0 ) q s 1(t 0 )) f (q 0 (0)) = ε f(q0 (0)) (q u 1 (t 0 ) q s 1(t 0 )) f(q 0 (0)) t (a, b), t (c, d) t (a, b) t (c, d) = ad bc M(t 0 ) = f(q 0 (0)) (q u 1 (t 0 ) q s 1(t 0 )) 70

M(t 0 ) ε 0 ε 0 ε < ε 0 ε p 0 (6.2) M(t 0 ) 0 p 0 [3], [9], (t, t 0 ) := f(q 0 (t t 0 )) (q u 1 (t, t 0 ) q s 1(t, t 0 )) = f(q 0 (t t 0 )) q u 1 (t, t 0 ) f(q 0 (t t 0 )) q s 1(t, t 0 ) u (t, t 0 ) := f(q 0 (t t 0 )) q u 1 (t, t 0 ), s (t, t 0 ) := f(q 0 (t t 0 )) q s 1(t, t 0 ) s (t, t 0 ) t d s dt (t, t 0) = Df(q 0 (t t 0 )) q 0 (t t 0 ) q s 1(t, t 0 ) +f(q 0 (t t 0 )) q s 1(t, t 0 ) q 0 (t t 0 ) ε = 0 (6.2) (6.3) q 0 (t t 0 ) = f(q 0 (t t 0 )). d s dt (t, t 0) = Df(q 0 (t t 0 )) f(q 0 (t t 0 )) q1(t, s t 0 ) +f(q 0 (t t 0 )) (Df(q 0 (t t 0 ))q1(t, s t 0 ) + g(q 0 (t t 0 )). 71

Df(q 0 (t t 0 ) f(q 0 (t t 0 )) q1(t, s t 0 ) +f(q 0 (t t 0 )) Df(q 0 (t t 0 ))q1(t, s t 0 ) = tr (Df(q 0 ))f(q 0 (t t 0 )) q1(t, s t 0 ) = tr (Df(q 0 )) s (t, t 0 ) d s dt (t, t 0) = tr (Df(q 0 )) s (t, t 0 ) + f(q 0 (t t 0 )) g((q 0 (t t 0 ), t). t 0 tr (Df(q 0 )) = (f 1 ) x (q 0 ) + (f 1 ) y (q 0 ) = 2 H x y 2 H x y = 0 d s dt (t, t 0) = f(q 0 (t t 0 )) g(q 0 (t t 0 )). d s dt (t, t 0) dt = lim t s (t, t 0 ) s (t 0, t 0 ) = lim t [ f(q 0 (t)) q s 1(t, t 0 ) ] s (t 0, t 0 ) = f(p 0 ) p 0 s (t 0, t 0 ) = 0 p 0 s (t 0, t 0 ) = s (t 0, t 0 ) s (t 0, t 0 ) = f(q 0 (t t 0 )) g(q 0 (t t 0 )) dt. t 0 u (t 0, t 0 ) = t0 f(q 0 (t t 0 )) g(q 0 (t t 0 )) dt. 72

M(t 0 ) = (t 0, t 0 ) = u (t 0, t 0 ) s (t 0, t 0 ) = = f(q 0 (t t 0 )) g(q 0 (t t 0 )) dt f(q 0 (t)) g(q 0 (t)) dt 6.2 6.2.1 (5.10) ς = 1 { u = w, ẇ = p 1 u + p 2 w + ςu 3 ɛ u 2 w, (6.4) ς = 1, (p 2, ɛ) = (0, 0) (5.10) H(u, w) = w2 2 p u 2 1 2 + u4 4 M(t 0 ) M(t 0 ) = = (w, p 1 u u 3 ) (0, p 2 w ɛu 2 w) dt (p 2 w 2 ɛu 2 w 2 ) dt 73

p 2 = ɛ u 2 w 2 dt. w 2 dt (p 2, ɛ) = (0, 0) ( {(u, w) ; H(u, w) = 0} ) w = ± p 1 u 2 u4 2 u > 0 (0, 0) w = p 1 u 2 u4 2 t t = t 0 ( 2p 1, 0) (0, 0) w = p 1 u 2 u4 2 t = t 0 ( 2p 1, 0) t du dt = w 74

M(t 0 ) = 0 p 2 = ɛ = ɛ = ɛ = ɛ t0 t0 2p 1 u 2 w 2 du w 2 du u 2 w du + 0 2p 1 0 2p 1 0 2p 1 0 = ɛ t 0 u 2 w du w du u 2 w du w du + w du t 0 u 2 p 1 u 2 u4 2 du + p 1 u 2 u4 2 du + u 2 = 4 5 ɛ p 1, (p 1 > 0). p 1 u 2 u4 2 du p 1 u 2 u4 2 du 0 2p1 u 2 0 2p1 ( ) p 1 u 2 u4 du 2 ( ) p 1 u 2 u4 du 2 p 2 = 4 5 ɛ p 1, p 1 > 0 M(t 0 ) = 0 6.2.2 ς = +1 (5.10) : { u = w, ẇ = p 1 u + p 2 w + u 3 ɛ u 2 w, 75

ς = 1, (p 2, ɛ) = (0, 0) (5.10) H(u, w) = w2 2 p u 2 1 2 u4 4 (p 2, ɛ) = (0, 0) (5.10) u > 0 ( p 1, 0) H = p 2 1/4 w = 1 2 (u 2 + p 1 ), u > 0 t ( p 1, 0) t = t 0 w ( p 1, 0) H = p 2 1/4 w = 1 2 (u 2 + p 1 ), u > 0 t = t 0 w t + ( p 1, 0) M(t 0 ) M(t 0 ) = = = p 1 (w, p 1 u + u 3 ) (0, p 2 w ɛu 2 w) dt (p 2 w 2 ɛu 2 w 2 ) dt p 1 p 2 w ɛ u 2 wdu p 2 = ɛ p 1 p 1 u 2 (u 2 + p 1 ) du p 1 p 1 (u 2 + p 1 ) du = ɛ 1 5 p 1, (p 1 < 0). 76

