Phase space Attractor mutual interaction selforganization jump hysteresis stabilize or destabilize Synchronization Self-excited Oscillation Nonlinear

Transcription

1 Phase space Attractor mutual interaction selforganization jump hysteresis stabilize or destabilize Synchronization Self-excited Oscillation Nonlinear Resonance Parametric Excitation tangent period doubling Neimark-Sacker Hopf Bifurcation tangent period doubling Parameter space

2 i 1 1. n ,

3 ii 9 2, bon voyage! le 1 er avril, 1998 Voici mon secret. Il est trés simple: on ne voit bien qu avec le cœur. L essentiel est invisible pour les yeux. L essentiel est invisible pour les yeux, répéta le petit prince, afin de se souvenir. And now here is my secret, a very simple secret: It is only with the heart that one can see rightly; what is essential is invisible to the eye. What is essential is invisible to the eye, the little prince repeated, so that he would be sure to remember. Le Petit Prince par Antoine de Saint-Exupéy

4 iii I vii

5 iv (Floquet-Lyapounov)

6 v (dissipative system in the large)

7 vi (σ, S )

8 I

9 system) (state) (dynamic system) (dynamical system) 1 (lumped system)

10 1 2 θ m T mg 1.1 g (Newton) l m

11 1 3 θ(t) J T (1.2) (1.1) J d2 θ dt 2 = T (1.1) J = ml 2, T = mgl sin θ (1.2) d 2 θ dt 2 + g sin θ = (1.3) l (1.3) 2 t = θ() dθ/dt() dθ/dt = ω (1.3) 1 dθ dt dω dt = ω = g l sin θ (1.4) θ ω θ ω [rad] 2π [rad] θ S 1 S 1 = {(x, y) R 2 x 2 + y 2 = 1} (1.5) (1.4) θ 2π ω R M (state space) M M = {(θ, ω) S 1 R θ S 1, ω R} (1.6) M (θ, ω) (1.4) (1.4) M (θ, ω) (ω, g sin θ) l (1.4) (vector field) M (1.6) 1.2 (1.4) (θ(), ω())

12 T s T d T s = Kθ T d = B dθ dt (1.7) (1.1) (1.2), (1.7) (1.8) J d2 θ dt 2 = T T s T d (1.8) J d2 θ dt 2 + B dθ ml 2 dt + K ml 2 θ + l sin θ = (1.9) g (1.9) (1.4) 1 dθ dt dω dt = ω = B ml 2 ω K ml 2 θ l g sin θ (1.1) (1.1) θ θ

13 1 5 K B θ T s, T d m mg T 1.3 M 2 R 2 M = {(θ, ω) R 2 θ, ω R} (1.11) q(t) = Cv C (t) λ(t) = Li L (t) i G (t) = g(v G (t)) = g 1 v G (t) + g 3 vg(t) 3 j(t) = J sin ωt (1.12) q, λ v i C L

14 1 6 i G j(t) i L L v C C i C G 1.4 i C = dq dt = C dv C dt (Faraday) v L = dλ dt = Ldi L dt (1.13) (1.14) v C i L 3 g 1 g 3 v G i G = g 1 v G (t) (1.15) g 1 (1.15) ω (Kirchhoff) v G (t) = v L (t) = v C (t) = v J (t) i G (t) + i L (t) + i C (t) = j(t) (1.16) (1.12), (1.13), (1.14) (1.16) (v C, i L ) dv C dt di L dt = 1 C [ i L g(v C ) + j(t)] = 1 L v C (1.17)

15 1 7 (v C, i L ) M 2 M = {(v C, i L ) R 2 v C, i L R} (1.18) J = (1.17) i L LC d2 v C dt 2 + L dg dv C dv C dt + v C = (1.19) (1.12) d 2 x dτ 2 ɛ(1 x2 ) dx dτ + x = (1.2) x = 3g3 g 1 v C, τ = 1 LC t (1.21) ɛ = g 1 L C (1.22) (1.2) (van der Pol) (1.17) v C [ d 2 ( ) ] 2 y dy dτ 2 ɛ dy 1 dτ dτ + y = (1.23) y = g 3 L g 1 C i L (1.24) (Rayleigh) (1.17) J (1.17) dv C dt di L dt = 1 C [ i L g(v C ) + J sin ωt] = 1 L v C (1.25)

16 1 8 t (non-autonomous system) (autonomous system) 1.3 t k k t N(k t) t N(k t) N(k) t = N ((k + 1) t) N(k t) t N(k t) (1.26) a N ((k + 1) t) = αn(k t) (1.27) α = 1 + a t t (1.27) N(k + 1) = αn(k) (1.28) N (1.28) (1.28) 1 k = N() (1.28) N(k) = α k N() (1.29) 1 α α N N α α = α (1 ɛn(k)) (1.3) ɛ (1.28) N(k + 1) = α (1 ɛn(k)) N(k) (1.31)

17 1 9 x(k) = ɛn(k) (1.32) x(k + 1) = α (1 x(k)) x(k) (1.32) 1.4 F (k + 2) = F (k + 1) + F (k); F (1) = F (2) = 1 (1.33) {F (k), k = 1, 2,...} F (k + 1) = G(k) G(k + 1) = F (k) + G(k) (1.34) (state) (phase) (phase space) n R n x i R n = {(x 1, x 2,..., x n ) x i R, i = 1, 2,..., n} (1.35) 1.1 M R n M R n R n 2 R 2 (phase plane) p p = J dθ dt = Jω (1.36)

18 1 1 (1.1) dp dt = T = mgl sin θ (1.37) dθ dt dp dt = p J = p ml 2 = mgl sin θ (1.38) (1.38) (1.4) t = t M R n 1 x(t ) = x M x(t) (t, x ) x 1.1 m l g M R n dx i dt = f i(t; x 1, x 2,..., x n ; λ 1, λ 2,..., λ m ), i = 1, 2,..., n (1.39) dx dt = f(t, x, λ) (1.4) t x λ t R, x = x 1 x 2. x n M, λ = λ 1 λ 2. λ m R m (1.41) f : R M R m R n ; (t, x, λ) f(t, x, λ) (1.42)

19 1 11 M (1.4) t dx dt = f(x, λ) (1.43) (1.43) t (1.43) dx dτ τ = t + t (1.44) = f(x, λ) (1.45) t = t x t = x t = f(x, λ) = (1.46) (equilibrium point) (1.43) 1.6 (1.38) (θ, p) = (, ) (π, ) 2 (1.4) t t = θ (1.47) (1.4) dx dθ dt dθ = f(t, x, λ) = 1 (1.48)

20 1 12 n + 1 (1.48) 2 1 L f(t + L, x, λ) = f(t, x, λ) (1.49) x(t + kl) = x(t) (1.5) (periodic solution) k Z Z = {..., 2, 1,, 1, 2,...} (1.51) n M R n x(k + 1) = f(k, x(k), λ), k Z (1.52) f : Z M R n M; (k, x, λ) f(k, x, λ) (1.53) f k 1 x(k) f x(k + 1) k = x x() = x x(1) = f(, x ) x(2) = f (1, x(1)) = f (2, f(, x )) (1.54) x(k + 1) = f (x(k)) (1.55)

21 1 13 x(k) f k f k x(k) = f (x(k 1)) = f (f (f ( f(x )))) = f k (x ) (1.56) }{{} k f x = f(x ) (1.57) (fixed point) f f 1 x( 1) = f 1 (x ), x( 2) = f 1 ( f 1 (x ) ) = f 2 (x ), (1.58) f f 1 f (Hénon) 2 x(k + 1) = 1 + y(k) ax 2 (k) y(k + 1) = bx(k) (1.59) (1.59) b J 2ax 1 J(x, y) = = b (1.6) b (1.59) x(k) = 1 y(k + 1) b y(k) = 1 + x(k + 1) + a b 2 y2 (k + 1) (1.61) f : R R; x f(x) = α (1 x)x (1.62)

22 W p W (p) = 1 ( ) 2 dθ 2 J = p2 dt 2J = V H p2 2ml 2 (1.63) V (θ) = mgl cos θ (1.64) H(θ, p) = W (p) + V (θ) = (1.38) dθ dt dp dt = H p = H θ p2 mgl cos θ (1.65) 2ml2 (1.66) H (1.66) dh dt = H dθ θ dt + H dp p dt = H H θ p + H H p θ = (1.67) H (θ(t), p(t)) = E (1.68) E (x, y) R n R n H : R n R n R; (x, y) H(x, y) (1.69) dx i dt dy i dt = H y i = H x i i = 1, 2,..., n (1.7)

23 1 15 (conservative system) (Hamilton) (1.7) ( ) T dx dt dy dt ( ) T H = y ( ) T (1.71) H = x n (x, y) n H H(x, y) = E M 2n 1 H(x, y) = E 2n 1 (1.71) 1 (dissipative system) (1.1) 2 1 H H(θ, ω) = 1 2 ω2 + K 2ml 2 θ2 + l cos θ (1.72) g F (1.1) F (ω) = B 2ml 2 ω2 (1.73) 2 2 dθ dt dω dt = H ω = H θ F ω (1.74) dh dt = H dθ θ dt + H dω ω dt = H F ω ω = B ml 2 ω2 (1.75) ω = θ θ 1.2 W e W e (v C ) = t v C i C dt = 1 2 C t v C dv C dt dt = 1 2 C vc v C dv C = 1 2 Cv2 C (1.76)

24 1 16 W m W m (i L ) = t v L i L dt = 1 2 L t i L di L dt dt = 1 2 L il i L di L = 1 2 Li2 L (1.77) vi (1.13), (1.14) W dw dt = W dv C v C dt W (v C, i L ) = 1 2 Cv2 C Li2 L (1.78) + W i L di L dt = v C{ i L g(v C )} + i L v C = v C g(v C ) (1.79) (1.17) (1.12) 3 dw dt = v C g(v C ) = g 1 v 2 C g 3 v 4 C (1.8) g 1 g 3 v C (1.78) (gradient) (gradient system) G : R n R; x G(x) (1.81) R dx i dt = G x i i = 1, 2,..., n (1.82) ( ) T dx G dt = = gradg (1.83) x G dg dt = n i=1 G dx n i x dt = ( ) 2 G (1.84) x i=1 i G

25 1 17 a G b i g1 i g2 g 1 C 1 v 1 v 2 C 2 g RC 1.5 a, b (1.85) C 1 dv 1 dt C 2 dv 2 dt F (v 1, v 2 ) = 1 2 G(v 1 v 2 ) 2 + = g 1 (v 1 ) G(v 1 v 2 ) = g 2 (v 2 ) G(v 2 v 1 ) C 1 dv 1 dt C 2 dv 2 dt v1 (1.85) v2 g 1 (v 1 )dv 1 + g 2 (v 2 )dv 2 (1.86) = F v 1 = F v 2 (1.87) F df dt = F dv 1 v 1 dt + F dv {( ) 2 ( ) 2 2 F F } v 2 dt = + (1.88) v 1 v

26 RC C dv 1 dt C dv 2 dt = g(v 1 ) G(v 1 v 2 ) = g(v 2 ) G(v 2 v 1 ) (1.89) v 1, v (1.2) dx dt dy dt = y = x + ɛ(1 x 2 )y (1.9) (x, y) ( x, y) 3 (1.12) (1.71) t t dx dt dy dt ( ) T H = y ( ) T (1.91) H = x R n (linear) dx dt = A(t)x (1.92) dx dt = A(t)x + B(t)u(t) (1.93)

27 1 19 x 1 u 1 (t) a 11 (t) a 12 (t) a 1n (t) b 11 (t) b 1m (t) x 2 u 2 (t) a x =, u(t) =, A(t) = 21 (t) a 22 (t) a 2n (t) b, B(t) = 21 (t) b 2m (t).. x n u m (t) a n1 (t) a n2 (t) a nn (t) b n1 (t) a nm (t) (1.94) (1.92) (homogeneous equation) (1.93) (1.93) 2 u(t) R m (nonlinear system) dx dt t = t x(t ) = x = f(t, x) (1.95) x(t) = ϕ(t, t, x ) (1.96) ξ(t) (1.95) x(t) = ϕ(t, t, x ) + ξ(t) (1.97) dx dt = d{ϕ(t, t, x ) + ξ(t)} = f(t, ϕ(t, t, x ) + ξ(t)) (1.98) dt f(t, ϕ(t, t, x ) + ξ(t)) = f(t, ϕ(t, t, x )) + f x x(t)=ϕ(t,t,x ) ξ(t) + (1.99) ξ(t) dξ(t) dt = A(t)ξ(t) (1.1) A(t) = f x x(t)=ϕ(t,t,x ) (1.11) (1.1) (1.96) (equation in variation, variational equation) (1.1) (1.96) (1.95) (linearized system)

28 (1.4) (1.38) (1.38) θ(t) = + ξ(t), p(t) = + η(t) sin( + ξ) = η + ξ, η dξ dt dη dt = 1 ml 2 η = mglξ (1.12) ɛ ɛ 1 dx dt = Ax + ɛf(t, x) (1.13) (1.4) (1.9) ɛ (1.12)

29 [O11] [O14] [O1] [O7] [O16] [D4] [D7] [D13] Abraham[D1] Andronov[N1] [N8] [N14] [N6] [N17] Stern[N18] [A1,2,3] [E8] [E2], [E7], [A14] Winfree[A17] Glass[A7], Hoppensteadt[A1],Pavlidis[A12], [A4] Gurel[A8], Berge[A5], [E9] [A13] [A9] Ccitanovic[A6], Moon[A11], Thompson[A15, 16] M S f E = F cos ωt (1.14) x f S f S = a 1 x + a 3 x 3 (1.15) f F = B dx dt (1.16)

30 1 22 f S, f F fe = F cos ω t M x (a) q(t) = Cv C (t) i L (t) = c 1 λ(t) + c 3 λ 3 (t) i G (t) = g 1 v G (t) j(t) = J cos ωt (1.17) (b) q(t) = Cv C (t) i L (t) = c 1 λ(t) + c 3 λ 3 (t) i G (t) = g (v G (t)) = g 1 v G (t) + g 3 vg(t) 3 j(t) = J cos ωt (1.18) x, y E I dx m τ x dt = x + w l u l, z x = f(x) = tan 1 x (1.19) l=1

31 1 23 u 1 w 1 x E z x w x w 1 u 1 y I z y u 2 w 2 u 2 w 2 (a) (b) 1.7 (a) (b) h a c x E h c x I h a b x E c E y h a c I y h I y c k (a) (b) (c) 1.8 m dy τ y dt = y + w l u l, z y = f(y) = tan 1 y (1.11) l=1 z x, z y tan 1 2 w l 1.8 a, b, c (1.32) x() x(1), x(2),... α 1.2 (1.9) ɛ.1,.5, 1., 2.

