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1 Tel : , Fax : tohru@ics.saitama-u.ac.jp URL : tohru / / p.1/66
2 (time series) ( 225 ) / / p.2/66
3 / / p.3/66
4 ?? / / p.3/66
5 ??? t ??? t / / p.4/66
6 ??? t ??? t / / p.4/66
7 ??? t ??? t / / p.4/66
8 Power 1 Power Frequency Frequency, / / p.5/66
9 x(n) x(3) x(2) x(1) n / / p.6/66
10 x(n) x(3) x(2) x(1) n x(n) x(n + 1) / / p.6/66
11 x(n) x(3) x(2) x(1) n / / p.7/66
12 x(n) x(3) x(n + 1) x(2) x(1) (x(1), x(2)) n x(n) / / p.7/66
13 x(n) x(3) x(n + 1) (x(2), x(3)) x(2) x(1) (x(1), x(2)) n x(n) / / p.7/66
14 x(n) x(3) x(n + 1) (x(2), x(3)) x(2) x(1) (x(1), x(2)) n x(n) / / p.7/66
15 x(n) x(3) x(n + 1) (x(2), x(3)) x(2) x(1) (x(1), x(2)) n x(n) / / p.7/66
16 x(n) x(3) x(n + 1) (x(2), x(3)) x(2) x(1) (x(1), x(2)) n x(n) x(n) x(n + 1) / / p.7/66
17 y(t+1).5 y(t+1) / / p.8/66
18 ??? t ??? t / / p.9/66
19 ??? t ??? t x(n + 1) = 4x(n)(1 x(n)) / / p.9/66
20 ??? y(t+1) t ???.5.4 y(t+1) t x(n + 1) = 4x(n)(1 x(n)) / / p.9/66
21 Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke / / p.1/66
22 Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke / / p.1/66
23 Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke / / p.1/66
24 Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke? / / p.1/66
25 x(n + 1) = 4x(n)(1 x(n)) / / p.11/66
26 1. 2. x(n + 1) = fµ(x(n)), x(n) R d, n Z (difference equation) (ordinary differential equation) (delay differential equation) (partial differential equation) x(n + 1) = f(x(n)) ẋ(t) = f(x(t)) ẋ(t) = f(x(t), x(t τ)) (autonomous system) ẋ(t) = f(x(t)) (non-autonomous system) ẋ(t) = f(x(t), t) (Input Output System) ẋ(t) = f(x(t), u(t)) / / p.12/66
27 1. x(n + 1) = fµ(x(n)), x(n) R d, n Z x() d x(n) 2. (a) (fixed point) (b) (limit cycle) (c) (torus) (d) (strange attractor) / / p.13/66
28 k R k /Z k R/Z (k 2) 1 k d λ i < (i=1,..., d) λ 1 = λ i < (i=2,...,d) λ i = (i=1,..., k) λ i < (i=k+1,.., d) λ i > (i=1,.., m 1) λ m = λ i < (i=m+1,.., d) / / p.14/66
29 1. (Orbital Instability) 2. (Long-term unpredictability and short-term predictability) 3. (Self-similarity) 4. (Non-periodicity) 5. (Boundedness) / / p.15/66
30 / / p.16/66
31 = / / p.16/66
32 = / / p.16/66
33 = / / p.16/66
34 = ( ) / / p.16/66
35 1. 2. ( ) (a) (b) (c) (d) / / p.17/66
36 y(t + 1) y(t + 2) KS / / p.18/66
37 x(n + 1) = fµ(x(n)) + η(n) y(n) = g(x(n)) + ξ(n) x(n) : n f : k µ : η(n) : ξ(n) : g : y(n) : n y(n) f? / / p.19/66
38 x(n + 1) = fµ(x(n)) + η(n) y(n) = g(x(n)) + ξ(n) x(n) : n f : k µ : η(n) : ξ(n) : g : y(n) : n YES! y(n) f? / / p.19/66
39 v(t) = {, d dt, dy2 (t),..., dm dt 2 dt m (Packard et al.,198) } / / p.2/66
40 v(t) = {, d dt, dy2 (t),..., dm dt 2 dt m (Packard et al.,198) } : (O. Rössler, 1976) dx = y z dt dy = x + ay dt dz = bx cz + xz dt (y, dy/dt) / / p.2/66
41 z(t) x(t) 1 15 / / p.21/66
42 8 z(t) x(t) t / / p.21/66
43 8 z(t) x(t) t x(t) / / p.21/66