M(t 0 ) = 0 ( p 1, 0) ( p 1, 0) ς = 1 [1] 7 [8] ẋ = α + x 2 + O(x 3 ) ẋ = α + x 2 (7.1) 77

ẏ = F (y, α) := α + y 2 + ψ(y, α), (7.2) ψ(y, α) = O(y 3 ). (7.2) (y, α) M M := {(y, α) ; F (y, α) = α + y 2 + ψ(y, α) = 0} F (0, 0) = 0, F (0, 0) = 1 0 α (0, 0) U g(0) = 0, F (y, g(y)) = 0 α = g(y) (y, α) U M = {(y, α) ; α = g(y)}. g(y) y = 0 F (y, g(y)) = 0 y d F F dg (F (y, g(y))) = (y, g(y)) + (y, g(y)) dy y α dy (y) = 0 y = 0 (g(0) = 0 F F (0, 0) = 0, y (0, 0) = 1 α dg dy (0) = 0 78

F (y, g(y)) y d 2 dy (F (y, g(y))) = 2 F 2 y (y, g(y)) + 2 F dg (y, g(y)) 2 y α dy (y) + F α (y, g(y)) d2 g (y) = 0. dy2 y = 0 2 F dg (0, 0) = 2, y2 F (0) = 0, dy d 2 g (0) = 2. dy2 M (y, α) = (0, 0) (0, 0) = 1 α M = {(y, α) ; α = g(y) = y 2 + O(y 3 )} G(y, α) = α y 2 = +O(y 3 ) y = a 1 (± α) + a 2 (± α) 2 + a 3 (± α) 3 + G(y, α) a 1, a 2, a 3,... G(y, α) = 0 y 1 (α), y 2 (α) y 1 (α) := α + O(α), y 2 (α) = α + O(α) (7.1) x 1 (α) = α, x 2 (α) = α α 1 α h α (x) { x α 0 h α (x) = a(α) + b(α)x α < 0 a(α) := (y 1 (α) + y 2 (α))/2, b(α) := (y 1 (α) y 2 (α))/(2 α). 79

h α h α (x j (α)) = y j (α), j = 1, 2 lim h α(x) = x. α 0 h α (7.1) (7.2) h a (7.1) (7.2) f(x, α) f : R 2 R (i) f(0, 0) = 0, (ii) f x (0, 0) = 0, (iii) f xx (0, 0) 0, (iv) f α (0, 0) 0. ẋ = f(x, α), x R, α R (7.3) η = β ± η 2, η R, β R ± f(x, α) = f 0 (α) + f 1 (α)x + f 2 (α)x 2 + O(x 3 ) ξ = x + δ ξ = ẋ = f 0 (α) + f 1 (α)(ξ δ) + f 2 (α)(ξ δ) 2 + 80

(iii) ξ = [f 0 (α) f 1 (α)δ + f 2 (α)δ 2 + O(δ 3 )] +[f 1 (α) 2f 2 (α)δ + O(δ 2 )]ξ +[f 2 (α) + O(δ)]ξ 2 +O(ξ) 3. f 2 (0) = 1 2 f xx(0, 0) 0. F (α, δ) := f 1 (α) 2f 2 (α)δ + O(δ 2 ) F (0, 0) = 0, α 1 α F δ (0, 0) = 2f 2(0) 0 F (α, δ(α)) = 0, δ(0) = 0 g(α, δ) = 0 δ = δ(α) α = 0 F (α, δ(α)) = 0 d F F dδ (F (α, δ(α))) = (α, δ(α)) + (α, δ(α)) dα α δ dα (α) = 0 δ(α) df 1 dα (0) + 2f 2(0) dδ (0) = 0. dα δ(α) = 1 df 1 2f 2 (0) dα (0)α + O(α2 ) δ = δ(α) (f j (α) α ) ξ = [f 0(0)α + O(α 2 )] + [f 2 (0) + O(α)]ξ 2 + O(ξ 3 ) (7.4) 81

= d/dα µ(α) µ = f 0(0)α + O(α 2 ) µ(0) = 0 (iv) µ (0) = f 0(0) = f α (0, 0) 0. G(µ, α) = µ f 0(0)α + O(α 2 ) G(0, 0) = 0, G α (0, 0) = f 0(0) 0 α(0) = 0, G(µ(α), α) = 0 α = α(µ) b(µ) = f 2 (0) + O(α(µ)) (iii) b(0) = f 2 (0) = 1 2 f xx(0, 0) 0. µ (7.4) ξ = µ + b(µ)ξ 2 + O(ξ 3 ) η = b(µ) ξ, β = b(µ) µ η = b(µ) ξ = b(µ) µ + b(µ) b(µ) η2 + O(η 3 ) = βη + b(µ) b(µ) η2 + O(η 3 ). 82

s = sign {b(µ)} µ 1 s = sign {b(µ)} = sign {b(0)} = sign {f xx (0, 0)}. η = βη + sη 2 + O(η 3 ), s = sign {f xx (0, 0)}. (7.3) η = β ± η 2 8 8.1 R 3 x 1 = x 2, x 2 = 0, ẏ = y + g(x 1, x 2 ) (8.1) [1] g : R 2 R C 2 - g(0, 0) = 0, g(x 1, x 2 ) = O(x 2 1 + x 2 2) (8.1) 83