32 24 2 n n n M R n M R n dx i dt = ẋ i = f i (x 1, x 2,..., x n ), i = 1, 2,..., n (2.1) ẋ = f(x) (2.2) t t R x x R n ẋ = dx dt f : M R n ; x f(x)

33 2 25 M R n f(x) = (2.3) (equilibrium point) (2.2) x R n (2.2) ξ(t) ξ(t) x (variation) x(t) = x + ξ(t) (2.4) (2.4) (2.2) ẋ(t) = ẋ + ξ(t) = f (x + ξ(t)) (2.5) ξ(t) f(x + ξ(t)) = f(x ) + f ξ(t) + = f ξ(t) + (2.6) x x x(t)=x x(t)=x ξ(t) 2 (2.5) (2.6) ξ(t) = Aξ(t) (2.7) A = Df(x ) = f (2.8) x x(t)=x Df(x ), f x f x x(t)=x (2.7) x (2.7) x (2.2) ξ(t) (2.7) 2.1 (1.9) (1.9) y =, x + ɛ(1 x 2 )y = (2.9) (x, y ) = (, ) x(t) = x + ξ = ξ, y(t) = y + η = η ξ = η η = ξ + ɛη (2.1)

34 (2.7) ẋ = Ax (2.11) A n n t =, x() = x x(t) = φ(t, x ) (2.12) x() = φ(, x ) = x (2.13) de At dt = Ae At (2.11) x(t) = φ(t, x ) = e At x (2.14) e At = exp(at) = I n + At 1! + A2 t 2 + = 2! A k t k k= k! (2.15) Φ(t) = e At (2.16) dφ(t) dt = AΦ(t) (2.17) Φ() = e A = I n (2.18) (2.11) Φ(t) (state transition matrix) [ ] Φ(t) = ϕ 1 (t) ϕ 2 (t) ϕ n (t) (2.19) ϕ j (t) j 1 {}}{ ϕ j () = e j = [ 1 ] T, j = 1, 2,..., n (2.2)

35 2 27 n (2.11) (2.14) x(t) = φ(t, x ) = Φ(t)x = c 1 ϕ 1 (t) + c 2 ϕ 2 (t) + + c n ϕ n (t) (2.21) x = [c 1, c 2,..., c n ] T (2.22) (2.22) (2.21) (2.11) ϕ j (t) n n A H Q = H 1 AH = diag[q 1 (µ 1 ) Q 2 (µ 2 ) Q r (µ r )] (2.23) µ j 1. µ j 1... µ.. j Q j (µ j ) =, j = 1, 2,..., r (2.24) µ j 1 µ j (2.11) x = Hy (2.25) (2.11) ẋ = Hẏ = Ax = AHy ẏ = H 1 AHy = Qy (2.26) 2 (1) A (2) A 1

36 2 28 A Ah = µh (2.27) χ(µ) = det(µi n A) = µ n + a 1 µ n a n 1 µ + a n = (2.28) A Ah i = µh i, i = 1, 2,..., n (2.29) H = [h 1 h 2 h n ], Q = diag[µ 1 µ 2 µ n ] (2.29) AH = HQ (2.3) (2.26) Q = H 1 AH = diag[µ 1 µ 2 µ n ] ẏ i = µ i y i, i = 1, 2,..., n (2.31) (2.31) y y i (t) = e µ it y i (), i = 1, 2,..., n (2.32) (2.11) x(t) = Hy(t) = [h 1 h 2 h n ]y(t) = y 1 (t)h 1 + y 2 (t)h y n (t)h n = y 1 ()e µ 1t h 1 + y 2 ()e µ 2t h y n ()e µ nt h n n = y i ()e µ it h i i=1 (2.33) (2.21) 1 h i h i (invariant subspace) 1 x() E i = {x R n x = ah i, a R}

37 2 29 (2.33) x(t) E i, t R (2.33) R n h i µ 1 µ 2 = µ 1 h 1 h 2 = h 1 (2.33) x(t) = c 1 e µ 1t h 1 + c 1 e µ 1t h1 + c 3 e µ 3t h c n e µ nt h n (2.34) c 1 (2.34) 1 2 (2.34) 1 2 µ 1 = ζ + jω, h 1 = h + jk, c 1 = c + jd (2.35) j = 1 ζ, ω h, k c 1 e µ 1t h 1 + c 1 e µ 1t h1 = 2R(c 1 e µ 1t h 1 ) = 2ce ζt (h cos ωt k sin ωt) 2de ζt (h sin ωt + k cos ωt) (2.34) 1 2 c 1 e ζt (h cos ωt k sin ωt) + c 2 e ζt (h sin ωt + k cos ωt) c 1, c 2 (2.34) x(t) = c 1 e ζt (h cos ωt k sin ωt) + c 2 e ζt (h sin ωt + k cos ωt) +c 3 e µ 3t h c n e µ nt h n (2.36) 2.3 (2.1) 2.2 χ(µ) = µ 2 ɛµ + 1 = (2.37) ɛ 2 2 µ 1,2 = ɛ ( ɛ ) ± j = ζ ± jω (2.38) 2

38 2 3 ζ = ɛ 2, ω = 1 ζ 2 h = 1, k = ζ ω ξ(t) = c 1 e ζt (h cos ωt k sin ωt) + c 2 e ζt (h sin ωt + k cos ωt) η(t) c = 1 e ζt cos ωt + c 2 e ζt sin ωt c 1 e ζt (ζ cos ωt ω sin ωt) + c 2 e ζt (ζ sin ωt + ω cos ωt) (2.39) A 1 n 1 µ 1. µ 1.. Q = H 1 µ... AH = = µi... n + N (2.4) 1... µ 1 µ N = (2.41)

39 2 31 N l N l.. =, l = 2, 3,..., n N n = (2.42) (2.25) (2.11) ẏ = Qy = (µi n + N)y (2.43) n 1 y(t) = e (µin+n)t y() = e µint e Nt y() = e µt t j N j y() j! j= t t 2 t n 1 1 1! 2! (n 1)! y 1 (). 1.. y 2 () = e µt.... t. 1! 1 y n () t t n 1 1 1! (n 1)! 1 = y 1 ()e µt + y 2 ()e µt. + + y n ()e µt.. t 1! 1 (2.44) (2.25) n µ e l = l [ T 1 ], l = 1, 2,..., n

40 2 32 Q µi n = {e 1 } (Q µi n ) 2 = {e 1, e 2 } (Q µi n ) n = {e 1, e 2,, e n } µ (2.1) ɛ = 2 (2.1) ξ = η η = ξ + 2η (2.45) (2.37) µ 1 = µ 2 = 1 (2.4) A = 1 = µ 1 I 2 + N = (2.45) ξ(t) = e At ξ() = e I2t e Nt ξ() = e t [I 2 + Nt] ξ() η(t) η() η() η() { = e t 1 + t 1 1 } ξ() = e t 1 t t ξ() η() t 1 + t η() (2.46) n d n x dt n + a d n 1 x 1 dt n 1 + +a d n 2 x 2 dt n a dx n 1 dt + a nx = (2.47) (2.47) x(t) = e µt (2.48)

41 2 33 (2.47) χ(µ) = µ n + a 1 µ n 1 + a 2 µ n a n 1 µ + a n = (2.49) (2.49) χ(µ) = (µ µ 1 )(µ µ 2 ) (µ µ n ) = (2.5) (2.47) x(t) = c 1 e µ 1t + c 2 e µ 2t + + c n e µ nt (2.51) (2.47) x 1 (t) = x(t) x 2 (t) = dx dt = dx 1 dt x 3 (t) = d2 x dt 2 = dx 2 dt x n (t) = dn 1 x dt n 1 = dx n 1 dt (2.52) (2.47) dx n dt = a 1 x n (t) a 2 x n 1 (t) a n 1 x ( t) a n x 1 (t) (2.53) (2.52) (2.53) x 1 1 x 1 d x x 2 dt =. 1. a n a 2 a 1 x n (2.47) A = 1 a n a 2 a 1 x n (2.54) (2.55) (2.49) (companion matrix)

42 (1.9) 1 θ + B ( ml θ K 2 + ml 2 + l ) θ = (2.56) g θ = ω ω = B ml 2 ω ( K ml 2 + l ) θ g (2.57) (1.1) x R n (2.2) 1 (2.7) (2.8) A = Df(x ) {µ 1, µ 2,..., µ n } = { µ i C det (µ i I n Df(x )) = } (2.58) x (hyperbolic) (simple) R(µ i ), i = 1, 2,..., n (2.59) (2.7) 2 A R(µ i ) > µ i R n E u A R(µ i ) < µ i R n E s (2.11)

43 2 35 R n x (tangent space) x R n (a) R n = E u E s, A(E u ) = E u, A(E s ) = E s (b) dim E u = #{µ i C R(µ i ) > }, dim E s = #{µ i C R(µ i ) < } (2.6) #{ } 2.6 (Lorenz) 3 ẋ = σ(y x) ẏ = rx y xz ż = bz + xy (2.61) σ, b >, r > 1 (2.8) σ σ A = r 1 (2.62) b χ(µ) = det(µi 3 A) = (µ + b){µ 2 + (1 + σ)µ + σ(1 r)} = (2.63) µ 1 = b < µ 2 = σ (σ ) rσ < 2 (2.64) µ 3 = σ (σ ) rσ > 2 µ 1 z µ 2, µ 3 h 2, h 3 σ (σ ) 2 h 2 = σ rσ, h 3 = 2 2 σ 1 2 σ (σ ) rσ 2

44 2 36 E s = {x R 3 x = a 1 h 2 + a 2 e 3, a 1, a 2 R} E u = {x R 3 x = a 3 h 3, a 3 R} (2.65) (2.2) t = u R n x(t) = ϕ(t, u), x() = ϕ(, u) = u (2.66) W u (x ) = {u R n lim t ϕ(t, u) = x } W s (x ) = {u R n lim t ϕ(t, u) = x } (2.67) x (unstable) (stable manifold) 2 x E u E s x dim E u = dim W u (x ) dim E s = dim W s (x ) (2.68) dim W u (x ) dim W s (x ) = x , 1.5, 1.6 (π, ) (1.38) (1.65) (π, ) H(θ, p) = p2 mgl cos θ = H(π, ) = mgl (2.69) 2ml2 p = ±2ml gl cos θ 2 (2.7) 2 (θ, p) R 2 (1.38) p = 2ml gl cos θ 2 (2.71) (π, ) p = 2ml gl cos θ 2 (2.72)

45 2 37 p E s W s -π π θ W u E u 2.1 (π, ) (π, ) (2.71) (2.72) ( π, ) (2.71) (π, ) ( π, ) (π, ) 2 (2.71), (2.72) 2 p = ml gl(θ π) p = ml gl(θ π) (2.73) µ 1 = g/l µ 2 = g/l :1 dim W u (x ) = m, dim W s (x ) = n m (2.74) m m O m n n + 1

46 2 38 (2.2) n + 1 O, 1 O,..., n O O (sink) no (source) 1O,, n 1 O (saddle) (2.2) (2.7) (2.7) (2.11) χ(µ) = det(µi n A) = µ n + a 1 µ n 1 + a 2 µ n a n 1 µ + a n = (2.75) (2.75) (Routh) (Hurwitz) 2 3 (2.75) χ(µ) = det(µi 2 A) = µ 2 + a 1 µ + a 2 = (2.76) O a 1 >, a 2 > 1O a 2 < 2O a 1 <, a 2 > (a 1, a 2 )- 2.2

47 2 39 a 2 2O O a 1 1O 2.2 Im Im Im Re µ 2 -ζ µ 1 ω -ω Re -ζ Re (a) (b) (c) 2.3 O (1) µ 1, µ (a) µ 1 < µ 2 <, (b) µ 1 = ζ + jω, µ 2 = ζ jω, ζ > (c) µ 1 = µ 2 = ζ (2.26) Q (a) µ 1, (b) ζ ω, ζ > µ 2 ω ζ (c 1) ζ, (c 2) ζ 1 ζ ζ 2.4 (a) (stable node) (b) (spiral or focus)

48 2 4 y 2 y 2 y 1 y 1 (a) (b) y 2 y 2 y 1 y 1 (c-1) 2.4 (c-2) O (c 1) (c 2) (2.33), (2.36), (2.44) (2) µ 2 < < µ 1 (2.26) Q µ 1 µ 2 (2.32) 2.5 (3) µ 1, µ 2 (4) (2.76) 2.2 (a) a 1 =, a 2 >, (b) a 2 =

49 2 41 Im y 2 µ 2 µ 1 Re y 1 (a) 2.5 (b) (a) 1 O (b) y 2 y 2 y 1 y 1 (a) (b) 2.6 (a) (b) (2.26) Q (a) ω, ω (b) µ (a) (b) e µt 2.6 (a) (center) A (b) y 1

50 2 42 Im Im Im Re Re Re (a) Im Im Re Re (b) 2.7 (a) O, (b) 1 O 3 O 2 O (2.11) χ(µ) = det(µi 3 A) = µ 3 + a 1 µ 2 + a 2 µ + a 3 = (2.77) O a 1 >, a 3 >, a 1 a 2 a 3 > 1O a 1 >, a 3 < a 1 <, a 3 <, a 1 a 2 a 3 > 2O a 1 <, a 3 > a 1 >, a 3 >, a 1 a 2 a 3 < 2O a 1 <, a 3 <, a 1 a 2 a 3 < 2.7

51 (Lagrange) (Maxwell) (Routh) (A. M. Lyapounov) x (2.2) x() = u x x(t) = ϕ(t, u) t ɛ > ϕ(t, u) x < ɛ u x < δ(t, ɛ) δ(t, ɛ) t ɛ ɛ δ δ ɛ t ẋ = x (2.78) x(t) = e t u, x() = u (2.79) t ɛ e t u < ɛ u < δ = ɛe t t δ t = 4 ɛ = 1 δ = e 4 < 1 17 ɛ t [, ) δ(ɛ)

52 2 44 x (2.2) x ɛ > δ > u x < δ u R n x(t) = ϕ(t, u) t ϕ(t, u) x < ɛ ɛ >, δ >, u x < δ, t, ϕ(t, u) x < ɛ (2.8) (asymptotic stability) x (asymptotically stable) (1) x (2) ɛ >, u x < ɛ, lim t x(t) = x 2 2 (attractivity) (2.8) ɛ >, δ >, u x < δ, τ, ϕ(τ, u) x > ɛ (2.81) x O (2.2) 1. (2.3) 2. (2.8) 3. (2.8) 4. (2.2)

53 O 1O 1O O O 2O 1O O 1O 2.8 (2.83) α = 4, δ = RC 1.1 (1.89) 3 g(v) = av + bv 3 (2.82) a, b > (2.82) (1.89) ẋ = αx x 3 δ(x y) ẏ = αy y 3 δ(y x) (2.83) α = a C, δ = G C, v 1 = C C b x, v 2 = b y (2.84) 2 αx x 3 δ(x y) = αy y 3 δ(y x) = (2.85) α = 4, δ = (x, y ) A = α δ 3x2 δ (2.86) δ α δ 3x 2

54 2 46 F(x, y) y x 2.9 (2.87) 2.8 F (x, y) = 1 2 α(x2 + y 2 ) (x4 + y 4 ) (x y)2 (2.87) (2.83) 2.9 ẋ = F x, ẏ = F y (2.88) (2.86)

55 2 47 y 3 O 1O 2 1 O 1O 2O O x O O O (2.83) α = 4, δ = 1 5. n n [O1] [O1] [O7] [O4] [O11] (local) 8 (center manifold) [D19] Matsumoto[D15]

56 (2.1) ɛ > 2.2 (2.57) x y (Brussellator) ẋ = a (b + 1)x + x 2 y ẏ = bx x 2 y (2.89) a, b 2.5 ) ẋ = (1 x2 2 y2 x ẏ = (1 x 2 y ) y (2.9) α δ 2.1

57 49 3 n n 4n 3.1 k Z R n x i (k + 1) = f i (x 1 (k), x 2 (k),..., x n (k)), i = 1, 2,..., n (3.1) x(k + 1) = f (x(k)) (3.2) k x k Z, x R n f : R n R n ; x f(x) f f(x) = x (3.3)

58 3 5 (fixed point) l f l- ( l-periodic point) l f(x ) = x 1, f(x 1 ) = x 2,..., f(x l 1 ) = x (3.4) l l- l- l l- l- l x(l) = f (x(l 1)) = f (f (f ( f(x )))) = f l (x ) (3.5) }{{} l x R n (3.2) ξ(k) (3.2) x(k) = x + ξ(k) (3.6) x(k + 1) = x + ξ(k + 1) = f (x + ξ(k)) (3.7) ξ(k) f (x + ξ(k)) = f(x ) + f ξ(k) + = x + f ξ(k) + (3.8) x x x=x x=x ξ)k) 2 (3.7) (3.8) ξ(k + 1) = Aξ(k) (3.9) A = Df(x ) = f (3.1) x x=x (3.9) x (variational equation) (3.9) x (3.2) (3.9) (3.1) A (3.9) (3.2) 3.1 (1.51) (1.51) x = 1 + y ax 2, y = bx (3.11)

59 3 51 a, (b 1) 2 + 4a > 2 ( b 1 + (b 1) (x 1, y 1 ) = 2 + 4a b(b 1) + b ) (b 1), 2 + 4a 2a 2a ( b 1 (b 1) (x 2, y 2 ) = 2 + 4a b(b 1) b ) (3.12) (b 1), 2 + 4a 2a 2a x(k) = x i + ξ(k), y(k) = y i + η(k), i = 1, 2 ξ(k + 1) = 2ax i ξ(k) + η(k) η(k + 1) = bξ(k) (3.13) (3.9) x(k + 1) = Ax(k) (3.14) (3.14) k = k = x() = x x(k) = φ(k, x ) (3.15) x() = φ(, x ) = x (3.16) k = 1, 2,... (3.14) x(1) = φ(1, x ) = Ax x(2) = φ(2, x ) = Ax 1 = AAx = A 2 x (3.17) x(k) = φ(k, x ) = Ax(k 1) = A k x (3.14) x(k) = φ(k, x ) = A k x (3.18) (3.18) 2.2 x(k) = µ k h (3.19)

60 3 52 (3.14) Ah = µh (3.2) (3.19) (3.2) 2.2 {µ 1, µ 2,, µ n } = { µ i C det(µ i I n A) = } x(k) = c 1 µ k 1h 1 + c 2 µ k 2h c n µ k nh n (3.21) c i h i µ i (3.18) 1 (3.21) x() E i = {c R n x = ah i, a R} 3.2 x(k + 1) = 2ax(k) + y(k) y(k + 1) = bx(k) (3.22) A A = 2a 1 (3.23) b χ(µ) = µi 2 A = µ 2 + 2aµ + b = (3.24) 2 (1) 2 a 2 > b µ 1 = a + a 2 b, µ 2 = a a 2 b (3.25) Ah = µ 1 h, [A µ 1 I 2 ] h 1 = h 1 1 = h 2 h 2 a + a 2 b Ah = µ 2 h, [A µ 2 I 2 ] h 1 = h 1 1 = a a 2 b h 2 h 2 (3.26)