44 8 z(t) x(t) t dy/dt x(t) / / p.21/66
45 8 z(t) x(t) t dy/dt x(t) : = / / p.21/66
46 (Takens, 1981; Packard et al.,198) v(t) = {, y(t + τ), y(t + 2τ),...,y(t + (m 1)τ)} y(t + 2τ) y(3) y(2) y(1) t 1. v(1) v(3) v(2) y(t + τ) 2. τ / / p.22/66
47 v(t) = {, y(t + τ), y(t + 2τ),...,y(t + (m 1)τ)} : (O. Rössler, 1976) t z(t) x(t) 1 15 x(t+2τ) x(τ) x(t+τ) / / p.23/66
48 (Takens, 1981) d M C 1 f : M M C 1 g : M R 1 m = 2d + 1 V : M R m V(x) = (g(x), g( f(x)), g( f 2 (x)),...,g( f m 1 (x))) =1 1& m = k + 1 ( ) M A m > 2d m > 2D Embedology (Sauer et al.,1991) m > D 2 (Ding et al, 1994) / / p.24/66
49 (Whitney, 1936) A d A R 2d+1 ( embedding ) C 1 C 1 F 1 1(one-to-one)+ (immersion)) = A F(A) / / p.25/66
50 1 1 A g / / p.26/66
51 A R 1 y 1 (t) / / p.27/66
52 A R 2 y 2 y 2 S y 1 y 1 / / p.28/66
53 A R 3 y 3 y 2 y 3 y 2 S y 1 y 1 / / p.29/66
54 R 3 S S / / p.3/66
55 R m d 1 d 2 D I = d 1 + d 2 m D I D I < D I < m > d 1 + d 2 d 1 = d 2 = d m > 2d / / p.31/66
56 (Sauer et al.,1991) A R k d m > 2d R k R m C 1 g = (g 1, g 2,...,g m ) R m / / p.32/66
57 (Sauer et al.,1991) A R k d m > 2d R k R m C 1 g = (g 1, g 2,...,g m ) R m / / p.32/66
58 (Sauer et al.,1991) A R k D m m > 2D R k R m C 1 1. A 2. A / / p.33/66
59 (Takens, 1981) d M C 1 f : M M C 1 g : M R 1 m = 2d + 1 V : M R m V(x) = (g(x), g( f(x)), g( f 2 (x)),...,g( f m 1 (x))) / / p.34/66
60 (Sauer et al.,1991) Φ R k U A U A D m > 2D τ> A pτ(3 p m) τ 2τ U C 1 V(x) = (g(x), g(φ τ (x)), g(φ 2τ (x)),...,g(φ (m 1)τ (x))) 1. A A / / p.35/66
61 (Sauer et al.,1991) f R k U A U A D m > 2D p m p p A p p/2 Df p U C 1 g 1. A V(x) = (g(x), g( f(x)), g( f 2 (x)),...,g( f m 1 (x))) 2. A / / p.36/66
62 R n S ( ) D D = lim ɛ log N(ɛ) log 1 ɛ N(ɛ) S ɛ n [, 1] ( ) n 1 ɛ = N(ɛ) = 3 n D = 1 3 ( ) n 1 ɛ = N(ɛ) = 2 n D = log 2 3 log 3 = / / p.37/66
63 x(n + 1) = fµ(x(n)) + η(n) y(n) = g(x(n)) + ξ(n) x f x V V F = V f V 1 v F v f F / / p.38/66
64 : dx dt dy dt dz dt = σx + σy = xz + rx y = xy bz σ = 1, r = 28, b = 8/3 / / p.39/66
65 1 5 4 z(t) x(t) x(t) / / p.4/66
66 y(t+1) y(t+2) y(t+4) y(t+8) y(t+1) y(t+2) y(t+2) y(t+4) y(t+1) y(t+2) y(t+4) y(t+8) y(t+1) y(t+2) y(t+2) y(t+4) y(t+1) y(t+2) y(t+4) y(t+8) y(t+1) y(t+2) y(t+2) y(t+4) / / p.41/66
67 y(t+1) y(t+2) y(t+4) y(t+8) y(t+1) y(t+2) y(t+2) y(t+4) y(t+1) y(t+2) y(t+4) y(t+8) y(t+1) y(t+2) y(t+2) y(t+4) y(t+1) y(t+2) y(t+4) y(t+8) y(t+1) y(t+2) y(t+2) y(t+4) / / p.42/66
68 1.5 x t 48[kHz] 12[bits] 1 / / p.43/66
69 x 1 4 x 1 4 x y(t+2).5 y(t+1).5 y(t+22) x y(t+1) x x 1 4 y(t+5) x x 1 4 y(t+11) x 1 4 y(t+3) x y(t+4) x x 3 (t) x x x 1 4 y(t+15) x x 1 4 y(t+2) x 1 4 x 2 (t) x 1 (t) 1 2 x 1 4 / / p.44/66
70 / / p.45/66
71 / / p.46/66
72 Orbital instability ɛ(t) = ɛ()e λt ɛ() ɛ() : λ : / / p.47/66
73 x(n + 1) = 4x(n)(1 x(n)) x() = x(t) t / / p.48/66
74 ɛ() ɛ(t) = ɛ()e λt / / p.49/66