χ : R 2 R C { 1 (x 2 χ(x 1, x 2 ) = 1 + x 2 2 ε 2 ), 0 (x 2 1 + x 2 2 (2ε) 2 ) G(x 1, x 2 ) = χ(x 1, x 2 )g(x 1, x 2 ) ε x 1 = x 2, x 2 = 0, ẏ = y + G(x 1, x 2 ), (8.2) (8.1) (x 1, x 2 ) < ε (8.2) x 1 (t) = z 1 + z 2 t, x 2 = z 2, z j = x j (0) (8.3) h (8.2) y(t) = h(x 1 (t), x 2 (t)) h(x 1 (t), x 2 (t)) d dt h(x 1(t), x 2 (t)) = h(x 1 (t), x 2 (t)) + G(x 1 (t), x 2 (t)) (8.4) h (x 1, x 2 )- e t t h lim h(x 1(t), x 2 (t))e t = 0 t 84

(8.4) h(x 1 (t), x 2 (t)) (8.2) t 0 (8.3) h(z 1, z 2 ) = 0 e s G(z 1 + z 2 s, z 2 )ds (8.5) (8.2) (x 1 (0), x 2 (0), y(0)) = (z 1, z 2, h(z 1, z 2 )) (8.2) y(t) = h(z 1, z 2 )e t + = t t e s t G(z 1 + z 2 s, z 2 )ds 0 e s t G(z 1 + z 2 s, z 2 )ds s s = s t x j (s) = x j ( s + t) = x j ( s), z j = x j (0), j = 1, 2 x j d x 1 d s = x 2, d x 2 d s = 0 (8.6) x 1 ( s) = z 1 + z 2 s, x 2 ( s) = z 2 z j = x j (0) = x j ( t) z 1 = x j ( t) = z 1 + z 2 ( t), z 2 = x 2 ( t) = z 2 85

= = t 0 e s t G(z 1 + z 2 s, z 2 )ds e s G( z 1 + z 2 ( t) + z 2 ( s + t), z 2 )d s 0 e s G( z 1 + z 2 s, z 2 )d s = h( z 1, z 2 ). h(x 1, x 2 ) (8.2) (8.1) g y (8.5) h(x 1, x 2 ) = 0 e s G(x 1 (s), x 2 (s), h(x 1 (s), x 2 (s)))ds (8.7) (8.7) [1] 8.2 γ = 0, ẋ = γ x + f(x, y), ẏ = γy + g(x, y), (γ > 0, 1 γ > 0) y = h(x) ẋ = γ x + f(x, h(y)), 1 γ > 0 γ γ < γ ε x, y < ε 86

[1] X, Y X Y L(X, Y ) L L(X,Y ) := sup Lu Y u X =1 [10], X = Y L(X, X) L(X) X Y k C k (X, Y ) F : X Y, F C k (X, Y ) ( ) F C k = max j=0,...k sup D j F (x) L(X j,y ) x X C k (X, Y ) η F η (R, X) { } ( ) F η (R, X) := u C 0 (R, X) ; u Fη = sup e ηt u(t) X < t R F η (R, X) Fη L : X Y im L im L := {Lu Y ; u X} Y. L ker L ker L := {u X ; Lu = 0} X. 87

X Y L L(X, Y ) ρ(l) ρ ; ρ := {λ C ; λi L : X Y is bijective}. I L σ(l) σ σ := C \ ρ. X, Y, Z X Y Y Z Z du dt = Lu + N(u) (8.8) L N L L(X, Z) k 2 0 X V N C k (V, Y ) N(0) = 0, DN(0) = 0 D 88

N(0) = 0 u(t) 0 (8.8) L σ σ = σ + σ 0 σ σ + = {λ σ; Re λ > 0}, σ 0 = {λ σ; Re λ = 0}, σ = {λ σ; Re λ < 0}, Re λ λ 2 γ, γ sup Re λ < γ, λ σ 0 < inf λ σ + Re λ, sup Re λ < γ λ σ + σ 0, σ + L L(X, Z) L σ 0, σ + Γ {λ ; Re λ < γ} σ 0 Dunford ([7], Section III. 4, [21], 1.3 ) P 0 = 1 (λi L) 1 dλ L(Z, X) 2πi P 0 Γ P 2 0 = P 0, P 0 Lu = LP 0 u for all u X 89

dim(im P 0 ) Γ + {λ ; Re λ > 0} σ + P + = 1 (λi L) 1 dλ L(Z, X) 2πi Γ + P + P h P h P h = I (P 0 + P + ) P 2 h = P 0, P h Lu = LP h u for all u X P 0 L(X, Y ) X Y Y Z P h L(X) L(Y ) L(Z) E 0 = im P 0 X, E + = im P + X, Z h = im P h Z Z = E 0 E + X h E := E 0 E + L 0, L +, L h L E 0, E +, X h u X u = u 0 + u h + u +, u 0 = P 0 u E 0, u + = P + u E +, u h = P h u Z h, 90

X h = P h X, Y h = P h Y (8.8) du 0 dt = L 0u 0 + P 0 N(u), du + = L + u + + P + N(u), dt du h dt = L hu h + P h N(u) cut-off χ : E R, = 0, + χ(u ) = { 1 for u 1 0 for u 2 χ(u ) [0, 1] for all u E E cut-off ε (0, ε 0 ] P 0 N ε (u), P + N ε (u), u = u 0 + u + + u h N0(u ε 0 + u + + u h ) = (P 0 N)(u 0 χ(u 0 /ε) + u + χ(u + /ε) + u h ), u 0 E 0, N+(u ε 0 + u + + u h ) = (P + N)(u 0 χ(u 0 /ε) + u + χ(u + /ε) + u h ), u + E +, Z = E 0 E + X h du 0 dt = L 0u 0 + N0(u), ε du + = L + u + + N ε dt +(u), du h dt = L hu h + P h N(u) (8.9) 91