61 3 53 x(k) = c 1 ( a + ) k 1 a 2 b y(k) a + + c 2 ( a ) k 1 a 2 b a 2 b a (3.27) a 2 b (2) 2 a 2 < b 2 µ 1 = a + j b a 2, µ 2 = a j b a 2, j = 1 (3.28) 1 Ah = µ 1 h, [ A ( a + j ) ] b a 2 I 2 h 1 = h 2 h 1 1 = a + j (3.29) b a 2 h 2 1 x(k) = ( a + j ) k 1 ( b a 2 y(k) a + j ) k = be jθ 1 b a 2 be jθ = = ( b ) k e jkθ e j(k 1)θ ( ) k cos kθ j sin kθ = b cos(k 1)θ j sin(k 1)θ ( ) k cos kθ ( ) k sin kθ b j b cos(k 1)θ sin(k 1)θ (3.3) x(k) ( ) k cos kθ ( ) k sin kθ = c 1 b + c 2 b (3.31) y(k) cos(k 1)θ sin(k 1)θ θ = tan 1 b a 2 a (3) a 2 = b µ 1 = µ 2 = a (3.32)

62 3 54 N N = A µ 1 I 2 = A + ai 2 = a 1 a 2 a N 2 =, AN = NA A k = ( ai 2 + N) k = ( a) k I2 k + k( a) k 1 N = ( a) k 1 + k ak (3.33) k a (3.34) 1 k x(k) = ( a) k 1 + k y(k) ak k a c 1 = c 1 ( a) k 1 + k k + c 2 ( a) k a (3.35) 1 k c 2 ak 1 k (3.14) A A A A det(a) > (3.36) det(a) < (3.37) det(a) = (3.38) rank A V 1 V V 1 = det(a)v (3.39) det(a) < 1 (3.4) A

63 A det(a) = det 2a 1 = b (3.41) b b > b < b = (3.22) 1 y = x x x(k + 1) = 2ax(k) (3.42) b < x R n (3.2) 1 (3.1) A = Df(x ) χ(µ) = det (µ i I n Df(x )) = µ n + a 1 µ n a n 1 µ + a n = (3.43) {µ 1, µ 2,..., µ n } = { µ i C det[µ i I n Df(x )] = } (3.44) A x (hyperbolic) (simple) A µ i, µ i = 1, i = 1, 2,..., n (3.45) µ i > 1 µ i R n E u µ i < 1 µ i R n E s (3.2) k = u R n x(k) = ϕ(k, u) (3.46)

64 3 56 E u W u x E s W s 3.1 x() = ϕ(, u) = u W u (x ) = {u R n lim k ϕ(k, u) = x } W s (x ) = {u R n lim k ϕ(k, u) = x } (3.47) x 2 x E u E s x 3.1 #{ } (a) dim E u = #{µ i C µ i > 1} (b) dim E s = #{µ i C µ i < 1} (3.48) (c) E u E s = R n, W u (x ) W s (x ) = x A (Hartman) (Grobman) 2 3 (1) A det A

65 3 57 (2) A L u = A E u= Df(x ) E u, L s = A E s= Df(x ) E s (3.49) det L u, det L s (3) dim W u (x ) = dim L u (2) (3) (3.48) D I (1) det L u > D (directly unstable subspace) (2) det L u < I (inversely unstable subspace) (3) det L s > D (directly stable subspace) (4) det L u < I (inversely stable subspace) dim W u (x ) = m (3.5) det A = det L u det L s > (3.51) det L u > m D det L u < m I m D m I (3.51) m = m = n D D I 2n md, m =, 1, 2,..., n; mi, m = 1, 2,..., n 1 (3.52)

66 (completely stable) D µ 1 < 1, µ 2 < 1 (directly unstable) 1D < µ 1 < 1 < µ 2 (inversely unstable) 1I µ 1 < 1 < µ 2 < (completely unstable) 2D µ 1 > 1, µ 2 > (completely stable) D µ 1 < 1, µ 2 < 1, µ 3 < 1 1 (directly unstable) 1D µ 1 < 1, µ 2 < 1, 1 < µ 3 1 (inversely unstable) 1I µ 1 < 1, 1 < µ 2 <, < µ 3 < 1 2 (directly unstable) 2D < µ 1 < 1, 1 < µ 2, 1 < µ 3 2 (inversely unstable) 2I µ 1 < 1, 1 < µ 2 <, 1 < µ 3 (completely unstable) 3D µ 1 > 1, µ 2 > 1, µ 3 > 1 A(P) A(R) A(C) C C P A(C) Q D 1D Q 1I 2D A(Q) R P R A(Q) A(R) A(P) (a) (b) (c) (d) (sink) (source) (saddle)

67 (3.43) (, 1) (Routh array) (Schur-Cohn) 2, 3 (1) A det A = ( 1) n a n > (3.53) (2) det L u (, 1) χ( 1) (a) n (a-1) χ( 1) > µ < 1 det L u > D (a-2) χ( 1) < µ < 1 det L u < I (b) n (b-1) χ( 1) > µ < 1 det L u < I (b-2) χ( 1) < µ < 1 det L u > D 2 χ(µ) = µ 2 + a 1 µ + a 2 = (3.54) (3.54) (a 1, a 2 )- 3.3 A A det A = det L u det L s < (3.55)

68 (completely stable) D < χ(1)χ( 1), < a 2 < 1 (directly unstable) 1D χ(1) <, < χ( 1), < a 2 (inversely unstable) 1I χ( 1) <, < χ(1), < a 2 (completely unstable) 2D < χ(1)χ( 1), 1 < a 2 χ(1)= 1D 2D a 2 1 D 1I χ(-1)= -1 1 a n mrd, m =, 1, 2,..., n 1; mri, m = 1, 2,..., n (3.56) (reverse) R , 1.8 (1.28) ( f(x) = α (1 x)x = α x 1 ) 2 + α 2 4 (3.57) α > x > 1/2 (1.59) (1.6) b >

69 3 61 x(k+1) x(k+1)=6(1-x(k))x(k) 1 x(k+1)=x(k) x 2 x 1 x 1 x(k) (3.2) x ɛ > δ > u x < δ u R n x(k) = ϕ(k, u) k ϕ(k, u) x < ɛ ɛ >, δ >, u x < δ, k, ϕ(k, u) x < ɛ (3.58) x (asymptotic stable) 2 (1) x (2) ɛ >, u x < ɛ, lim ϕ(k, u) = x (3.59) k x(k + 1) = 6 (1 x(k)) x(k) (3.6) 3.4 x = 1

70 l l n 2n Levinson[P8] [N7] [P11] [D18] [E3] x R n (3.2) l- l- ξ(k + l) = Aξ(k) (3.61) l 1 A = D l f(x ) = Df(x i ) (3.62) i=

71 3 63 x i (3.4) f(x ) = x 1, f(x 1 ) = x 2,..., f(x l 1 ) = x (3.63) 3.2 K 95% 5% k K x(k) K y(k) a, b 3.6 x(k + 1) = y(k) + ax(k) y(k + 1) = x 2 (k) + b (3.64)

72 64 4 5, R n ẋ = f(x) (4.1) t t R x x M R n f : M R n ; x f(x) (4.1) L x(t + L) = x(t), t R (4.2) L L 2L, 3L,...

73 4 65 L (4.1) x(t) = ϕ(t, x ), x() = ϕ(, x ) = x (4.3) (4.2) x (4.3) (4.2) x(l) = ϕ(l, x ) = x() = x (4.4) L x γ(x ) = {x R n x(t) = ϕ(t, x ), t [, L]} (4.5) (periodic orbit) (closed orbit) (4.4) (4.3) (orbit) γ(x ) = {x R n x(t) = ϕ(t, x ), t R} (4.6) γ + (x ) = {x R n x(t) = ϕ(t, x ), t [, )} (4.7) (positive semi-orbit) γ (x ) = {x R n x(t) = ϕ(t, x ), t (, ]} (4.8) (negative semi-orbit) γ(x ) = γ + (x ) γ (x ) (4.9) L limit cycle ẋ = y + δ(a 2 x 2 y 2 )x ẏ = x + δ(a 2 x 2 y 2 )y (4.1) δ = x(t) = R cos(t + θ), y(t) = R sin(t + θ) (4.11)

74 4 66 y A x 4.1 (4.1) R θ δ R θ (4.11) (x, y) (R, θ) ẋ = Ṙ cos(t + θ) R(1 + θ) sin(t + θ) ẏ = Ṙ sin(t + θ) R(1 + θ) cos(t + θ) (4.12) R θ Ṙ = δ(a 2 R 2 )R θ = (4.13) θ(t) = θ R δ > < R < A Ṙ > A < R Ṙ < R = R = A (4.1) x(t) = A cos(t + θ ), y(t) = A sin(t + θ ) (4.14) (4.14) (4.14) (orbitally stable) 4.1 (4.1) ẍ ɛ(a 2 x 2 ẋ 2 )ẋ + x = (4.15)

75 4 67 V h V = -mgl cos θ θ y -π π θ 4.2 (4.14) ẍ ɛ(1 x 2 )ẋ + x = (4.16) ẍ ɛ(1 ẋ 2 )ẋ + x = (4.17) , 1.5, 1.6, 2.7 (1.65)

76 4 68 (1) h < h < mgl (θ, p) H(p, θ) = θ cos 1 p2 mgl cos θ = h < mgl (4.18) 2ml2 h mgl, p 2ml 2 (h + mgl) (4.19) (2) h h > mgl H(p, θ) = p2 mgl cos θ = h > mgl (4.2) 2ml2 p = ± 2ml 2 (h + mgl cos θ) (4.21) 4.2 L (4.3) ξ(t) x(t) = ϕ(t, x ) + ξ(t) (4.22) (4.22) (4.1) dx(t) dt = {dϕ(t, x ) + ξ(t)} dt = f (ϕ(t, x ) + ξ(t)) (4.23) ξ(t) ξ(t) f ϕ(t, x ) dξ(t) dt = A(t)ξ(t) (4.24) A(t) = Df (ϕ(t, x )) = f x (ϕ(t, x )) (4.25) dϕ(t, x ) dt = f (ϕ(t, x )) (4.26)

77 4 69 (4.25) (4.24) L A(t + L) = A(t), t R (4.27) (4.24) (4.24) (4.2) (Poincaré) L (4.3) x Π 1 Π q : R n R; x q(x) (4.28) Π = {x R n q(x) = } (4.29) x Π x(t) = ϕ(t, x ) (4.3) Π q(x ) x f(x ) = f(x ) grad q(x ) (4.31) f (4.31) x Π (4.3) (local cross section) x Π Π Π T T : Π; x ϕ(τ, x) (4.32) τ x Π ϕ(t, x) Π τ x τ (first return time)

78 4 7 φ(t, x 1 ) R n Π x x 1 x = φ(l, x ) x 2 x 2 = φ(τ(x 1 ), x 1 ) φ(t, x ) 4.3 T (Poincaré map) (return map) 4.3 T T (x ) = x (4.33) T τ(x ) = L (4.34) (4.3) T x 1: (4.1) (4.13) Π x Π = {(x, y) R 2 y =, x > } (4.35) x x (4.13) 1 2π Π x 1 Π (4.13) 1 R 2 (t) = A Ke 2δA2 t (4.36)

79 4 71 T(x 2 ) x 2 x 2 1 A 2 x (4.37) K R x x Π x 2 1 = A 2 x 2 x 2 + (A2 x 2 )e 4πδA2 = T (x 2 ) (4.37) Π T T A 2 δ > R n ẋ = f(t, x) (4.38) f L f(t + L, x) = f(t, x) (4.39) x R n (4.38) x(t) = ϕ(t, x ), x() = ϕ(, x ) = x (4.4) K (4.38) ϕ(t + K, x ) = ϕ(t, x ), t R (4.41)

80 4 72 R n x ϕ(t, x ) R n t = x 1 t 4.5 t = L K K = L (fundamental harmonic solution) K = kl k 1/k (subharmonic solution of order k) (4.39) T T : R n R n ; x x 1 = ϕ(l, x ) (4.42) (stroboscopic mapping) L (time L map) 4.5 T (4.38) 4.4 (Duffing) 1 ẍ + kẋ + f(x) = B cos t (4.43) k > f(x) f(x) ± (x ± ) (4.43) 1 2π 2 ẋ = y ẏ = ky f(x) + B cos t (4.44)

81 ẋ = A(t)x (4.45) (4.45) x R n A(t) t L A(t + L) = A(t) (4.46) (4.45) Ẋ = A(t)X (4.47) (4.47) (principal matrix solution) Φ(t) Φ() = I n (4.47) Φ(t) (4.45) (4.47) x(t) = Φ(t)c X(t) = Φ(t)C (4.48) x() = c, X() = C Ẋ(t + L) = A(t + L)X(t + L) = A(t)X(t + L) (4.49) X(t + L) (4.47) X(t + L) Φ(t) X(t + L) = Φ(t)D (4.5) X(t + L) = Φ(t)D = X(t)C 1 D = X(t)F (4.51) (4.48) X(t) C D F F X(t) (fundamental matrix) Φ(t) E Φ(t + L) = Φ(t)E, E = Φ(L) (4.52) Φ(t + L) = Φ(t)Φ(L) (4.53) X(t) F X(t + L) = Φ(t + L)C = Φ(t)Φ(L)C = X(t)C 1 EC (4.54)

82 4 74 F = C 1 EC (4.55) (4.55) (det X(t) ) χ(µ) = det(µi n Φ(L)) = (4.56) µ (characteristic multiplier) ν = 1 L log e µ (characteristic exponent) ν 2πj/L Θ(t) Θ(t + L) = Θ(t)J (4.57) Θ(t) (4.47) (normal solution) (4.56) J J = diag(µ 1, µ 2,..., µ n ) Θ(t) Θ(t) = (4.45) [ ] x (1) (t) x (2) (t) x (n) (t) x (i) (t + L) = µ i x (i) (t), i = 1, 2,..., n (4.58) (Floquet) 1 µ i h i Eh i = µ i h i, i = 1, 2,..., n (4.59) EH = HJ (4.6) H = [h 1 h 2 h n ], J = diag(µ 1, µ 2,..., µ n ) Θ(t) = Φ(t)H (4.61) Θ(t + L) = Φ(t + L)H = Φ(t)Φ(L)H = Θ(t)H 1 EH = Θ(t)J

83 4 75 (4.1) 1 1 (4.1) L (4.3) (4.1) (4.3) (4.24) (4.24) Φ(t) (4.3) (4.1) ϕ(t, x ) = f (ϕ(t, x )) (4.62) ϕ(t, x ) = f (ϕ(t, x )) x ϕ(t, x ) = A(t) ϕ(t, x ) (4.63) ϕ(t, x ) (4.24) Φ(t) ϕ(t, x ) = Φ(t) ϕ(t, x ) (4.64) ϕ(t + L, x ) = Φ(t + L) ϕ(t, x ) = Φ(L) ϕ(t, x ) (4.65) Φ(t ) = I n ϕ(t, x ) ϕ(t + L, x ) = ϕ(t, x ) (4.66) (4.65) (4.66) Φ(L) ϕ(t, x ) = ϕ(t, x ) (4.67) Φ(L) n 1 (Liouville) det Φ(L) = det Φ()e R L tracea(τ)dτ = e R L tracea(τ)dτ (4.68) A(t) = [a ij (t)] (trace) trace A(t) = (4.56) (4.68) n a ij (t) (4.69) i=1 det Φ(L) = µ 1 µ 2 µ n = e R L tracea(τ)dτ > (4.7)

84 µ 1 = 1 µ 2 µ n = e R L tracea(τ)dτ > (4.71) ẋ = f(x, y) ẏ = g(x, y) (4.72) L x(t) = φ(t), y(t) = ψ(t) (4.73) ξ = a(t)ξ + b(t)η η = c(t)ξ + d(t)η (4.74) a(t) = f f (φ(t), ψ(t)), b(t) = (φ(t), ψ(t)) x y c(t) = g g (φ(t), ψ(t)), d(t) = (φ(t), ψ(t)) x y (4.75) (4.74) µ (4.71) µ < 1 L (a(τ) + d(τ)) dτ = ( ) L µ = exp (a(τ) + d(τ)) dτ L (4.76) { f g } (φ(τ), ψ(τ)) + x y (φ(τ), ψ(τ)) dτ < (4.77) (Floquet-Lyapounov) Φ(t) Φ(L) Φ(L) = e KL (4.78) K A(t), Φ(L) K F (t) = Φ(t)e Kt (4.79)