75 Lorenz (1969) Tong & Lim (198) Priestly (1985) Sano & Sawada (1985) Eckmann et al (1985,86) Farmer & Sidorowich (1987) Broomhead & Lowe (1987) Casdagli (1989) Sugihara & May (199) Mees (1991) Mees et al.(1993) Judd & Mees (1995) Lapedes & Farber (1987) Weigend et al (199) Wolpert & Miall (199) Sauer (1993) Cao et al. (1995)... Sugihara & May (199) Tsonis & Elsner (1992) Ikeguchi & Aihara (1993,97), MDL log log semi log / / p.5/66
76 = / / p.51/66
77 / / p.52/66
78 / / p.52/66
79 / / p.52/66
80 / / p.52/66
81 / / p.52/66
82 / / p.52/66
83 / / p.52/66
84 / / p.53/66
85 ẑ(t) z(t) P (z(t) z)(ẑ(t) ẑ) R 1 = t=1 P P (z(t) z) 2 (ẑ(t) ẑ) 2 t=1 t=1 ẑ(t) z(t) E 1 = P (z(t) ẑ(t)) 2 t=1 P (z(t) z) 2 t=1 / / p.54/66
86 z(t) = z(t + 1) z(t), ẑ(t) = ẑ(t + 1) z(t) z(t) ẑ(t) ( ) R 2 = P ( z(t) z)( ẑ(t) ẑ) t=1 P P ( z(t) z) 2 ( ẑ(t) ẑ) 2 t=1 t=1 z(t) ẑ(t) S e = 1 P t=1 I( z(t) ẑ(t)) P / / p.55/66
87 (Sugihara & May, 1991) (Wales, 1991) / / p.56/66
88 Correlation (%) Henon Ikeda Measles Chickenpox Cobalt Correlation (%) Henon Ikeda Measles p m / / p.57/66
89 1/ f α (Tsonis & Elsner, 1992) ɛ(t) ɛ()e λt log ɛ(t) λt ɛ(t) t H log ɛ(t) H log t = log log, semi log / / p.58/66
90 (TI & Aihara, 1994,97) z(t) : ẑ(t) : z(t) = z(t + 1) z(t) : ẑ(t) = ẑ(t + 1) z(t) : {z(t), ẑ(t)} R 1 { z(t), ẑ(t)} R 2 R 1, R 2 R 1 1, R 2 / / p.59/66
91 1 (a) 1 (b) predicted predicted 1 2 actual 2 actual (c).2 (d) predicted.1.2 predicted actual actual / / p.6/66
92 Correlation (%) Correlation (%) Correlation (%) Embedding dimension Embedding dimension Embedding dimension (a) (b) A.dat (c) Correlation (%) Embedding dimension Embedding dimension Correlation (%) Correlation (%) Embedding dimension (d) (e) NYEX (f) / / p.61/66
93 (a) (b) 4. / / p.62/66
94 / / p.63/66
95 23 ( ) / / p.64/66
96 1. ( ) 2. ( ) / / p.65/66
97 ( ) 17: 723 A4 ( [ ]) / / p.66/66
(a) (b) (c) 4. (a) (b) (c) p.2/27
338 8570 255 Tel : 048 858 3577 Fax : 048 858 3716 Email : tohru@ics.saitama-u.ac.jp URL : http://www.nls.ics.saitama-u.ac.jp/ tohru Copyright (C) 2002, Tohru Ikeguchi, Saitama University. All rights reserved.
IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................
Tel : , Fax : URL : tohru / / p.1/12
338 8570 255 Tel : 048 858 3577, Fax : 048 858 3716 Email : tohru@ics.saitama-u.ac.jp URL : http://www.nls.ics.saitama-u.ac.jp/ tohru / / p.1/12 / / p.2/12 1 0.8 x(t) 0.6 0.4 0.2 0 0 10 20 30 40 50 60
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20010916 22;1017;23;20020108;15;20; 1 N = {1, 2, } Z + = {0, 1, 2, } Z = {0, ±1, ±2, } Q = { p p Z, q N} R = { lim a q n n a n Q, n N; sup a n < } R + = {x R x 0} n = {a + b 1 a, b R} u, v 1 R 2 2 R 3
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3 3. (stochastic differential equations) { dx(t) =f(t, X)dt + G(t, X)dW (t), t [,T], (3.) X( )=X X(t) : [,T] R d (d ) f(t, X) : [,T] R d R d (drift term) G(t, X) : [,T] R d R d m (diffusion term) W (t)