L 0, L +, L h e Lht w X C 3 e γt w X, w Z h, t > 0, e L +t w X C 1 e γ t w X, w E +, t R, r > 0 C 2 (r) e L 0t w X C 2 (r)e r t w X, w E 0, t R. γ cu γ cu = max{r, γ } N C 1 (X, Y ) γ cu < γ ε 0 u 0 X, u + X < ε 0 h C 0 (E 0 E + ; Z h ) : W cu loc := {(u 0 + u + + u h ) Z; u h = h(u 0 + u + )} (8.8) h(0, 0) = 0. u = u 0 + u + + u h E 0 E + X h = X du 0 dt = L 0u 0 + N0(u), ε u 0 E 0, du + = L + u + + N ε dt +(u), u + E +, du h dt = L hu h + P h N(u), u h X h (8.10) 92

(w 0 (t, w 0 0 + w 0 + + ψ), w + (t, w 0 0 + w 0 + + ψ)) w 0 (0) = w 0 (0, w 0 0 + w 0 + + ψ) = w 0 0, w + (0) = w + (0, w 0 0 + w 0 + + ψ) = w 0 + ẇ 0 = L 0 w 0 + N ε (w 0 + w + + ψ(w 0 + w + )), w 0 E 0 ẇ + = L + w + + N ε +(w 0 + w + + ψ(w 0 + w + )), w + E + (8.11) j E F C k,1 C k,1 (E; F ) := { C k,1 w C k,1 (E; F ); w j,lip D j w(x) D j w(y) := sup x,y E,x y x y E <, 0 j k w; C k,1 (E, F ) := w; C k (E; V ) + max 0 j k w j,lip. E = F C k,1 (E; E) = C k,1 (E) ψ(0, 0) = 0 ψ C 0,1 b (E; Z h ) T (T ψ)(w 0 0 + w 0 +) = 0 e L hs (P h N)(w 0 (s, w 0 0 + w 0 + + ψ) + w + (s, w 0 0 + w 0 + + ψ) + ψ(w 0 + w + )) ds T T h = h T h (8.8) C 0 - } 93

p p 1 ψ ψ 1 < p, ψ(w 0 + w + ) ψ( w 0 + w + ) Zh < p 1 ( w 0 w 0 E0 + w + w + E+ ) ψ N(u), w 0 E0 < ε, w + E+ < ε k (ε), = 0, +, h N (w ε 0 + w + + ψ) Y0 k (ε)ε, = 0, +, P h N h (w 0 + w + + ψ) Y+ k h (ε)ε, N (w ε 0 + w + + w h ) N ( ε w 0 + w + + w h ) Y0 κ (ε)( w 0 w 0 E0 + w + w + E+ + w h w h Yh ), = 0, +, P h N(w 0 + w + + w h ) P h N( w 0 + w + + w h ) Yh κ h (ε)( w 0 w 0 E0 + w + w + E+ + w h w h Yh ) T ψ(w0 0 + w+) 0 0 C 3 εκ h (ε) 0 e γs ds = C 3 εκ(ε)/γ. w 0 (t) = w 0 (t, w 0 0 + w + 0 + ψ( w 0 0 + w +)), 0 w + (t) = w + (t, w 0 0 + w + 0 + ψ( w 0 0 + w +)) 0 94

N(u) t 0 ψ w + (t, w 0 0 + w 0 + + ψ(w 0 + w + )) w + (t, w 0 0 + w 0 + + ψ( w 0 + w + )) E+ C 1 e γ t w 0 0 w 0 0 E0 +C 1 κ + (ε) 0 t e γ (s t) { w 0 w 0 E0 + w + w + E+ + ψ(w 0 0 + w 0 +) ψ( w 0 0 + w 0 +) Zh } ds C 1 e γ t w 0 0 w 0 0 E0 +C 1 κ + (ε)(1 + p 1 ) 0 t e γ (s t) { w 0 w 0 E0 + w + w + E+ } ds. w 0 (t, w 0 0 + w 0 + + ψ(w 0 0 + w 0 +)) w 0 (t, w 0 0 + w 0 + + ψ( w 0 + + w 0 0)) E0 C 2 (r)e rt w 0 0 w 0 0 E0 +(1 + p 1 )C 2 (r)κ 0 (ε) 0 t e r(s t) { w 0 w 0 E0 + w + w + E+ } ds. 95

w 0 w 0 E0 + w + w + E+ 2 max{c 1, C 2 (r)}e max{γ,r}t { w 0 0 w 0 0 E0 + w 0 + w 0 + E+ } +2(1 + p 1 ) max{c 1 κ + (ε), C 2 (r)κ 0 (ε)} 0 t e max{γ,r}(s t) { w 0 w 0 E0 + w + w + E+ } ds. w 0 w 0 E0 + w + w + E+ 2C 4 { w0 0 w 0 0 X + w+ 0 w + 0 X }e γt, (8.12) C 4 = max{c 1, C 2 (r)}, γ = max{γ, r} + 2κ(ε)(1 + p 1 )C 4, κ(ε) = max{κ 0 (ε), κ + (ε)}. ε γ, γ p 1 ψ,c 4 L 0, L + γ = max{γ, r} + 2κ(ε)(1 + p 1 )C 4 < γ 96