85 4 77 F (t + L) = Φ(t + L)e K(t+L) = Φ(t)Φ(L)e KL e Kt = Φ(t)e Kt = F (t) Φ(t) F (t) e Kt Φ(t) = F (t)e Kt (4.8) (4.47) Φ(t) F (t), K Φ(t) = F (t)e K t (4.81) K F (t) F (t + 2L) = F (t) 2L n n K Φ(L) 2 = e 2LK (4.82) K F (t) = Φ(t)e K t (4.83) F (t + 2L) = Φ(t) { Φ(L) } 2 e K (t+2l) = Φ(t)e 2K L e 2K L e K t = Φ(t)e K t = F (t) (4.84) (4.81) 2 (4.8) F (t) x = F (t)y (4.85) (4.45) K ẏ = Ky (4.86) (4.8), (4.85) 4.6 (Mathieu) 2 ẍ + (a + b cos 2t)x = (4.87)

86 4 78 (4.87) ẋ = y ẏ = (a + b cos 2t)x (4.88) t = u = (x, y ) R 2 x(t) = ϕ(t, u ), y(t) = ψ(t, u ) (4.89) π π T : R 2 R 2 ; u = (x, y ) u 1 = (ϕ(π, u ), ψ(π, u )) (4.9) Φ(t) = ϕ 1(t) ψ 1 (t) ϕ 2 (t), Φ() = I 2 (4.91) ψ 2 (t) u 1 = Φ(π)u (4.92) T = Φ(π) (4.93) (4.93) χ(µ) = det (µi 2 Φ(π)) = µ ϕ 1(π) ψ 1 (π) ϕ 2 (π) µ ψ 2 (π) = µ 2 { ϕ 1 (π) + ψ 2 (π) } µ + det Φ(π) = (4.94) det Φ(π) = det Φ()e R π tracea(τ)dτ = 1 (4.95) A(t) (4.88) (4.94) χ(µ) = µ 2 { ϕ 1 (π) + ψ 2 (π) } µ + 1 = (4.96) (4.96) m = ϕ 1 (π) + ψ 2 (π) (4.97) 3

87 4 79 (1) m > 2 < µ 1 < 1 < µ 2 µ 1, µ 2 h 1, h 2 H = [h 1 h 2 ] Θ(t) = Φ(t)H, Θ(t + π) = Θ(t) µ 1 (4.98) µ 2 ν 1 = 1 π log µ 1, ν 2 = 1 π log µ 2 (4.99) (4.98) Θ(t) = F (t) (4.1) e ν 2t eν 1t F (t + π) = F (t) F (t) = Θ(t) e ν 1t = Φ(t)H e ν 1t (4.11) e ν 2t e ν 2t (2) m < 2 µ 1 < 1 < µ 2 < Θ(t) = Φ(t)H, Θ(t + 2π) = Θ(t) µ2 1 (4.12) µ 2 2 ν 1 = 1 2π log µ2 1, ν 2 = 1 2π log µ2 2 (4.13) Θ(t) = F (t) (4.14) e ν 2t eν 1t 2π F (t + 2π) = F (t) (4.15) (4.87)

88 4 8 2 m=-1 15 m= bm= a (4.87) m 1/2 (parametric excitation) 4.6 a, b (4.97) m m a = k 2, k = 1, 2,... a = 1 (3) m < 2 µ 1, µ 2 = e jθ { 4 { ϕ θ = tan 1 1 (π) + ψ 2 (π) } 2 } ϕ 1 (π) + ψ 2 (π) (4.16) (4.93) (4.95)

89 n 2n md, m =, 1, 2,..., n; mi, m = 1, 2,..., n 1 D n 1 n md, m =, 1, 2,..., n 1; mi, m = 1, 2,..., n 2 2(n 1) D 1 1 D I m D, m I 2 l- m D l, m I l l- T l { m Dl, m D1, l..., m Dl 1 l } (4.17)

90 4 82 md l k+1 = T ( md l k), k mod l 3 1 l- l 1 (4.17) 1 (group) l- l- g i md l k, i = 1, 2,..., g 4 D 1 D l D S 1 D D 1 I I 2 D U D 2 1 S, 2 S (4.44) ẋ = y, ẏ =.1y x cos t (4.18) S 2 S D D 1 D x D (t) =.922 cos t sin t.149 cos 3t sin 3t +.1 cos 5t +.8 sin 5t + x 1S (t) =.3219 cos t sin t.9 cos 3t +.3 sin 3t + (4.19) x 2S (t) = cos t sin t cos 3t sin 3t.2 cos 5t +.32 sin 5t.2 cos 7t S

91 4 83 y 1 2 S D 1 S -1 1 x Duffing (4.18) 4.1 (x, y ) µ 1, µ 2 D (.917,.3812) 3.763,.142 ( 1 D) 1 S (.3228,.36).586 ±.436j ( D) 2 S (1.1381,.7446).171 ±.71j ( D)

92 4 84 z S 1 2 I y S 2 2 x 4.8 Rössler (4.11) 4.8 (Rössler) ẋ = y z ẏ = x + ay ż = bx cz + xz (4.11) a =.6, b =.5, c = y =. I 2 S1, 2 S

93 [O11] [O14] Hale[O5] Yakubovich [O15] McLachalan[O1] Champbell[O3] [N14] Stern[N18] Migulin[N12] [N8] Rössler[P9] (4.32) x k > T 4.4 (4.68) 1 (1.7) (1.82) 4.1 (4.42)

94 RLC (forcing term) 5.1 RLC i(t) v(t) C dv dt L di dt = i = Ri v + e(t) (5.1) v + 2ζ v + ω 2 v = ω 2 e(t) (5.2)

95 5 87 R L i e(t) C v 5.1 RLC ζ = R 2L, ω = 1 LC (5.3) A. ( e(t) = ) 2 v + 2ζ v + ωv 2 = (5.4) χ(µ) = µ 2 + 2ζµ + ω 2 = (5.5) ω > ζ µ 1 = µ 2 = ζ + j ω 2 ζ2 = ζ + jω d (5.6) ω d = ω 2 ζ2 (5.4) v(t) = V e ζt cos(ω d t + θ) i(t) = C v = Ie ζt cos(ω d t + θ δ) (5.7) V, θ I = CV ζ 2 + ω 2 d = ω CV, δ = tan 1 ω d ζ (5.8) v(t) i(t) (5.1) 1 B. e(t) = E (5.2) v + 2ζ v + ω 2 v = ω 2 E (5.9)

96 5 88 v s (t) E (5.7) (5.1) v s (t) = E (5.1) C. e(t) = E cos ωt v + 2ζ v + ωv 2 = ωe 2 cos ωt (5.11) 1 (5.11) v s (t) = V s cos(ωt φ) (5.12) V s = ωe 2, φ = tan 1 2ζω (ω 2 ω2 ) 2 + (2ζω) 2 ω 2 (5.13) ω2 (5.11) v(t) = V e ζt cos(ω d t + θ) + v s (t) (5.14) 1 (5.12) (5.11) ζ > 2π/ω D. (5.12) v s (t) (5.13) ω, ζ ζ ω ω π ζ =, ω = ω (resonance) (nonresonance) (5.13) ν = ω, Q = ω ω 2ζ = ω L R = 1 L R C V s E G G = V s E = ω ω (ω ω ω ω 1 ) 2 ( ) = 2 2ζ + ω 1 (ν 1) 2 + ( ) (5.15) 2 ν Q

97 Q=1 G Q=5 Q=3 Q=2 Q= ν 5.2 Q Q (5.15) 5.2 ν = 1 G = Q Q G (5.11) v + ωv 2 = ωe 2 cos ω t (5.16) v(t) = V cos(ω t + θ) ω te sin ω t (5.17) t 5.3 (secular term) E. (5.2) 2π/ω e(t) e(t) = E + (E ck cos kωt + E sk sin kωt) (5.18) k=1

98 5 9 v t 5.3 (5.16) E = E ck = ω π E sk = ω π 2π ω ω e(τ)dτ 2π 2π ω 2π ω e(τ) cos kωτdτ e(τ) sin kωτdτ (5.19) (a) ζ C (5.4) (b) ζ = v + ω 2 v = ω 2 { E + } (E ck cos kωt + E sk sin kωt) k=1 (5.2) v + ω 2 v = (5.21) 2π/ω (5.21) (b-1) ω = ω (harmonic resonance) (5.2)

99 5 91 v t 5.4 E c1 = ω π E s1 = ω π 2π ω 2π ω e(τ) cos ω τdτ = e(τ) sin ω τdτ = (5.22) (5.2) 2π/ω ( Eck v s (t) = E + 1 k 2 cos kω t + E ) sk 1 k 2 sin kω t k=2 M, N (5.23) v(t) = M cos ω t + N sin ω t + v s (t) (5.24) (b-2) ω/ω n/m ( ) ω 2 v(t) = M cos ω t + N sin ω t + E + E ck ω 2 cos kωt + ω2 E sk k2 ω2 ω 2 sin kωt k2 ω2 k=1 (5.25) M, N 2π/ω (M = N = ) M N ω ω 2 (quasi-periodic function) (beat oscillation) 5.4 (b-3) ω/ω = n/m (subharmonic resonance) (5.25) ω = nω/m v(t) = M cos m n ωt + N sin m n ωt + E + k=1 ( m 2 E ck m 2 k 2 cos kωt + m2 n2 ) E sk m 2 k 2 sin kωt n2 (5.26)

100 5 92 2π = ω m 2π n ω = n 2π m ω m = 1 1/n- n = 1 m- mω m 1, n ɛ ( ɛ 1 ) ɛ n ẋ = Ax + f(t) + ɛf (x, ẋ, t) (5.27) n ẋ = Ax + ɛf (x, ẋ) (5.28) ɛ = (weakly nonlinear system) (quasi linear system) A. ɛ ẍ + Ω 2 x = ɛf (x, ẋ, ɛ, t) (5.29) 2π f(t + 2π) = f(t), F (x, ẋ, ɛ, t + 2π) = F (x, ẋ, ɛ, t) (5.3) F (5.29) ɛ x(t) = x (t) + ɛx 1 (t) + ɛ 2 x 2 (t) + ɛ 3 x 3 (t) + (5.31)

101 5 93 ɛ F (x, ẋ, ɛ, t) = F (x, ẋ,, t) + ɛ[f 1 1x 1 + F 1 1ẋ 1 + F 1 1] + ɛ 2[ F 1 1x 2 + F 1 1ẋ F 2 2x F 2 2ẋ F 2 11x 1 ẋ 1 + F 2 11x 1 + F 2 11ẋ F p+q+r F x p ẋ q ɛ r (x, ẋ,, t) = Fpqr p+q+r (5.32) (5.31) (5.29) (5.32) ɛ ɛ : ẍ + Ω 2 x = f(t) ] ɛ 1 : ẍ 1 + Ω 2 x 1 = F (x, ẋ,, t) = F 1 ɛ 2 : ẍ 2 + Ω 2 x 2 = F 1 1x 1 + F 1 1ẋ 1 + F 1 1 = F 2 (5.33) ɛ 2 : ẍ 3 + Ω 2 x 3 = F 1 1x 2 + F 1 1ẋ F 2 2x F 2 2ẋ 2 1 +F11x 2 1 ẋ 1 + F11x F11ẋ F 2 2 = F 3 1 F k x 1, x 2,..., x k 1 F k k ẍ k + Ω 2 x k = F k (5.34) f(t) f(t) = a + (a k cos kt + b k sin kt) (5.35) k=1 (1) Ω (5.33) 1 x (t) = a Ω 2 + k=1 ( ak b Ω 2 cos kt + k2 (5.33) 2 ) k Ω 2 sin kt = ϕ(t) (5.36) k2 ẍ 1 + Ω 2 x 1 = F (ϕ(t), ϕ(t),, t) (5.37)

102 5 94 2π x (t) x k (t) (5.31) ɛ (ϕ(), ϕ()) (2) Ω m m 2 Ω 2 = ɛa (5.38) (5.35) mω a m = ɛα, b m = ɛβ (5.39) (5.27) ẍ + m 2 x = g(t) + ɛg(x, ẋ, ɛ, t) (5.4) g(t) = f(t) (a m cos mt + b m sin mt) = a + (a k cos kt + b k sin kt) k=1,k m G(x, ẋ, ɛ, t) = F (x, ẋ, ɛ, t) + (α cos mt + β sin mt) + ax (5.41) (5.4) (5.31) (5.32), (5.33) ɛ ẍ + m 2 x = g(t) (5.42) x (t) = M cos mt + N sin mt + a m 2 + = M cos mt + N sin mt) + ϕ(t) k=1,k m a k b k ( m 2 cos kt + k2 m 2 sin kt) k2 (5.43) M, N x (t) (5.43) x 1 (t) ẍ 1 + m 2 x 1 = G(x, ẋ,, t) = G 1 (5.44) (5.44) P (M, N ) = Q(M, N ) = 2π 2π G(x, ẋ,, τ) sin mτdτ = G(x, ẋ,, τ) cos mτdτ = (5.45)

103 5 95 M, N (5.43) (5.44) G(x, ẋ,, t) = G 1 (t) = a 1 + x 1 (t) = M 1 cos mt + N 1 sin mt + a 1 m 2 + M 1, N 1 x 2 (t) k=1,k m k=1,k m (a k1 cos kt + b k1 sin kt) (5.46) a k1 ( m 2 cos kt + b k1 k2 m 2 sin kt) (5.47) k2 ẍ 2 + m 2 x 2 = G 1 1x 1 + G 1 1ẋ 1 + G 1 1 = G 2 (5.48) M k, N k (5.45) P P det M N Q Q (5.49) M N (5.31) (x 1 (), ẋ 1 ()) , 4.7 ẍ + ɛζẋ + Ω 2 x + ɛcx 3 = B cos t (5.5) ẍ + Ω 2 x = B cos t ɛ(ζẋ + cx 3 ) (5.51) x(t) = x (t) + ɛx 1 (t) + ɛ 2 x 2 (t) + ɛ 3 x 3 (t) + (5.52) (1) Ω 1 (5.33) x (t) = x 1 (t) = B Ω 2 1 cos t ζb (Ω 2 1) 2 sin t 3cB 3 4(Ω 2 1) 4 cos t cb 3 4(Ω 2 1) 3 (Ω 2 cos 3t 9) (5.53)

104 5 96 (2) Ω 1 1 Ω 2 = ɛa, B = ɛb (5.54) (5.51) ẍ + Ω 2 x = ɛ(b cos t + ax ζẋ cx 3 ) (5.55) (5.52) ɛ : ẍ + x = ɛ 1 : ẍ 1 + x 1 = b cos t + ax ζẋ cx 3 (5.56) x (t) = M cos t + N sin t (5.57) (5.56) 2 { ẍ 1 + x 1 = ζm + (a 34 ) } cr2 N sin t + { ( a 3 ) } 4 cr2 M ζn + b cos t c(n 2 3M 2 )N sin 3t c(3n 2 M 2 )M cos 3t (5.58) r 2 = M 2 + N 2 P (M, N ) = ζm + Q(M, N ) = (a 34 cr2 ) N = (a 34 cr2 ) M ζn + b = (5.59) x 1 (t) = M 1 cos t + N 1 sin t 1 32 c(n 2 3M 2 )N sin 3t 1 32 c(3n 2 M 2 )M cos 3t (5.6) (5.59) { ( a 3 ) } 2 4 cr2 + ζ 2 r 2 = b 2 (5.61) (5.61) (b, r)- (a, r)- 5.5, 5.6 b a r

105 ζ =.1 1O O r O b 5.5 (5.61) (a = c = 1.) b = 1..5 O 1O r b = O a 5.6 (5.61) (c = 1, ζ =.2)

106 5 98 (5.31) (5.38) - (5.4) (5.31) x(t) = x (t) + ɛx 1 (t) + ɛ 2 x 2 (t) + ɛ 3 x 3 + = ϕ (t) (5.62) x(t) = ϕ (t) + ξ(t) (5.63) ξ + m 2 ξ = ɛ G x ξ + ɛ G ẋ ξ (5.64) ξ (1) (t) = ξ (1) (t) + ɛξ(1) 1 (t) + ɛ2 ξ (1) 2 (t) + (5.65) ξ (2) (t) = ξ (2) (t) + ɛξ(2) 1 (t) + ɛ2 ξ (2) 2 (t) + ξ (1) ξ (2) () = 1, ξ(1) () =, ξ(2) () =, ξ(1) k (1) () = ξ k () =, (5.66) (2) () = 1, ξ(2) k () = ξ k () =, k = 1, 2,... (5.65) (5.64) ɛ ξ (1) + m 2 ξ (1) = ξ (2) + m 2 ξ (2) = ξ (1) 1 + m 2 ξ (1) 1 = ɛ G(x, ẋ,, t) ξ (1) + ɛ G(x, ẋ,, t) x ẋ ξ(1) ξ (2) 1 + m 2 ξ (2) 1 = ɛ G(x, ẋ,, t) ξ (2) + ɛ G(x, ẋ,, t) x ẋ ξ(2) (5.67) (5.67) 2 ξ (1) = cos mt, ξ (2) = 1 sin mt (5.68) m 2 ξ (1) 1 (t) = 1 [ ] t G(x, ẋ,, τ) cos mτ m G(x, ẋ,, τ) sin mτ sin m(t τ)dτ m x ẋ ξ (2) 1 (t) = 1 [ ] t G(x, ẋ,, τ) m 2 sin mτ + m G(x, ẋ,, τ) cos mτ sin m(t τ)dτ x ẋ (5.69)

107 5 99 µ ξ χ(µ) = (1) (2π) ξ (2) (2π) ξ (1) (2π) µ ξ (2) (2π) = µ2 + a 1 µ + a 2 = (5.7) a 1 = { ξ (1) (2π) + ξ (2) (2π) } a 2 = ξ (1) (2π) ξ (2) (2π) ξ (2) (2π) ξ (1) (2π) (5.71) (5.71) (5.65) a 1 = 2 ɛ { ξ (1) (2) 1 (2π) + ξ a 2 = 1 + ɛ { ξ (1) (2) 1 (2π) + ξ 1 (2π)} +ɛ 2{ ξ (1) 2 (2π) + ξ (2) 2 1 (2π)} ɛ 2{ ξ (1) 2 (2π) + ξ(1) 1 (2π) ξ (2) 1 (2π) + ξ (2) 2 (2π)} (1) (2π) ξ(2) 1 (2π) ξ 1 (2π)} + (5.72) χ( 1) = 1 a 1 + a 2 = 4 + ɛ{ } + > χ(1) = 1 + a 1 + a 2 = ɛ 2{ ξ (1) 1 (2π) ξ (2) 1 a 2 = 1 + ɛ { ξ (1) (2) 1 (2π) + ξ 1 (2π)} + } (1) (2π) ξ(2) 1 (2π) ξ 1 (2π) + ɛ 3 { } + (5.73) (5.69) ξ (1) 1 (2π) = 1 [ ] 2π G(x, ẋ,, τ) cos mτ m G(x, ẋ,, τ) sin mτ sin mτdτ m x ẋ = 1 P m M ξ (1) 1 (2π) = 2π = Q M ξ (2) 1 (2π) = 1 m 2 2π [ G(x, ẋ,, τ) cos mτ m G(x, ẋ,, τ) sin mτ x ẋ = 1 P m 2 N ξ (2) 1 (2π) = 1 2π m = 1 Q m N ] cos mτdτ [ ] G(x, ẋ,, τ) sin mτ + m G(x, ẋ,, τ) cos mτ sin mτdτ x ẋ [ G(x, ẋ,, τ) sin mτ + m G(x, ẋ,, τ) cos mτ x ẋ ] cos mτdτ (5.74)

108 A ( D) < χ(1), < a 2 < 1 < det A, tracea < ( 1 D) χ(1) <, < a 2 det A < ( 2 D) < χ(1), 1 < a 2 < det A, < tracea (5.43) (5.45) (5.73) χ(1) = ɛ2 m 2 det A + ɛ3 { } + a 2 = 1 + ɛ m tracea + ɛ3 { } + (5.75) P M A = Q M P N Q (5.76) N ɛ χ( 1) > I (5.75) (5.59) tracea = 2ζ < det A = a 2 + ζ 2 3acr c2 r 4 (5.77) (5.61) db 2 dr 2 = a2 + ζ 2 3acr c2 r 4 = det A (5.78) db 2 /dr 2 > db 2 /dr 2 < 3 r (resonant state) (non-resonant state)

109 ɛ 2 ẍ + x = ɛg(x, ẋ, ɛ, t) (5.79) G(x, ẋ, ɛ, t + 2π) = G(x, ẋ, ɛ, t) (5.8) (5.79) 1 ẋ = y ẏ = x + ɛg(x, y, ɛ, t) (5.81) ɛ = (5.81) x(t) = u cos t + v sin t y(t) = u sin t + v cos t (5.82) ɛ ( < ɛ 1) x(t) = u(t) cos t + v(t) sin t y(t) = u(t) sin t + v(t) cos t (5.83) (5.81) (u, v) ẋ(t) = u(t) cos t + v(t) sin t u(t) sin t + v(t) cos t ẏ(t) = u(t) sin t + v(t) cos t u(t) cos t v(t) sin t (5.84) (5.81) u(t) cos t + v(t) sin t = u(t) sin t + v(t) cos t = ɛg(u cos t + v sin t, u sin t + v cos t, ɛ, t) (5.85) u, v u(t) = ɛg(u cos t + v sin t, u sin t + v cos t, ɛ, t) sin t v(t) = ɛg(u cos t + v sin t, u sin t + v cos t, ɛ, t) cos t (5.86) u, v (5.86) (5.81) ɛ ɛ u, v u, v cos t, sin t (5.86) 2

110 5 12 (5.86) (5.86) [, 2π] u(t) = ɛ 2π G(u cos τ + v sin τ, u sin τ + v cos τ, ɛ, τ) sin τdτ 2π v(t) = ɛ 2π G(u cos τ + v sin τ, u sin τ + v cos τ, ɛ, τ) cos τdτ 2π (5.87) (5.87) (method of averaging) (5.86) t 2π t Fourier G(x, y, ɛ, t) sin t = ϕ (u, v) + G(x, y, ɛ, t) cos t = ψ (u, v) + k=1 k=1 (5.87) { } ϕ ck (u, v) cos kt + ϕ sk (u, v) sin kt { } ψ ck (u, v) cos kt + ψ sk (u, v) sin kt u(t) = ɛϕ (u, v) v(t) = ɛψ (u, v) (5.88) (u, v ) (5.83) (5.81) (5.88) u = A cos ϑ v = A sin ϑ (5.89) A, ϑ x = u cos t + v sin t = A cos t cos ϑ + A sin t sin ϑ = A cos(t ϑ) y = u sin t + v cos t = A sin t cos ϑ + A cos t sin ϑ = A sin(t ϑ) 2π u = Ȧ cos ϑ ϑa sin ϑ = ɛ G(x, y, ɛ, τ) sin τdτ 2π v = Ȧ sin ϑ ϑa cos ϑ = ɛ 2π G(x, y, ɛ, τ) cos τdτ 2π Ȧ = ɛ 2π G (A cos(τ ϑ), A sin(τ ϑ), ɛ, τ) sin(τ ϑ)dτ = ɛφ(a) 2π ϑ = ɛ 2π G (A cos(τ ϑ), A sin(τ ϑ), ɛ, τ) cos(τ ϑ)dτ = ɛψ(a) 2π (5.9)

111 5 13 v 2 1 S R N u (5.95) (a = c = 1, b =.3, ζ =.1) A (5.9) 1 A 2 (5.81) Φ(A) = (5.91) ϑ(t) = ɛψ(a )t + ϑ (5.92) { } x(t) = A cos (1 ɛψ(a )) t ϑ { } (5.93) y(t) = A sin (1 ɛψ(a )) t ϑ (5.93) (5.91) (5.9) 1 A dφ(a )/da < dφ(a )/da > (5.55) ẋ = y ẏ = x + ɛ(b cos t + ax ζy cx 3 ) (5.94)

112 5 14 f(a) f(a) = A 1 A (5.98) (5.83) (5.87) (5.88) { u = ɛ ζu + (a 34 ) } 2 cr2 v = ɛ P (u, v) 2 { ( v = ɛ a 3 ) } (5.95) 2 4 cr2 u ζv + b = ɛ2 Q(u, v) P, Q (5.89) (5.95) M, N u, v (5.95) ẍ + ɛ(α βx 2 + x 4 )ẋ + x = (5.96) ɛ, α, β > (5.96) ẋ = y (5.97) ẏ = x ɛ(α βx 2 + x 4 )y (5.9) Ȧ = ɛ 2 A ( α β 4 A A4 ) = ɛφ(a) ϑ = (5.98)

113 β > 2 6α A 1 = 1 3 β β 2 24α, A 2 = 1 3 β + β 2 24α (5.99) 2 A 1 A 2 (5.96) 5.8 (5.98) [N3] [D17] 5.2 (Malkin) [N11] Malkin[N11] Rouche Mawhin[O12] Andronov[N1] Minorsky[N13] Stoker [N19] [N7] A(1) (1) 1 (5.36) ϕ(t) x(t, ξ, η, ɛ) = ϕ(t) + ξa(t) + ηb(t) + ɛc(t) + (5.1) x(, ξ, η, ɛ) = ϕ() + ξ, ẋ(, ξ, η, ɛ) = ϕ() + η (5.11)

114 5 16 ξ, η ϕ(), ϕ() ɛ = ϕ(t) ɛ (5.1) (5.29) (5.1) A(t), B(t) (5.11) A(t), B(t) (2) (5.1) G 1 (ξ, η, ɛ) = x(2π, ξ, η, ɛ) x(, ξ, η, ɛ) = G 2 (ξ, η, ɛ) = ẋ(2π, ξ, η, ɛ) ẋ(, ξ, η, ɛ) = (5.12) A(t), B(t) (5.12) ξ, η ɛ ξ, η ɛ 5.2 ɛ d 2 x dt 2 + x = ɛf ( x, dx dt τ = ωt ( ω 2 d2 x dτ 2 + x = ɛf x, ω dx ) dτ 2π ) (5.13) (5.14) x(τ) = x (τ) + ɛx 1 (τ) + ɛ 2 x 2 (τ) + ω = 1 + ɛω 1 + ɛ 2 ω 2 + (5.15) x () = x () + ɛx 1() + ɛ 2 x 2() + = x () = x 1() = x 2() = = (5.16) τ 5.3 ẍ + x = ɛ(1 x 2 )ẋ (5.17) 5.4 ẍ + (a + ɛ cos 2t)x = (5.18) a = k 2 + ɛa 1 + ɛ 2 a 2 + x C (t) = cos kt + ɛx 1 (t) + ɛ 2 x 2 (t) + x S (t) = sin kt + ɛx 1 (t) + ɛ 2 x 2 (t) + (5.19)

115 5 17 k =, 1, 2 a 1, a 2, x 1 (t), x 2 (t) 5.5 (5.59) c < ẍ ɛ(1 x 2 )ẋ + Ω 2 x = B cos t (5.11) Ω 1 1 Ω = ɛa, B = ɛb ẍ ɛ(1 ẋ 2 )ẋ + x + ɛx 3 = ɛb cos νt (5.111) (5.81) 1

116 M R n dx dt = f(x, λ) (6.1) t x λ t R, x = x 1 x 2. x n λ 1 λ 2 M, λ = R m. λ m f : M R m R n ; (x, λ) f(x, λ) 2 f(x, λ) = (6.2)

117 6 19 (6.1) M (6.2) x (6.2) (1) (2) (3) 1 1 x(k + 1) = x(k) + h(k) Df(x(k), λ)h(k) = f(x(k), λ) (6.3) x(k + 1) = x(k) [ Df(x(k), λ)] 1f(x(k), λ) (6.4) Df(x(k), λ) = f (x(k), λ) (6.5) x (6.2) (6.4) (6.3) 2 h(k) 1

118 6 11 (6.4) det (Df(x(k), λ)) = (6.6) 1 (6.2) (5.94) ẋ = y ẏ = x + ɛ(b cos t + ax ζy cx 3 ) (6.7) (5.83) x(t) = u(t) cos t + v(t) sin t y(t) = u(t) sin t + v(t) cos t (5.95) u = ζu (a 34 ) cr2 v = f(u, v, λ) v = (a 34 cr2 ) u ζv + b = g(u, v, λ) (6.8) (6.9) (6.7) r 2 = u 2 + v 2 (6.1) t ɛt/2 ɛ/2 2 λ = (b, ζ) R 2 (6.11) (5.95) a, c a = c = 1 a b ζ (6.9) f(u, v, λ) = ζu g(u, v, λ) = (a 34 cr2 ) v = (a 34 cr2 ) u ζv + b = (6.12) (6.12) (6.7) (6.8) 6.1

119 (6.9) (6.7) r (6.8) O D 1O 1D r r b r (b, ζ) (1) (6.12) 2 b 2 (5.61) r 2 { ( 1 3 ) } 2 4 r2 + ζ 2 r 2 = b 2 (6.13) r 2 3 r (ζ, b) (6.13) 6.1 (ζ, b) 3 ζ b (b, r) (ζ, b) 1 3 ζ b r 6.2 ζ =.1 r b = b r b = b 2 2 r b r r b b = b 1 b 1 < b < b 2 r 3 2 1

120 6 112 r 1 1 1O O r b 1.5 ζ b 1 b 1 3 ζ (b, ζ) r

121 6 113 r O 1O O b=b 1 b=b 2 b 6.2 ζ =.1 b 6.3 b = b 1 b b = b b ζ (2) (6.12) f f Df(u, λ) = u v g g = ζ uv (u2 + 3v 2 ) 1 3 u v 4 (3u2 + v 2 ) ζ 3 (6.14) 2 uv f u g u f v g v = ζ r r4 = (6.15)

122 6 114 v v v u u u (a) b =.6 (b) b =.12 (c) b =.2 v v u u (d) b =.44 (d) b = ζ =.1 f f (u(k), v(k)) u g g (u(k), v(k)) u (u(k), v(k)) v h 1 (k) f (u(k), v(k), λ) = (6.16) (u(k), v(k)) h 2 (k) g (u(k), v(k), λ) v u(k + 1) = u(k) + h 1 (k) v(k + 1) = v(k) + h 2 (k) (6.17) (3) b r b =.1 r = b

123 6 115 b (6.15) b = b 2 b = 1.5 b (6.15) b = b 1 b b (1) *1 (4) 2 b ζ 2 3 (6.15) (u, v) (b, ζ) 4 f(u, v, b, ζ) = ζu (1 34 ) r2 v = g(u, v, b, ζ) = (1 34 ) r2 u ζv + b = h(u, v, b, ζ) = f u g u f v g v = ζ r r4 = (6.18) ζ (x, y, b) (6.18) (3) (6.15) (6.18) f u g u h u f v g v h v f ζ + 3 b 2 uv (u2 + 3v 2 ) g b = (3u2 + v 2 ) ζ 3 2 uv 1 h 6u + 27 b 4 r2 u 6v r2 v (6.19) 3 f, g 2 ζ (6.18) C (6.19) C (6.18) 4 4 (x, y, b, ζ) (bifurcation diagram) *1 (6.13) b r r b

124 C ζ b M R n ẋ = f(t, x) (6.2) 2π f(t + 2π, x) = f(t, x)

125 6 117 F (t, x, ẋ) = f(t, x) ẋ (6.21) (6.21) 2π x(t) x(t) = ϕ(t) = a + (a k cos kt + b k sin kt) (6.22) k=1 (6.22) (a, b) = (a, a 1, b 1,...) (6.22) (6.21) F (t, ϕ(t), ϕ(t)) = F c + (F ck cos kt + F sk sin kt) (6.23) k=1 (6.21) F c (a, b) = 1 2π F ck (a, b) = 1 π F sk (a, b) = 1 π 2π 2π 2π F (τ, ϕ(τ), ϕ(τ))dτ = F (τ, ϕ(τ), ϕ(τ)) cos kτdτ = k = 1, 2,... F (τ, ϕ(τ), ϕ(τ)) sin kτdτ = (6.24) (6.24) (6.24) (6.22) m m x(t) = ϕ(t) = a + (a k cos kt + b k sin kt) (6.25) k=1 m (6.21) m (6.24) (2m + 1)n F c (a, b) = 1 2π F ck (a, b) = 1 π F sk (a, b) = 1 π 2π 2π 2π F (τ, ϕ(τ), ϕ(τ))dτ = P c (F ) = F (τ, ϕ(τ), ϕ(τ)) cos kτdτ = P ck (F ) = F (τ, ϕ(τ), ϕ(τ)) sin kτdτ = P sk (F ) = k = 1, 2,..., m (6.26)

126 6 118 m (a, b) = (a, a 1, b 1,..., a m, b m ) (6.27) P c, P ck, P sk,... (6.27) (a, b) (6.26) F a = F c a + F a j = F c a j + F b j = F c b j + m ( Fck a k=1 m ( Fck a j k=1 m ( Fck k=1 b j cos kt + F sk a cos kt + F sk a j cos kt + F sk b j ) sin kt = F x ) sin kt = F x x + F a ẋ ẋ = F a x F cos jt j sin jt ẋ ) sin kt = F F sin jt + j cos jt x ẋ (6.28) ( ) F c F = P c a x ( ) ( ) F ck F F sk F = P ck, = P sk, k = 1, 2,..., m a x a x ( ) F c F F = P c cos jt j sin jt, j = 1, 2,..., m a j x ẋ ( ) F c F F = P c sin jt + j cos jt, b j x ẋ ( ) F ck F F = P ck cos jt j sin jt, j, k = 1, 2,..., m a j x ẋ ( ) F sk F F = P sk cos jt j sin jt, a j x ẋ ( ) F ck F F = P ck sin jt + j cos jt, b j x ẋ ( ) F sk F F = P sk sin jt + j cos jt b j x ẋ (6.29) (6.25) (6.27) F (t, ϕ m (t), ϕ m (t)), F x (t, ϕ m(t), ϕ m (t)), F ẋ (t, ϕ m(t), ϕ m (t)) 6.2 ẍ +.1ẋ + x 3 =.3 cos t (6.3)