T ψ(w 0 0 + w 0 +) T ψ( w 0 1 + w 0 2) 0 0 C 3 e γs P h N(w 0 + w + + ψ) P h N( w 0 + w + + ψ) X ds 0 C 3 κ h (ε) e γs { w 0 w 0 E0 + w + w + E+ + ψ(w 0 + w + ) ψ( w 0 + w + ) Zh } ds 0 C 3 κ h (ε)(c 4 + p 1 ){ w0 0 w 1 0 E0 + w+ 0 w 2 0 E+ } e (γ γ)s ds C 3 κ(ε)(c 4 + p 1 )(γ γ) 1 { w 0 0 w 0 1 E0 + w 0 + w 0 2 E+ }. ψ 1, ψ 2 C 0,1 (E 0 E +, Z h ) ψ j Lip < p 1 T ψ 1 T ψ 2 0 0 C 3 κ 0 (ε) e γs { ψ 1 ψ 2 0 + w 0 (s, ψ 2 ) w 0 (s, ψ 2 ) E0 + w + (s, ψ 2 ) w + (s, ψ 2 ) E+ } ds C 3 κ 0 (ε) ψ 1 (w 0, w + ) ψ 2 (w 0, w + ) 0 + I 1, 0 I 1 := e γs( w 0 (s, w0, 0 w+, 0 ψ 1 ) w 0 (s, w0, 0 w0, 0 ψ 2 ) E0 + ) + w + (s, w0, 0 w+, 0 ψ 1 ) w + (s, w0, 0 w0, 0 ψ 2 ) E+ ds. 97

ψ j (w 0, w + ) X sup ψ j (w 0 + w + ) Zh = ψ j 0, j = 1, 2. w 0 E 0,w + E + w (t, ψ j ) = w (t, w 0 0 + w 0 + + ψ j ), = 0, +, w + (t, ψ 1 ) w + (t, ψ 2 ) E+ 0 t C 1 κ + (ε) e L +(s t) {N ε +(w 0 (s, ψ 1 ) + w + (s, ψ 1 ) + ψ 1 ) N+(w ε 0 (s, ψ 2 ) + w + (s, ψ 2 ) + ψ 2 )} ds 0 t e γ (s t) { w 0 (s, ψ 1 ) w 0 (s, ψ 2 ) E0 E+ + w + (s, ψ 1 ) w + (s, ψ 2 ) E+ + ψ 1 ψ 2 0 }ds C 1 κ + (ε) ψ 1 ψ 2 0 (γ ) 1 (e γ t 1) +C 1 κ + (ε) 0 t e γ (s t) { w 0 (s, ψ 1 ) w 0 (s, ψ 2 ) E0 + w + (s, ψ 1 ) w + (s, ψ 2 ) E+ }ds 98

w 0 (t, ψ 1 ) w 0 (t, ψ 2 ) E0 0 t C 2 (r)κ 0 (ε) e L 0(s t) {N ε (w 0 (s, ψ 1 ) + w + (s, ψ 1 ) + ψ 1 ) N ε (w 0 (s, ψ 2 ) + w + (s, ψ 2 ) + ψ 2 )} ds 0 t E0 e r(s t) { w 0 (s, ψ 1 ) w 0 (s, ψ 2 ) E0 + w + (s, ψ 1 ) w + (s, ψ 2 ) E+ + ψ 1 ψ 2 0 }ds C 2 (r)κ 0 (ε) ψ 1 ψ 2 0 (r) 1 (e rt 1) +C 2 (r)κ 0 (ε) 0 t e r(s t) { w 0 (s, ψ 1 ) w 0 (s, ψ 2 ) E0 + w + (s, ψ 1 ) w + (s, ψ 2 ) E+ }ds t 0 w 0 (t, ψ 1 ) w 0 (t, ψ 2 ) E0 + w + (t, ψ 1 ) w + (t, ψ 2 ) E+ 2C 4 κ(ε) max{γ 1, r 1 }(e γcut 1) ψ 1 ψ 2 0 +2C 4 κ(ε) 0 t e γ cu(s t) { w 0 (s, ψ 1 ) w 0 (s, ψ 2 ) E0 + w + (s, ψ 1 ) w + (s, ψ 2 ) E+ }ds w 0 (s, ψ 1 ) w 0 (s, ψ 2 ) E0 + w + (s, ψ 1 ) w + (s, ψ 2 ) E+ 2C 4 κ(ε)(min{γ, r}) 1 ψ 1 ψ 2 0 (e γ cut 1)e γ 2t, γ 2 = γ cu + 2C 4 κ(ε). 99

I 1 C 4 κ(ε)(min{λ +, r}) 1 ψ 1 ψ 2 0 0 = C 4 κ(ε)(min{λ +, r}) 1 ψ 1 ψ 2 0 (λ γ 2 ) 1. ε γ γ cu 2C 4 κ(ε) > 0 e (γ C 4κ(ε))s + e (γ γcu 2C 4κ(ε))s ds γ max{γ, r} > 2 max{c 1, C 2 (r)} max{κ 0 (ε), κ + (ε)} (8.13) T C 0,1 (E 0 E +, Z h ) (u 0 +u + +u h ) = 0 X (8.8) h (8.8) (u 0 +u + +u h ) = 0 X u 0 = N1(u ε 0 + u + + u h ), u + = N2(u ε 0 + u + + u h ), u h = h(u 0 + u + ), u 0 (0) = 0, u + (0) = 0, u h (0) = 0 N ε 0(0, 0, h(0, 0)) = N ε +(0, 0, h(0, 0)) = h(0, 0) = 0. (8.13) γ cu = max{r, γ } < γ 100