127 π x(t) (6.3) x(t + π) x(t) = x(t + π) { x(t) = a2k+1 cos(2k + 1)t + b 2k+1 sin(2k + 1)t } (6.31) k= 2 x(t) = a 1 cos t + b 1 sin t + a 3 cos 3t + b 3 sin 3t (6.32) (6.32) (6.3) cos t, sin t, cos 3t, sin 3t 2 F c1 (a 1, b 1, a 3, b 3 ) = a 1 +.1b ] [a 1 (r r 2 4 3) + a 3 (a 2 1 b 2 1) + 2b 3 a 1 b 1.3 = F s1 (a 1, b 1, a 3, b 3 ) = b 1.1a ] [b 1 (r r 2 4 3) + b 3 (a 2 1 b 2 1) 2a 3 a 1 b 1 = F c3 (a 1, b 1, a 3, b 3 ) = 9a 3 +.3b [ ] a 1 (a 2 1 3b 2 4 1) + 3a 3 (2r1 2 + r3) 2 = F s3 (a 1, b 1, a 3, b 3 ) = 9b 3.3a [ ] b 1 (3a 2 1 b 2 4 1) + 3b 3 (2r1 2 + r3) 2 = (6.33) r 2 1 = a b 2 1, r 2 3 = a b 2 3 (6.31) 3 1 : (a 1, b 1, a 3, b 3 ) = (.923,.347,.149,.24) 2 : (a 1, b 1, a 3, b 3 ) = (.3219,.349,.9,.3) 3 : (a 1, b 1, a 3, b 3 ) = (1.1198,.5246,.29,.673) m m (harmonic balance method) (equivalent linearization)

128 ẍ + kẋ + c 1 x + c 3 x 3 = B cos t (6.34) x(t) = u cos t + v sin t (6.35) (6.34) cos t sin t F c (u, v) = ( 1 + c ) c 3r 2 u kv B = F s (u, v) = ku + ( 1 + c ) c 3r 2 v = (6.36) r 2 = u 2 + v 2 (5.59) (6.34) (6.35) x(t) = u(t) cos t + v(t) sin t (6.37) (6.37) ẋ = u(t) cos t + v(t) sin t u(t) sin t + v(t) cos t ẍ = 2 u(t) cos t + 2 v(t) sin t u(t) cos t v(t) sin t (6.34) u(t), v(t) ü, v, k u, k v, sin 3t, cos 3t cos t sin t { u = 1 ku (1 c 1 34 ) } 2 c 3r 2 v { ( v = 1 1 c 1 3 ) } (6.38) 2 4 c 3r 2 u kv + B r 2 = u 2 + v 2

129 M R n ẋ = f(t, x, λ) (6.39) t x λ t R, x = x 1 x 2. x n λ 1 λ 2 M, λ = R m. λ m f : R M R m R n ; (t, x, λ) f(t, x, λ) (6.39) ẋ i = f i (t, x 1, x 2,..., x n, λ 1,..., λ m ), i = 1, 2,..., n (6.4) λ = λ t = t x(t ) = x (6.39) x(t) = ϕ(t, t, x, λ ) (6.41) x i (t) = ϕ(t, t, x 1,..., x n, λ 1,..., λ m ), i = 1, 2,..., n (6.42) (6.41) t t x λ f (??) (6.41) ϕ(t, t, x, λ ) = ϕ(t, t, x, λ ) t = f(t, ϕ(t, t, x, λ ), λ ) (6.43)

130 6 122 ϕ(t, t, x, λ ) = x (6.44) (6.43) x d ϕ(t, t, x, λ ) = f(t, ϕ(t, t, x, λ ), λ ) ϕ(t, t, x, λ ) (6.45) dt x x x (6.44) ϕ(t, t, x, λ ) x = I n (6.46) 1 1 (6.45) (6.46) (6.45), (6.46) d ϕ i = dt x j δ ij δ n k=1 ϕ i (t, t, x, λ ) x j δ ij = f i x k ϕ k x j, = δ ij { 1 i = j i j i, j = 1, 2,..., n (6.47) 2 (6.45), (6.46) x 2 d 2 ϕ dt x 2 = f(t, ϕ, λ ) 2 ϕ x x f(t, ϕ, λ ) ϕ ϕ x 2 x x 2 ϕ(t, t, x, λ ) x 2 = (6.48) (6.47) d 2 ϕ i = dt x j x l n k=1 f i 2 ϕ k + x k x j x l n n k=1 p=1 2 ϕ i (t, t, x, λ ) x j x l =, i, j, l = 1, 2,..., n 2 f i x k x p ϕ k x j ϕ p x l, (6.49)

131 (6.43) λ d ϕ dt λ = f(t, ϕ, λ ) ϕ x λ + f λ (6.44) ϕ(t, t, x, λ ) λ (6.5) = (6.51) 1 1 (6.5) (6.51) (6.5), (6.51) d ϕ i = dt λ q n k=1 ϕ i (t, t, x, λ ) λ q = f i x k ϕ k λ q + f i λ q, i = 1, 2,..., n, q = 1, 2,..., m (6.52) (6.45), (6.46) d dt 2 ϕ x λ = f(t, ϕ, λ ) x 2 ϕ(t, t, x, λ ) x λ = d 2 ϕ i n f i 2 ϕ k n = + dt x j λ q x k x j λ q k=1 2 ϕ x λ + 2 f ϕ ϕ x 2 x λ + 2 f x λ n k=1 p=1 2 f i x k x p ϕ k x j ϕ p λ q + 2 ϕ i (t, t, x, λ ) x j λ q =, i, j = 1, 2,..., n, q = 1, 2,..., m n k=1 ϕ x 2 f i x k λ q ϕ k x j, (6.53) (6.54) ẋ = f(t, x, y, λ) ẏ = g(t, x, y, λ) x(t) = φ(t, t, x, y, λ ) y(t) = ψ(t, t, x, y, λ ) (6.55) (6.56)

132 6 124 x(t ) = φ(t, t, x, y, λ ) = x (6.57) y(t ) = ψ(t, t, x, y, λ ) = y (6.56) A = f x g x B = A x = C = A y = f y g y 2 f φ x f ψ x x y x 2 g φ x g ψ x x y x 2 f φ x f ψ y x y y 2 g φ x g ψ y x y y 2 f φ + 2 f ψ x y x y 2 x 2 g φ + 2 g ψ x y x y 2 x 2 f φ + 2 f ψ x y y y 2 y 2 g φ + 2 g ψ x y y y 2 y (6.58) 1 d dt d dt φ φ x ψ = A x ψ, x x φ φ y ψ = A y ψ, y y φ x ψ x φ y ψ y t=t t=t 1 = = 1 (6.59) 1 φ φ f d λ dt ψ = A λ ψ + λ g, λ λ λ φ λ ψ λ t=t = (6.6) 2 d dt 2 φ 2 φ φ x 2 2 ψ = A x 2 2 ψ + B x ψ, x 2 x 2 x 2 φ x 2 2 ψ x 2 t=t =

133 6 125 d dt 2 φ 2 φ φ x y 2 ψ = A x y 2 ψ + C x ψ, x y x y x d dt 2 φ 2 φ φ y 2 2 ψ = A y 2 2 ψ + C y ψ, y 2 y 2 y 2 φ x y 2 ψ x y 2 φ y 2 2 ψ y 2 t=t t=t = = (6.61) 2 d dt d dt 2 φ 2 φ φ 2 f x λ 2 ψ = A x λ 2 ψ + B λ ψ + x λ 2 g x λ x λ λ x λ 2 φ 2 φ φ 2 f y λ 2 ψ = A y λ 2 ψ + C λ ψ + x λ 2 g y λ y λ λ x λ φ + 2 f ψ x y λ x φ + 2 g ψ x y λ x φ + 2 f ψ y y λ y φ + 2 g ψ y y λ y,, 2 φ x λ 2 ψ x λ 2 φ y λ 2 ψ y λ t=t t=t = = (6.62) R n ẋ = f(t, x) (6.63) f L f(t + L, x) = ft, x) (6.64) u R n (6.63) x(t) = ϕ(t, u), x() = ϕ(, u) = u (6.65) L (6.63) ϕ(t + L, u) = ϕ(t, u), t R (6.66)

134 6 126 T t : R n R n ; u u 1 = ϕ(l, u) (6.67) T (6.63) F (u) = T (u) u = ϕ(l, u) u = (6.68) ϕ(l, u) (6.63) u [, L] (6.68) DF (u) = DT (u) I n = ϕ(l, u) u I n (6.69) T DT (u) (6.45) t = t = L u(k + 1) = u(k) + h(k) DF (u(k)) h(k) = F (u(k)) (6.7) (6.3) ẋ = y ẏ =.1y x cos t (6.71) t = (x, y ) (6.56) x(t) = φ(t, x, y ) y(t) = ψ(t, x, y ) (6.72) (6.68) F 1 (x, y ) = x(2π) x = φ(2π, x, y ) x (6.73) F 2 (x, y ) = y(2π) y = ψ(2π, x, y ) y F 1 F 1 x DF (x, y ) = y F 2 F = 2 x y φ(2π, x, y ) x 1 φ(2π, x, y ) y ψ(2π, x, y ) x ψ(2π, x, y ) y 1

135 (6.49) (6.71) (6.49) n ẋ = f(x), x R n (6.74) t = x() = x (6.74) x(t) = ϕ(t, x ) x(t) = ϕ(t, x ) (6.74) Π Π q : R n R; x q(x) (6.75) Π = { x R n q(x) = } (6.76) T (4.31) T : Π Π; x ϕ(τ, x) (6.77) τ T (x ) x = ϕ (τ(x ), x ) x = (6.78) x Π q (ϕ (τ(x ), x )) = (6.79) x τ Π p p : Π Σ; x = (x 1, x 2,..., x n ) u = (u 1, u 2,..., u n 1 ) = p(x) u 1 = p 1 (x) = x 1,..., u n 1 = p n 1 (x) = x n 1 (6.8) Π q x n h : Σ Π; u x = h(u) x 1 = h 1 (u) = u 1,..., x n 1 = h n 1 (u) = u n 1, x n = h n (u) (6.81)

136 6 128 φ(t, x 1 ) R n Π h p R n-1 Σ u 1 = h(x 1 ) u = h(x ) u = P(u 2 1 ) u 2 = h(x 2 ) x x 1 x = φ(l, x ) x 2 x 2 = φ(τ(x 1 ), x 1 )=T(x 1 ) φ(t, x ) P T (6.78), (6.79) P : Σ Σ; u u 1 = P (u) = p T h(u) (6.82) F 1 (u, τ) = P (u) u = p (ϕ (τ, h(u))) u = F 2 (u, τ) = q (ϕ (τ, h(u))) = (6.83) (u, τ) u τ (6.83) 2 F 1 F 1 P u τ F 2 F = u I n 1 2 q u τ u P τ q τ (6.84)

137 6 129 P u P τ = p T h [ ] ϕ(τ, h(u)) x x u = I n 1 x = p ϕ [ ] x τ = I n 1 f(ϕ(τ, h(u))) q u = q ϕ(τ, h(u)) h x x u = q ϕ(τ, h(u)) x x q τ = q ϕ x τ = q f(ϕ(τ, h(u))) x I n 1 h n u I n 1 h n u,, (6.85) ẋ = f 1 (x, y, z) ẏ = f 2 (x, y, z) ż = f 3 (x, y, z) (6.86) Π = {(x, y, z) R 3 q(x, y, z) = z a = } (6.87) a xy- Σ = {(u, v) R 2 u = x, v = y} (6.88) (6.8), (6.81) p : Π Σ; x = (x, y, z) u = (u, v) = (x, y) h : Σ Π; u = (u, v) x = (x, y, a) (6.89) (x, y, z ) (6.86) x(t) = φ 1 (t, x, y, z ), y(t) = φ 2 (t, x, y, z ), z(t) = φ 3 (t, x, y, z ) (6.9) (6.83) F 1 (u, v, τ) = φ 1 (τ, u, v, a) u = F 2 (u, v, τ) = φ 2 (τ, u, v, a) v = F 3 (u, v, τ) = φ 3 (τ, u, v, a) a = (6.91)

138 (4.11) a =.6, b =.5, c = 1.6 y =. µ 1 = 1 (x, z ) (µ 2, µ 3 ) I (1.4357, ) ( 1.446,.119) ( 1 I) S 2 1 (1.1331, 2.635) (.3162,.17) ( D 2 ) S 2 2 (1.362, ) (6.84) F 1 u F 2 u F 3 u F 1 v F 2 v F 3 v F 1 τ F 2 τ F 3 τ = φ 1 x 1 φ 1 y φ 2 x φ 3 x f 1 φ 2 y 1 f 2 φ 3 f 3 y (6.92) y =. (6.91) (4.11) ξ = η ζ η = ξ +.6η ζ = (.5 + z )ξ (1.6 x )ζ (6.93) (6.9) 1, 3 (6.93) [1,, ] T [,, 1] T

139 [C6] Rouche Mawhin[O12] [P6] [N8] [N14] [N15] [N17], Schmidt Tondl[N16], Ueda[N23] (5.94) ẍ + x = ɛ(b cos t + ax.1ẋ cx 3 ) (6.94) (1.51) x(k + 1) = 1 + y(k) ax 2 (k) y(k + 1) = bx(k) (6.95) (a, b) 6.3 ẍ ɛ(1 x 2 )ẋ + x = (6.96) x = ɛ = ẍ ɛ(1 ẋ 2 )ẋ + x 3 = B cos νt (6.97) x(t) = u(t) cos νt + v(t) sin µt (6.98)

140 ẍ + kẋ + (a + b cos 2νt)x + x 3 = (6.99) (6.98) 6.1 (6.12) 6.2 (6.33) 6.3 (6.73)

141 ,

142 7 135 i G i C i L i G i = g(v) G G C v R v 7.1 LCRG (a) (b) M R n dx dt = f(x, λ), x Rn, λ R m (7.1) V : M R; x V (x) (7.2) M x V (x) V (positive semi definite) V () = V (x) > (positive definite) (negative semi definite) (negative definite) V (x) V (x) < 2 M V M x V = dv (x) dt = V dx x dt = V f(x, λ) (7.3) x V (7.1) 7.1 LCRG 7.1(a) C dv dt L di dt = Gv + i = v Ri (7.4)

143 7 136 W W (v, i) = 1 2 Li Cv2 (7.5) dw dt = Li di dv + Cv dt dt = Ri2 Gv 2 (7.6) W (b) i G = g(v G ), v G g(v G ) (7.7) C dv dt L di dt = g(v) + i = v Ri (7.8) (7.5) dw dt = Li di dv + Cv dt dt = Ri2 vg(v) (7.9) 1 (7.7) 1 3 v, i (7.8) (dissipative system in the large) D- R n ẋ = f(t, x, λ) (7.1)

144 7 137 λ R m (7.1) t L f(t + L, x, λ) = f(t, x, λ) (7.11) t = t = x() = x (7.1) x(t) = ϕ(t, x ), x() = ϕ(, x ) = x (7.12) (7.1) (7.1) D- D- (1) R (7.12) t x(t) R (2) N x R (7.12) t > LN x(t) R D- x R (7.1) 7.3 D- 2 (Levinson) D- ẍ + f(x, ẋ)ẋ + g(x) = e(t) (7.13) f g (1) e(t) (2) x xg(x) > g(x) G(x) = x g(x)dx g(x)/g(x) x 1/ x (3) a x a f(x, y) M > x a f(x, y) m, m > ẍ + kẋ + c 1 x + c 3 x 3 = B + B cos t (7.14)

145 7 138 ẍ ɛ(1 x 2 )ẋ + x = B cos νt (7.15) - ẍ ɛ(1 x 2 )ẋ + c 1 x + c 3 x 3 = B + B cos νt (7.16) D- [O9, P3] D- 1 L (7.1) λ f(x, λ) = (7.17) (7.17) 2 (bifurcation) (bifurcation value) µ λ = λ dr(µ) (7.18) dλ λ=λ λ 1 λ m m 1 1 λ λ R x R n (7.17) x (7.17) Df(x, λ) = f(x, λ) x = A(λ) (7.19) χ(µ) = det(µi n A(λ)) = µ n + a 1 µ n a n 1 µ + a n = (7.2)

146 7 139 O 1O 2O 3O 4O R(µ i ), i = 1, 2,..., n (7.21) µ i = (7.17) ko + k+1 O k =, 1, 2,..., n 1 (7.22) (7.22) λ = λ (1) λ < λ 2 k O k+1 O (2) λ = λ k O k+1 O 1 (3) λ < λ ko + k+1 O (7.23) ko + k+1 O (7.24) (7.22)

147 7 14 kd k+1d ko k+2o ko k+2o λ=λ* λ λ=λ* λ 7.3 (a) (b) (turning point) (tangent bifurcation) *1 2 x (7.22) k k =, 1,..., n 1 n (7.2) χ() = det( A(λ)) = a n = (7.25) R n R R n R (7.17) (7.25) (7.17) n 1 (7.25) 1 n 1 (7.25) 1 1 (Hopf) 1 2 (Hopf bifurcation) ko k+2 O + LC( k D), k =, 1, 2,..., n 2 ko + LC( k+1 D) k+2 O, k =, 1, 2,..., n 2 (7.26) LC( k D) k D 7.3 *1 (7.22) k = k = n 1

148 7 141 (7.26) 1 (super critical) 2 (sub-critical) 2 (7.26) k n 2 n k = χ(jω) = det(jωi n A(λ)) = (7.27) 2 ω 1 ω χ(jω) = (jω) 2 + jωa 1 + a 2 = ω 2 + a 2 =, ωa 1 = a 1 =, a 2 > (7.28) ω = a 2 3 χ(jω) = (jω) 3 + a 1 (jω) 2 + jωa 2 + a 3 = ω 2 a 1 + a 3 =, ω 2 + a 2 = a 1 a 2 + a 3 =, ω = a 2 (7.29)

149 (7.2) ω 2 a 1 = ω = a 2, a 2 > 3 a 1 a 2 + a 3 = ω = a 2, a 2 > 4 a a 2 1(a 4 a 2 ) = ω = r a4 a 1, a 4 a 1 > x 1 = ax 1 + bx 2 + x 2 = cx 1 + dx 2 + m=2 m=2 1 m! (x 1D 1 + x 2 D 2 ) m f(x 1, x 2 ) 1 m! (x 1D 1 + x 2 D 2 ) m g(x 1, x 2 ) (7.3) a = f x 1, b = f x 2, c = g x 1, d = g x 2, D 1 = x 1, D 2 = x 2, D p 1 Dq 2 = p+q x p 1 xq 2, D p 1 Dq 2 f = p+q f x p 1 xq 2 = f pq, D p 1 Dq 2 g = p+q g x p 1 xq 2 = g pq (7.3) y 1 = ωy 2 + P (y 1, y 2 ) y 2 = ωy 1 + Q(y 1, y 2 ) (7.31) P (y 1, y 2 ) = P k (y 1, y 2 ) = P k (y 1, y 2 ), Q(y 1, y 2 ) = k=2 k l= Q k (y 1, y 2 ) k=2 p k l,l y k l 1 y l 2, Q k (y 1, y 2 ) = k l= q k l,l y k l 1 y l 2 V (y 1, y 2 ) = 1 2 (y2 1 + y 2 2) + V k (y 1, y 2 ), V k (y 1, y 2 ) = k=3 k l= v k l,l y k l 1 y l 2 (7.32)

150 7 143 V k (y 1, y 2 ) [O16, N2, D12] V 3 (y 1, y 2 ) V (y 1, y 2 ) = D 1 (y y 4 2) + (7.33) V (y 1, y 2 ) = D 2 (y y 2 2) 2 + (7.34) D 2 = 3 4 D 1 = 1 8 D = 1 [ 3(p 3 + q 3 ) + p 12 + q { }] (7.35) 2(p 2 q 2 p 2 q 2 ) p 11 (p 2 + p 2 ) + q 11 (q 2 + q 2 ) ω (7.33), (7.34) (7.35) O 2 O D 1 (7.26) 3 (1) D < O 2 O + LC( D) (7.36) (2) D > O + LC( 1 D) 2 O (7.37) (3) D = 2 D (7.3) D = 1 { } a(f 21 + g 12 ) + b(f 3 + g 21 ) c(f 12 + g 3 ) 2b 1 { + 2b(a 2 ab( f2 2 + f 2 g 11 + f 11 g 2 + g 2 g 2 + 2g 2 + bc) 11) +ac(f 2 f 2 + 2f f 11 g 2 + f 2 g 11 g 2 2) b 2 (f 2 g 2 + g 2 g 11 ) } +c 2 (f 11 f 2 + f 2 g 2 ) + (2a 2 bc)(f 2 f 11 g 11 g 2 ) (7.38) (7.1) (7.1) x(t) = ϕ(t, x, λ), x() = ϕ(, x, λ) = x (7.39)

151 7 144 T : R n R n ; x T (x) = ϕ(l, x, λ) (7.4) (7.1) 2π T x T (x ) = (7.41) x R n χ(µ) = det(µi n A(λ)) = µ n + a 1 µ n a n 1 µ + a n = (7.42) DT (x ) = T (x ) x = ϕ(l, x, λ) x = A(λ) (7.43) (7.42) (tangent bifurcation) χ(1) = det(i n A(λ)) = 1 + a a n 1 + a n = (7.44) kd + k+1 D, k =, 1,..., n 1 ki + k+1 I, k = 1, 2,..., n 2 (7.45) (period doubling bifurcation) χ( 1) = det( I n A(λ)) = ( 1) n + a 1 ( 1) n 1 + a n 1 + a n = (7.46) kd k+1 I + 2 k D 2, k =, 1,..., n 2 kd k 1 I + 2 k D 2, k = 2, 3,..., n ki k+1 D + 2 k D 2, k = 1, 2,..., n 1 ki k 1 D + 2 k D 2, k = 1, 2,..., n 1 (7.47) k D 2 D 2- k

152 7 145 D 1D 2D 3D 4D 1I 2I 3I 7.4 (Neimark-Sacker bifurcation) 1 χ(e jθ ) = det(e jθ I n A(λ)) = e jnθ + a 1 e j(n 1)θ + + a n 1 e jθ + a n = (7.48) T θ θ kd k+2 D + ICC, k =, 1,..., n 2 kd + ICC k+2 D, k =, 1,..., n 2 (7.49) ICC (invariant closed curve) ICC ICC 2 ICC (7.1) , 3 2 (7.48) e j2θ + a 1 e jθ + a 2 = cos 2θ + a 1 cos θ + a 2 =, sin 2θ + a 1 sin θ =

153 7 146 θ a 2 = 1, 2 < a 1 < 2 (7.5) 3 a 3 (a 3 a 1 ) + a 2 = 1, 2 < a 3 a 1 < 2 (7.51) (branching) (pitch fork bifurcation) ko k+1 O + 2 k O, k =, 1,..., n 1 ko k 1 O + 2 k O, k = 1, 2,..., n kd k+1 D + 2 k D, k =, 1,..., n 1 kd k 1 I + 2 k D, k = 1, 2,..., n ki k+1 I + 2 k I, k = 1, 2,..., n 2 ki k 1 I + 2 k I, k = 2, 3,..., n 1 (7.52) (7.53) C RC (2.78) α, δ = 1 αx x 3 (x y) = αy y 3 (y x) = (7.54)

154 7 147 (x, y) (y, x), (x, y) ( x, y) (x, y) ( y, x) (1) α = y = x O 1 O + 2 O (2) α = 2 y = x 1 O 1O 2 O O (3) α = 3 y = x 2 1 O 1O O O y = x 2 O (7.17) (7.25) F (x, λ 1 ) = f(x, λ) = G(x, λ 1 ) = χ() = det( A(λ)) = a n = (7.55) (x, λ 1 ) R n R λ 1 m λ R m 1 (7.17) (7.27) F (x, λ 1, ω) = f(x, λ) = [ ] G 1 (x, λ 1, ω) = R det(jωi n A(λ)) = [ ] G 2 (x, λ 1, ω) = I det(jωi n A(λ)) = (7.56)

155 (7.54) α α < 1 O(, ) < α < 2 3 1O(, ), O( α, α), O( α, α) 2 < α < 3 5 2O(, ), O( α, α), O( α, α) 1O( α 2, α 2), 1 O( α 2, α 2) 3 < α 9 2O(, ), O( α, α), O( α, α) O( α 2, α 2), O( α 2, α 2) ( 1 1O ( 1 1O 1O 2 ( α 3 + α + 1), 1 2 ( α 3 ) α + 1), 2 ( α 3 + α + 1), 1 2 ( α 3 ) α + 1), ( 1 2 ( α 3 + α + 1), 1 2 ( α 3 ) α + 1), ( 1O 1 2 ( α 3 + α + 1), 1 2 ( α 3 ) α + 1) R( ), I( ) (x, λ 1, ω) R n R R 7.1 ω (7.56) 2 1 (x, λ 1 ) R n R A. (7.41) (7.44) F (x, λ 1 ) = x T (x) = G(x, λ 1 ) = det(i n A(λ)) = 1 + a a n 1 + a n = (7.57) (x, λ 1 ) R n R

156 7 149 (7.41) (7.46) F (x, λ 1 ) = x T (x) = G(x, λ 1 ) = det( I n A(λ)) = ( 1) n + a 1 ( 1) n 1 + a n 1 + a n = (7.58) (x, λ 1 ) R n R (7.41) (7.48) F (x, λ 1, ω) = x T (x) = [ ] G 1 (x, λ 1, θ) = R det(e jθ I n A(λ)) = [ ] G 2 (x, λ 1, θ) = I det(e jθ I n A(λ)) = (7.59) (x, λ 1, θ) R n R R (7.59) 2 θ 1 (x, λ 1 ) R n R B T (6.77) x R n T (6.78) χ(µ) = det(µi n A(λ)) = µ n + a 1 µ n a n 1 µ + a n = (7.6) DT (x ) = T (x ) x = ϕ(l, x, λ) x = A(λ) (7.61) λ 1 1 χ(1) = det(i n A(λ)) = 1 + a a n 1 + a n = (7.62) (7.6) χ(µ) = µ n + a 1 µ n a n 1 µ + a n = (µ 1)(µ n 1 + b a µ n b n 2 µ + b n 1 ) = (7.63) k b k = 1 + a 1 + a a k = 1 + a l (7.64) l=1

157 7 15 χ A (µ) = µ n 1 + b 1 µ n b n 2 µ + b n 1 = (7.65) (7.65) χ A (1) = 1 + b b n 2 + b n 1 = (7.66) (7.6) µ µ = 1 (7.66) dχ dµ (1) = n + (n 1)a a n 1 = (7.67) (7.65) χ A ( 1) = ( 1) n 1 + b 1 ( 1) n 2 + b n 2 + b n 1 = (7.68) µ = 1 χ( 1) = ( 1) n + a 1 ( 1) n 1 + a n 1 + a n = (7.69) (7.65) χ A (e jθ ) = (e jθ ) n 1 + b 1 (e jθ ) n b n 2 e jθ + b n 1 = (7.7) µ = 1 χ(e jθ ) = (e jθ ) n + a 1 (e jθ ) n a n 1 e jθ + a n = (7.71) (6.83) (7.67) χ A (1) = b = a = (7.72) 3

158 7 151 (7.67) χ A (1) = a 2 + 2a = (7.73) (7.68) χ A ( 1) = a = (7.74) (7.7) cos 2θ + (a 1 + 1) cos θ + a 2 + a = sin 2θ + (a 1 + 1) sin θ = (7.75) θ a 1 + a 2 =, 3 < a 1 < 1 (7.76) (synchronization) (1.22) (1) (1.22) L di dt C dv dt = v = i g(v) + j(t) (7.77) i G = g(v G ) = g 1 v G + g 3 v 3 G (7.78) j(t) = J cos ωt (7.79)

159 7 152 (7.77) x = Li, y = Cv (7.8) (7.77) dx dt dy dt = 1 LC y = 1 x + g ( 1 1 g ) 3 LC C g 1 C y2 y + J cos ωt C (7.81) τ = 1 LC t = ω t, ω = 1 LC (7.82) (7.81) dx dτ dy dτ = y = x + g ( 1 1 g ) 3 ω C g 1 C y2 y + J cos ω τ ω C ω (7.83) ɛ = g 1 ω C = g 1 L C, γ = g 3 g 1 C, B = J C (7.83) τ t g 1, ν = ω ω (7.84) dx dt dy dt = y = x + ɛ ( 1 γy 2) y + ɛb cos νt (7.85) (2) (7.85) x(t) = u(t) cos νt + v(t) sin νt y(t) = u(t) sin νt + v(t) cos νt (7.86) [ ( u = ɛ γr2 [ v = ɛ 2 σu + ) ] u σv (1 34 γr2 ) v + B ] (7.87) r 2 = u 2 + v 2, σ = 2(ν 1) ɛ (7.88)

160 7 153 (7.88) ɛ/2 6.1 u = (1 34 γr2 ) u σv = f(u, v) v = σu + (1 34 ) γr2 v + B = g(u, v) (7.89) (3) (7.89) (7.89) (1 34 γr2 ) u σv = σu + (1 34 ) γr2 v = B (7.9) 2 [(1 34 γr2 ) 2 + σ 2 ] r 2 = B 2 (7.91) (6.13) f f u v g g = γ(3u2 + v 2 ) 3 2 γuv σ 3 u v 2 γuv + σ 1 3 (7.92) 4 γ(u2 + 3v 2 ) (u, v) f f χ(µ) = u v g g = µ 2 + a 1 µ + a 2 = (7.93) u v a 1 = 3γr 2 2 a 2 = σ γr γ2 r 4 = σ 2 + (1 34 γr2 ) (1 94 γr2 ) (7.94) (4) 7.5 (7.91) (σ, B, r) γ γ = 4/3 B (σ, r) a 2 = a 1 = (σ, B)

161 7 154 r O a 2 a 1 B Q O -.5 O P.5 1 σ 1.5 B Q -.5 σ P (7.91).6.5 Q O P O+2 1 O.4 B.3.2 2O O+ 1 O+ 2 O 2O O P 2O.1 O+ 1 O+ 2 O σ (a) (b) 7.6 (a) P (b)

162 H 2 C 2 D H 1.5 Q P C 1 B D T 2 D+ 1 D+ 2 D T 3 T 1 2D ν 7.7 (7.85) ɛ =.2 (5) a 2 = t 1, t 2, t 3 c 1, c 2 a 1 =, a 2 > h 1, h 2 O 2 O + LC( D) a 1 =, a 2 < a 1 = P Q 2 2 B =, σ = σ 1 1 3γr 2 /4 = (6) P Q ν (7.85) (7.86)

163 7 156 ν (7.86) 2 O + 1 O P Q 2O + 1 O (7) (7.85) 7.3.2(A) (7.85) 7.7 T 1, T 2, T 3 C 1, C 2 H 1, H 2 D 2 D + ICC 7.6 (ICC) D

164 [O8] Hale[O5] Rouche et al.[o13] [O17] D- Lefschetz[O9] Cartwright[P3] [N7] Guckenheimer Holmes[D8] Hale Koak[D1] Marsden McCracken [D14] Andronov et al.[n1, D2, 3] Neimark[D18] 198 [P5] [P15] Kubicek and Marek[C3], Parker and Chua[C5] Kuznetsov[D12] (1.9) (6.9) u = ζu v = (a 34 cr2 ) v (a 34 cr2 ) u ζv + b (7.95) V (u, v) = 1 2 (u2 + v 2 ) (7.96) V < (7.89) B (σ, r) (7.91) 7.6 BVP(Bönhffer van der Pol) ẋ = c ( x 1 ) 3 x3 + y ẏ = 1 c (x + by a) (7.97) c >, b >

165 , ẋ = f(x, y, λ) ẏ = g(x, y, λ) (8.1) (x, y) R 2 λ R m 1 2 (8.1) M M M

166 8 159 M (8.1) x(t) = ϕ(t, x ), x() = ϕ(, x ) = x (8.2) x(t) = x(t), ϕ(t, x ) = φ(t, x, y ) y(t) ψ(t, x, y ) (8.2) γ(x ) = {x R 2 x(t) = ϕ(t, x ), t R} γ + (x ) = {x R 2 x(t) = ϕ(t, x ), t [, )} γ + (x ) = {x R 2 x(t) = ϕ(t, x ), t (, ]} (8.3) (8.2) α ω ω(x ) = ω(γ(x )) = ωlimit(x ) = ωlimit(γ(x )) = ϕ(t, x ) (8.4) τ t τ x γ(x ) ω (ω-limit set) α(x ) = α(γ(x )) = αlimit(x ) = αlimit(γ(x )) = ϕ(t, x ) (8.5) τ α (α-limit set) (8.1) α M M ω N M R 2 x N ϕ(t, x ) N (invariant set) (ω) (α) t τ A α 1 O 1 ω LC 1