N C 2 (X, Y ) γ cu < γ ε h (i) : h C 1,1 (E 0 E +, Z h ). (ii) h w (0, 0) = 0, w E, = 0, +, (i) : h(w0 0 + w+), 0 wj 0 E 0 w0 0 w + (w 0 (t, w + ), w + (t, w + )) ẇ 1 = L 1 w 0 + N ε 1(w 0 + w + + h(w 0, w + )), ẇ 2 = L 2 w + + N ε 2(w 0 + w + + h(w 0, w + )) (8.14) w (0) = w 0 := w (0, w 0 0 + w 0 + + h(w 0 0 + w 0 +)), = 0, + (w 0 0, w 0 +) 101

T (1) (T (1) ψ (1) )(w 0 0 + w 0 +) = 0 e L hs N (1) (w 0 (s, w 0 0 + w 0 + + h) + w 0 (s, w 0 0 + w 0 + + h) + h, ψ (1) ) ds, N (1) (w 0 + w + + h, ψ (1) ) = ψ (1) P hn v (w 0 + w + + v) + P hn w 0 (w 0 + w + + h) + P hn w + (w 0 + w + + h) ( w 0 ( w 0 0 v=h (s, w 0 0 + w 0 + + v) + ψ (1) w 0 v (s, w0 0 + w 0 + + v) w + (s, w 0 w+ 0 0 + w+ 0 + v) + ψ (1) w + v (s, w0 0 + w+ 0 + v) ) v=h ) v=h K(ε) K(ε) 0 as ε 0 K(ε) < γ γ ε T (1) C 1 (E 0 E +, Z h ) K(ε) N (1) ψ (1), P h N, P h N v, w, P h N w, w, w 0 C 1, C 2 (r), C 3, r, γ w, ( = 0, + ) v T (1) ε (T (1) ) (j) ψ Lip ψ Lip, (j = 0, 1) 102

σ E 0 a > 0 w0 0 + aσ E 0 ζ(w0 0 + ; a, σ) := {h( w0 0 + aσ) + ) h(w0 0 + )}/a, θ(w 0 + w + + h; a, σ) := {P h N(w 0 + + w + + + h + ) P h N(w 0 + w + + h)}/a, h + := h(w 0 (t, (w 0 0 + aσ) + w 0 +) + w + (t, w 0 0 + aσ + w 0 +)), w + := w (t, (w 0 0 + aσ) + w 0 + + h + ), = 0, +. ζ(w 0 0 + w 0 + ; a, σ) = 0 e L hs {θ(w 0 + w + + h; a, σ) + σn (1) (w 0 + w + + h, ζ/σ) σn (1) (w 0 + w + + h, ζ/σ)} ds. N C 2 (X; Y ) h, a 0 m(a) := sup θ(w 0 + w + + h; a, σ) σn (1) (w 0 + w + + h, ζ/σ) X 0. t [0, ) C 5 (a) G(a) := C 3 0 sup w 0 j <ε ζ(w 0 0 + w 0 +; a, σ) σψ (1) (w 0 0 + w 0 +) X e γs σn (1) 1 (w 0 + w + + h, ζ/σ) σn (1) 1 (w 0 + w + + h, ψ (1) ) X ds + C 3 γ m(a) C 5 (a)g(a) + C 3 γ m(a). G(a) 0 as a 0 103

ψ (1) h(w0 0 + w+) 0 w0 0 ψ (1) (w0 0 + w+) 0 h w 0 0 (w 0 0 + w 0 +) = ψ (1) (w 0 0 + w 0 +). (ii); w (s, 0 + 0 + h(0, 0)) = 0, ( = 0, + ), DN(0) = 0 ψ (1) T (1) h w 0 0 (0, 0) = 0. w + N C k (X; Y ), k = 1, 2, h C k 1 b (E 0 E +, Z h ) γ cu < γ h C k (E 0 E +, Z h ), k 2 (8.8) w 0 (t) + w + (t) E 0 E + ẇ 0 = L 0 w 0 + N ε 1(w 0 + w + + h(w 0, w + )), ẇ + = L + w + + N ε 2(w 0 + w + + h(w 0, w + )), (8.15) (8.10) u t u 0 (t) = w 0 (t) + O(e µt ), u + (t) = w + (t) + O(e µt ), u h (t) = h(u 0 (t), u + (t)) + O(e µt ) 104

µ (8.15) v(t) := u h (t) h(u 0 (t) + u + (t)) X h v = u h D u0 h(u 0 + u + ) u 0 D u+ h(u 0 + u + ) u + = L h u h + P h N(u 0 + u + + u h ) D u0 h(u 0 + u + )(L 0 u 0 + N0(u ε 0 + u + + u h )) D u+ h(u 0 + u + )(L 0 u 0 + N+(u ε 0 + u + + u h )) L h [h(u 0 + u + )] + L h [h(u 0 + u + )] (8.16) (8.17) w h = h(w 0 + w + ) t L h w h + P h N(w 0 + w + + h(w 0 + w + )) = D w0 h(w 0 + w + )(L 0 w 0 + N0(w ε 0 + w + )) +D w0 h(w 0 + w + )(L 0 w 0 + N0(w ε 0 + w + )). Lw h = L h h(w 0, w + ) L h h(u 0 + u + ) (8.10) L h h(u 0 + u + ) = P h N(u 0 + u + + h(u 0 + u + )) +D u0 h(u 0 + u + )(L 0 u 0 + N0(u ε 0 + u + )) +D u0 h(u 0 + u + )(L 0 u 0 + N0(u ε 0 + u + )). (8.16) v v = L h v + Q(u 0 + u + + v), 105

Q(u 0 + u + + v) = D u0 h(u 0 + u + ){N ε (u 0 + u + + h(u 0 + u + )) N ε (u 0 + u + + (v + h(u 0 + u + )))} +D u+ h(u 0 + u + ){N+(u ε 0 + u + + h(u 0 + u + )) N+(u ε 0 + u + + (v + h(u 0 + u + )))} +P h N(u 0 + u + + (v + h(u 0 + u + ))) P h N(u 0 + u + + h(u 0 + u + )). h C k (E 0 E +, Z h ), k 2 u, ũ E, ( = 0, + ) u h X h δ(0) = 0 δ(ε) Q(u 0 + u + + v)) Y δ(ε) v Xh. v(t) Xh C 3 v(0) Xh e γt + C 3 δ(ε) t 0 e γ(t s) v(s) Xh ds. v(t) Xh C 3 v(0) Xh e (γ C 3δ(ε))t. (8.18) u 3 h(u 0 + u + ) Xh C 3 u 3 (0) h(u 0 (0) + u + (0)) Xh e (γ C 3δ(ε))t φ := u w, ( = 0, +), φ v { φ = L φ + R (φ 0 + φ + + v), = 0, +, v = L h v + Q( (φ 0 + w 0 ) + (φ + + w + ) + v ), (8.19) R (φ 0 + φ + + v) = N ε ( (w 0 + φ 0 ) + (w + + φ + ) + (v + h(w 0 + φ 0 + w + + φ + ) ) ) N ε (w 0 + w + + h(w 0 + w + )), = 0, +. 106

η 0 8.19 { F η (R, E) := η (γ, γ) T 0,, T + w C 0 (R, E); w Fη := sup e ηt w(t) E < t [0, ) φ 0 F η (R, E 0 ), φ + F η (R, E + ) (T 0 φ 0 )(t) := (T + φ + )(t) := t t e L 0(t s) R 0 (φ 0 + φ + + v) ds, e L +(t s) R + (φ 0 + φ + + v) ds }. T [T(φ 0 + φ + )](t) = (T 0 φ 0 )(t) + (T + φ + )(t) φ 0, φ + T (8.19) F η (R, E) T F η (R, E) N0, ε N+ ε R 0 (φ 0 + φ + ) E0 κ 0 (ε){(1 + p 1 )( φ 0 E0 + φ + E+ ) + v Xh }, R + (φ 0 + φ + ) E+ κ + (ε){(1 + p 1 )( φ 0 E0 + φ + E+ ) + v Xh }, 107

(8.18) T 0 φ 0 Fη κ 0 (ε) sup e ηt e L0(t s) R 0 (φ 0 + φ + + v) ds t 0 t E0 κ 0 (ε)c 2 (r)(1 + p 1 ) { } sup e ηt e r(s t) ( φ 0 E0 + φ + E+ + v Xh )} ds t 0 t κ 0 (ε)c 2 (r)(1 + p 1 ) { } sup e ηt e r(s t) ηs e ηs ( φ 0 E0 ) + φ + E+ + v Xh ) ds t 0 t κ 0 (ε)c 2 (r)(1 + p 1 ) sup e {( φ ηt 0 Fη (R,E 0 ) + φ + Fη (R,E + ) + v Fη (R,X h )) t 0 t } e r(s t) ηs ds κ 0(ε)C 2 (r)(1 + p 1 ) ( φ 0 Fη (R,E η r 0 ) + φ + Fη (R,E + ) + v Fη (R,X h )) < γ < η < γ (8.18) T + φ + Fη κ + (ε) sup e ηt e L +(t s) R 0 (φ 0 + φ + + v) ds t 0 t E+ κ + (ε)c 1 (1 + p 1 ) { } sup e ηt e γ (s t) ( φ 0 E0 + φ + E+ + v Xh )} ds t 0 t κ + (ε)c 1 (1 + p 1 ) sup e {( φ ηt 0 Fη (R,E 0 ) + φ + Fη (R,E + ) + v Fη (R,X h )) t 0 t } e γ (s t) ηs ds κ +(ε)c 1 (1 + p 1 ) ( φ η γ 0 Fη (R,E 0 ) + φ + Fη (R,E + ) + v Fη (R,X h )) < 108

T F η (R, E) T ε φ 0, φ + φ 0, φ +, v(0) = ṽ(0) v(t), ṽ(t) X h Q(φ 0 + φ + + v) Q( φ 0 + φ + + ṽ) Y max{ v Xh, ṽ Xh } D φ0 h(φ 0 + φ + ) D φ0 h( φ 0 + φ + ) Xh + max{ v Xh, ṽ Xh } D φ+ h(φ 0 + φ + ) D φ+ h( φ 0 + φ + ) Xh + P h N(φ 0 + φ + + v + h(φ 0 + φ + )) P h N(φ 0 + φ + + h(φ 0 + φ + )) P h N( φ 0 + φ + + ṽ + h( φ 0 + φ + )) + P h N( φ 0 + φ + + h( φ 0 + φ + )) Xh 2p 2 κ(ε) max{ v Xh, ṽ Xh }( φ 0 φ 0 E0 + φ + φ + E+ ) +κ h (ε)[2(1 + p 1 )( φ 0 φ 0 E0 + φ + φ + E+ ) + v ṽ Xh ]. κ(ε) = max{κ 0 (ε), κ + (ε)}, p 2 = max{ D u0 h(u 0 + u + ) Lip, D u+ h(u 0 + u + ) Lip } φ = φ 0 + φ + E := E 0 E +, φ = φ 0 + φ + E := E 0 E + t v(t) ṽ(t) Xh κ h (ε)c 3 e γ(t s) v(s) ṽ(s) Xh ds + I 2, 0 I 2 2C 3 (1 + p 1 ) max{κ(ε), κ h (ε)} t 0 (8.18) e γ(t s) (p 2 max{ v(s) Xh, ṽ(s) Xh } + 1) φ(s) φ(s) E ds. max{ v(s) Xh, ṽ(s) Xh } C 3 v(0) Xh 109