167 8 16 LC LC 2O A 1O 2 2O 1O 1 O M 8.1 γ + = ω(γ + ) γ + x x ω(γ + ) γ + (x ) M ω ω(γ + (x )) ω(γ + (x )) γ + (x ) ω(γ + (x )) ω(γ + (x )) 8.2

168 8 161 y y γ + y=f(x) y 1 y 2 y 3 γ + =ω(γ + ) x ẍ + f(x)ẋ + g(x) = (8.6) (Linard) f(x) f() < g(x) x xg(x) > F (x) = x f(x)dx, G(x) = x g(x)dx (8.7) x ± F (x) ± F G (8.6) ẋ = y F (x) ẏ = g(x) (8.8) (8.8) y 8.3 f() < 2 (8.8) 1 y (8.8) (5.96) 8.4(a) α ( 1 D)

169 D D O O (a) (b) 8.4 ω ( D) β α (a) α ω (separatrix) 8.5(b) 1 α ω (saddle connection) 8.5(c) α ω (separatrix loop) (Andronov) (Pontriagin) (8.1) ẋ = f(x, y) + ξ(x, y) ẏ = g(x, y) + η(x, y) (8.9)

170 8 163 ω α 1O 1O 1 α ω 1O 2 (a) (b) 1O ω α (c) 8.5 (b) (c) λ (8.9) (8.1) (8.9) M (8.1) ξ, η 1 (coarsed system) (Lefschetz) (structual stability) (8.1) 3 (1) (2) (3) 3 (De Baggis) 8.4 (2)

171 8 164 (a) (b) (c) (3) 8.6 (8.1) 1 O(x ) Df(x ) = f(x ) x g(x ) x f(x ) y g(x ), a 2 = y f(x ) x g(x ) x f(x ) y g(x ) < (8.1) y ( f(x ) a 1 = trace(df(x )) = + g(x ) ) x y (8.11) 1 O(x ) 8.6 (b) a 1 > a 1 < (8.11) a 1 =

172 P dx dt dy dt = y [ ] (8.12) = x + ɛ (1 γy 2 )y cx 3 + B cos νt (7.85) 1.4(b) (7.86) [ ( u = ɛ 1 3 ) 2 4 γr2 u (σ 34 ) ] cr2 v [ ( v = ɛ σ 3 ) 2 4 cr2 u + (1 34 ) ] (8.13) γr2 v + B r 2 = u 2 + v 2, σ = 2(ν 1) ɛ (8.14) (8.13) [( 1 3 ) 2 4 γr2 + (σ 34 ) 2 ] cr2 r 2 = B 2 (8.15) χ(µ) = µ 2 + a 1 µ + a 2 = (8.16) a 1 = 3γr 2 2, a 2 = σ (γ + cσ)r (γ2 + c 2 )r γ = c = , 7.6 P P a 1 = a 2 = (8.17) 2 2 P (8.13) ẋ = y ẏ = λ 1 + λ 2 x + x 2 xy (8.18)

173 P R B 1 P Q σ 8.7 (8.13) (2 O+ 1 O; ) ( O; ) P ( O+ 1 O+ 2 O; D) ( O+ 1 O+ 2 O; ) 8.8 P

174 8 167 (8.13) (8.18) P (λ 1, λ 2 ) (8.18) 2 1 (Bogdanov) (Takens) 8.7 P R 8.6 P Q 2 [P1, P6] π ẋ = f(t, x, y, λ) ẏ = g(t, x, y, λ) f(t + 2π, x, y, λ) = f(t, x, y, λ) g(t + 2π, x, y, λ) = g(t, x, y, λ) (8.19) (8.2) (8.19) 7.1 D- 2π T : R 2 R 2 ; x = (x, y ) x 1 = (x 1, y 1 ) = (φ(2π, x, y ), ψ(2π, x, y )) (8.21) x(t) = ϕ(t, x ) = x(t) = φ(t, x, y ), x() = ϕ(, x ) = x (8.22) y(t) ψ(t, x, y )

175 8 168 T l- (8.19) 2π 2lπ ( D) ( 1 D) ( 1 I) ( 2 D) 4 1 D 1 I 2 2 T T D- M M x γ(x ) = {x R 2 x(k) = T k (x ), k Z} γ + (x ) = {x R 2 x(k) = T k (x ), k = 1, 2,...} γ + (x ) = {x R 2 x(k) = T k (x ), k = 1, 2,...} (8.23) T M ω(x ) = ω(γ(x )) = ωlimit(x ) = α(x ) = α(γ(x )) = αlimit(x ) = k Z + l k k Z + l k T l (x ) T l (x ) (8.24) M N T (N) = N T (invariant set) { ω( 1 D) = W s ( 1 D) = x R 2 lim k { α( 1 D) = W u ( 1 D) = x R 2 } x(k) = lim T k (x) = 1 D k lim x(k) = lim T k (x) = 1 D k k } (8.25) ω ( ω-branch) α (α-branch) ω α ω ω α α ω α x M R 2 α ω x P, Q α(x) = P, ω(x) = Q (8.26) x (doubly asymptotic point) Q P

176 8 169 x α ω P ω α x ω P α α ω α ω Q ω α (a) 8.9 (b) α 1 D ω a b d c b' a' c' d' H P = Q (homoclinic point) P Q (heteroclinic point) D α ω H H H (8.19) H T T γ(h) α ω H

177 8 17 y A + B + C + D + B A 1D C x D 8.11 (8.28) ABCD A + B + C + D + T T 1 1 D ω R(abcd) ω α L T L R(abcd) R (a b c d ) T L (R(abcd)) = R (a b c d ) (8.27) T (horse shoe map) R(abcd) R (a b c d ) T L 1 T

178 8 171 α D H C H -1 ω H -2 A 1 D H 2 H 1 B D T L 8.7 ẋ = y, ẏ =.2y x cos t.8 (8.28) T 8.11 (x, y) = ( 1.278,.8358) (µ 1, µ 2 ) = (.1862, ) 1 D α ω ABCD T A + B + C + D + T (chaos) D α ω R(ABCD) ω α H H 2 = T 2 (H ), H 1 = T 1 (H ), H, H 1 = T (H ), H 2 = T 2 (H )

179 8 172 D C D C D C A B A B A + B + C + D A B (a) (b) (c) 8.13 T R(ABCD) T T T (a), (b), (c) R(ABCD) T R(ABCD) 2 (1) { } # R(ABCD) T k (R(ABCD)) = 2 k, k = 1, 2,... (8.29) (2) T T 1 1 { } R(ABCD) T (R(ABCD)) (a) (b) 4 2 {R(ABCD) T (R(ABCD))} T

180 8 173 {R(ABCD) T (R(ABCD))} T (R(ABCD)) 1 T 1 {R(ABCD) T (R(ABCD))} 1 T 11 {R(ABCD) T (R(ABCD))} 1 T (R(ABCD)) T (b) 8.13(c) R k = R(ABCD) T k (R(ABCD)) 2 k k 1 k 1 { k S k {}}{ } = s 1 s 2 s k s i = 1; i = 1, 2,..., k (8.3) k 8.13(c) T R = R(ABCD) T k (R(ABCD)) (8.31) k=1 { S {}}{ } = s 1 s 2 s i = 1; i = 1, 2,... (8.32) R S R k = R(ABCD) T k (R(ABCD)) k S k = { k {}}{ } s k s 2 s 1 s i = 1; i = 1, 2,..., k (8.33) T 1 R = R(ABCD) T k (R(ABCD)) (8.34) k=1

181 8 174 D - D C 1 C - B - (a) A - A B D C (b) A B D C A B (c) 8.14 T 1 S = { {}}{ } s 2 s 1 s i = 1; i = 1, 2,... (8.35) R(ABCD) T R = R(ABCD) T k (R(ABCD)) T k (R(ABCD)) (8.36) k=1 k=1 2 S = { {}}{{}}{ } s 2 s 1 s 1 s 2 s i = 1; i = ±1, ±2,... (8.37) 1 1

182 B - D d c C b- C - D - c- d- A - A a b B a T (8.36) T (8.37) 8.15 R(ABCD) T 1 (R(ABCD)) T 1 (R(ABCD)) A a d D T AadD T 1 (R(ABCD)) C c b B CcbB T 1 (R(ABCD)) R(ABCD) a b c d abcd (1) A a d D s 2 s 1 T 1 s 2 s 1 (2) C c b B s 2 1 s 1 T 1 s 2 1s 1

183 (shift) σ σ : S S ; s = s 2 s 1 s 1 s 2 σ(s) = s 2 s 1 s 1 s 2 (8.38) (T, R ) R T (σ, S ) S σ (symbolic dynamics) (Cantor set) (8.36) (Cantor set) C I = [, 1] 3 1 (1/3, 2/3) 2 {[, 1/3], [2/3, 1]} 4 {[, 1/9], [2/9, 1/3], [2/3, 7/9], [8/9, 1]} C (8.31) (8.34) C C (dense in itself) [, 1] C (8.32) (8.35) (8.37) T 2 3 t = t t t i +, t 2 (8.39) 3i S = { t 1 t 2 t i = 2; i = 1, 2, } (8.4) C C C C C [, 1]

184 , 3 f : S S S; ( s 1 s 2, t 1 t 2 ) s 1 t 1 s 2 t 2 (8.41) g : S [, 1]; t = t 1 t 2 g(t) = 1 ( t t ) (8.42) C 1:1 n (nest) (self similar) (fractal) (σ, S ) 2 S R S σ σ S σ 2 {, } S (8.43)

185 σ( ) = σ( ) = (8.44) T D I 2-2 { 11 11, } S (8.45) 2 1 σ( ) = σ( ) = (8.46) 2 k k k k k (8.44) 1 = = 1111 (8.47) σ 2 s = s n s 1 s 1 s n t = t n t 1 t 1 t n (8.48) d(s, t) = k= δ k 2 k (8.49) δ k s k = t k 1 s k t k d(s, t) S 2 S σ (1) (2) (3)

186 8 179 (8.41) 2 [, 1] 3 σ 2 (8.48) (8.49) k 2 k k T k 2 T k (sensitive dependence on initial conditions) (1), (2), (3) (chaotic state) (attractor) ω 8.8 ẋ = y, ẏ = ky x cos t (8.5) k α ω 8.17 k = α ω 8.3

187 8 18 y y 2 S D 1 S x D 1 S 2 S x (a) k =.1 (b) k =.5 y y D 1 S 2 S x D 1 S 2 S x (c) k =.5 (d) k = 8.17 (8.5)

188 k 2 2k 2k 4k D 2k 1 I 2k + 2 D 2k+1, k =, 1, 2,... (8.51) k k (universality) γ γ = (8.52) γ 8.9 ẋ = y, ẏ =.1y x cos t + B (8.53) B (1) (8.53) (8.21) (a) T 1, T 2 C 1, C P 1, P 2 1 I 8.19(b) P 1 (8.51) P1 2, P1 4, P1 8 2, 4, 8 P T 2 1, T 2 2, T 2 3 P P 2 2

189 P 1 P C 1 D B.2. 1I D 2 D+ 1 D T T 1 2 P 1 1I+ 1 D+ D 1I+ 1 D+ D -.2 P C B 8.18 (8.53) k = P 1 4 P 1 8 B T T 2 2 P 2 2 T B 8.19 (8.53) k =.1

190 8 183 (2) B =.75 B B B < (a) - (f) (f) D ω (g) I α D ω T : R n R n ; x x 1 = T (x ) (8.54) (8.19) L x R n γ + (x ) = { x R n x(k) = T k (x ), k =, 1, 2,... } (8.55) k x(k) x x(k) x = DT k (x ) = x T k (x ) = k DT (x(l)) (8.56) D / x (8.56) { µ1 (k), µ 2 (k),..., µ n (k) } = { µ i (k) C det(µ i (k)i n DT k (x )) } (8.57) l=1 t (8.55) m i (k) = k µ i (k), m i = lim m k i(k) = lim µi (k), i = 1, 2,..., n (8.58) k k

191 8 184 y y D D S 2 1 I S S 4 1 I x I x S 2 2 S 4 4 I 2 2 S 4 2 (a) (b) y y D D Ch 4 1 Ch 4 3 I 2 1 Ch 2 1 I x I x Ch 4 2 Ch 4 4 Ch 2 2 I 2 2 (c) (d) y y D D Ch Ch x x (e) (f) 8.2 (8.53) B =.75 (a) B =.15 (b) B =.185 (c) B =.195 (d) B =.197 (e) B =.199 (f) B =.217

192 8 185 y D I x (g) 8.21 (8.53) B =.75 (g) B =.23 (8.55) L ν i (k) = 1 kl log µ i(k), ν i = lim ν 1 i(k) = lim k k kl log µ i(k), i = 1, 2,..., n (8.59) (Lyapounov exponent) x T (8.55) 8.1 (8.59) ν 1 > ν 2 > > ν n (8.6) ν 1 1 e() R n 1 DT (x(l)), l = 1, 2,...

193 T L k D, k I k k ν i >, i = 1,..., k ν i <, i = k + 1,..., n (ICC) ν 1 =, ν i <, i = 2,..., n ν 1 >, 1 n 1 ν i <, i = 2,..., n f() = e(), e() = 1 f(1) = DT (x )e(), e(1) = e() f(1) f(1) f(k) = DT (T k 1 x ))e(k 1), e(k) = e() f(k) f(k) (8.61) e(1) = e() e() f(1) = f(1) f(1) DT (x )e() f(2) = DT (T (x ))e(1) = e() f(1) DT (T (x )) DT (x )e() = e() f(1) DT 2 (x )e() e(2) = e() f(2) f(2) = e() 2 f(1) f(2) DT 2 (x )e() (8.62) f(k) = e() k 1 k 1 i=1 f(i) DT k (x )e() e(k) = e() f(k) f(k) = e() k k i=1 f(i) DT k (x )e() (8.63) 1 = e(k) f(k 1) = e() DT k (x )e() f(k) DT k 1 (x )e()

194 8 187 f(k) e() = DT k (x )e() DT k 1 (x )e() (8.64) k i=1 log f(k) e() = k i=1 DT i k (x )e() DT i 1 (x )e() = log DT i (x )e() DT i 1 (x )e() = log DT k (x )e() e() e k k i=1 1 ν 1 = lim k kl k i=1 log f(i) e() = lim 1 k kl k i=1 log DT k (x )e() e() (8.65) (8.61) (8.65) n n DT (x(l)), l = 1, 2,... n 1 n e 1 (), e 2 (),..., e n () DT (x ) f 1 (1) = DT (x )e 1 (),..., f n (1) = DT (x )e n () (8.66) f l (1), l = 1,..., n e 1 (1), e 2 (1),..., e n (1) g 1 (1) = f 1 (1), e 1 (1) = g 1(1) g 1 (1) g 2 (1) = f 2 (1) (f 2 (1), e 1 (1)) e 1 (1), e 2 (1) = g 2(1) g 2 (1) g n (1) = n 1 f n (1) (f n (1), e i (1)) e 1 (1), e n (1) = g n(1) g n (1) i=1 (8.67) (8.66) (8.67) e i (k), f i (k), g i (k); i = 1, 2,..., n, k = 1, 2,... det[f 1 (k) f 2 (k) f n (k)] = det[g 1 (k) g 2 (k) g n (k)] = g 1 (k) g 2 (k) g n (k) (8.68)

195 ν P D D 2 P 2 P 4 P 8 D 4 P D E B 8.22 (8.53) k =.1 (8.59) det[f 1 (k) f 2 (k) f n (k)] = m 1 (k) m 2 (k) m n (k) (8.68) (8.69) = e ν 1(k)kL e ν 2(k)kL e ν n(k)kl = e (ν 1(k)+ν 2 (k)+ +ν n (k))kl ν 1 (k) + ν 2 (k) + + ν n (k) = 1 kl (8.69) n log g i (k) (8.7) k g i (k) i=1 ν i (k) = 1 kl log g i(k), i = 1, 2,..., n (8.71) ν i = lim ν 1 i(k) = lim k k kl log g i(k), i = 1, 2,..., n (8.72) (8.71) (8.53) b =.75 B.1.22 B

196 8 189 B = P D P < B < P 2 2 D 2 B = P 2 2 B > P B = E ω µ 1 µ 2 = µ 2 e 2πk = (e 2πν ) 2 ν 1 = ν 2 = 2πk 4π = k 2 = ω ω R n 1 1 A (domain of attraction) DA(A) { } DA(A) = x R n lim T k (x) = A k (8.73) ω 2 ω (a), (b) (a) ω(d) ω(i) (b) α ω 2 4 (Fuzzy boundary)

197 8 19 (a) (b) 8.23 (??) 1.4 < x <.9, 1. < y < 1. (a) k =.1, B =.75, B =.15, (b) k =.1, B =.75, B =.185