I 2 2C 3 (1 + p 1 ) max{κ(ε), κ h (ε)} 2C 3 (1 + p 1 ) max{κ(ε), κ h (ε)}(1 + C 3 p 2 v(0) Xh ) t 0 e γ(t s) ηs e ηs φ(s) φ(s) E ds. 2C 3 (1 + p 1 ) max{κ(ε), κ h (ε)}(1 + C 3 p 2 v(0) Xh ) φ φ Fη(R,E) t 0 e γ(t s) ηs ds γ < η < γ I 2 2C 3(1 + p 1 ) max{κ(ε), κ h (ε)} (1+C 3 p 2 v(0) Xh ) φ γ η φ Fη (R,E)(e ηt e γt ). C 6 (ε) := 2C 3(1 + p 1 ) max{κ(ε), κ h (ε)} η γ (1 + C 3 p 2 v(0) Xh ) v(t) ṽ(t) Xh C 6 (ε) φ φ Fη (R,E)(e ηt e γt ) t +κ h (ε)c 3 e γ(t s) v(s) ṽ(s) Xh 0 ds. v(t) ṽ(t) Xh C 6 (ε) φ φ Fη(R,E)(e ηt e γt )e C 3κ h (ε)t T + φ + T + φ+ E+ (t) κ(ε)(1 + p 1 )C 1 e γ (s t) { φ φ E + v ṽ Xh } ds. t 110

t t e γ (s t) v ṽ Xh ds C 6 (ε) φ φ Fη(R,E) t e γ (s t) (e ηs e γs )e C 3κ h (ε)s ds C 6 (ε) φ φ Fη (R,E) { e { η+c 3 κ h } (ε)}t γ η + C 3 κ h (ε) + e{ γ+c3κh(ε)}t γ γ + C 3 κ h (ε) 2C 6 (ε) φ φ e { η+c 3κ h (ε)}t Fη(R,E) η γ e γ (s t) φ φ E ds φ φ Fη (R,E) t e γ (s t) ηs ds φ φ Fη (R,E) e ηt η γ φ φ Fη (R,E) e { η+c 3κ h (ε)}t η γ φ φ Fη (R,E) e { η+c 3κ h (ε)}t η γ T + φ + T + φ+ E+ (t)e { η+c 3κ h (ε)}t κ(ε)(1 + p 1 )C 1 (2C 6 (ε) + 1) φ φ Fη (R,E) η γ (T + φ + )(t) γ < η < γ η F η (R, E + ) ε > 0 γ < η C 3 κ h (ε) < γ F η (R, E + ) F {η C3 κ h (ε)}(r, E + ). η := η C 3 κ h (ε) γ < η := η C 3 κ h (ε) < γ 111

T + φ + T + φ+ F η (R,E + ) (2C 6 (ε) + 1) κ(ε)(1 + p 1 )C 1 φ η γ φ F η (R,E). C 1 C 2 (r) γ r T 0 φ 0 T 0 φ0 E+ (t)e { η+c 3κ h (ε)}t κ(ε)(1 + p 1 )C 2 (r) (2C 6(ε) + 1) φ φ Fη(R,E) η r T 0 φ 0 T 0 φ0 F η (R,E 0 ) κ(ε)(1 + p 1 )C 2 (r) (2C 6(ε) + 1) φ η r φ F η (R,E) γ cu = max{r, γ } T + φ + T + φ+ F η (R,E 0 ) κ(ε)(1 + p 1 )C 1 (2C 6 (ε) + 1) η γ cu φ φ F η (R,E), (8.20) T 0 φ 0 T 0 φ0 F η (R,E 0 ) κ(ε)(1 + p 1 )C 2 (r) (2C 6(ε) + 1) η γ cu φ φ F η (R,E). (8.21) (8.20) (8.21) max{r, γ } = γ cu < C 3 κ(ε) < γ ε γ cu < η < γ η T : F η (R, E) F η (R, E) 112

S : X X S : (w 0 0 + w 0 + + v(0)) (u 0 (t) + u + (t) + v(0)) S w = w 0 + w + E + E 0 = E, u = u 0 + u + E w(0) w(0) S(w(0) + v(0)) S( w(0) + v(0)) (u(t) + v(0)) = S(w(0) + v(0)), (ũ(t) + v(0)) = S( w(0) + v(0)) w(t), w(t) w(0) = w(0) w(t; w(0)) = w(t; w(0)), t 0. φ(t) = u(t) w(t) E, φ(t) = ũ(t) w(t) E w(t) = u(t) φ(t), w(t) = ũ(t) φ(t). w(t) w(t) E u(t) ũ(t) E + φ(t) φ(t) E. 113

u(t) = ũ(t) w(t) w(t) E φ(t) φ(t) E. w(0) w(0) w(t) w(t) (8.12) γ > α > γ cu + 2 max{κ 0 (ε), κ + (ε)}(1 + p 1 ) max{c 1, C 2 (r)} α t e αt w(t) w(t) E. γ cu < η < γ η φ(t) F η (R, E) γ cu < α < γ α e αt φ(t) φ(t) E <. w(t) = w(t), t 0 w(0) w(0) u(t) ũ(t). (8.8) L σ = σ + σ 0 + σ + σ = σ + σ σ 114

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