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. p.1/27
1. 2. 3. (a) (b) (c) 4. (a) (b) (c) 5. 6. p.2/27
, 2000 p.3/27
, 2000 (?) p.3/27
, 2000 (?) p.3/27
, 2000 (?), 2002 p.3/27
(time series) ( 225 ) p.4/27
p.5/27
?? p.5/27
1 0.9 0.8 0.7 0.6??? 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400 450 500 1 t 0.9 0.8 0.7 0.6??? 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400 450 500 t p.6/27
1 0.9 0.8 0.7 0.6??? 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400 450 500 1 t 0.9 0.8 0.7 0.6??? 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400 450 500 1. t p.6/27
1 0.9 0.8 0.7 0.6??? 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400 450 500 1 t 0.9 0.8 0.7 0.6??? 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400 450 500 1. 2. t p.6/27
10 2 10 2 10 1 10 1 Power 10 0 Power 10 0 10 1 10 1 10 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Frequency 10 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Frequency, p.7/27
x(n) x(3) x(2) x(1) n p.8/27
x(n) x(3) x(2) x(1) n x(n) x(n + 1) p.8/27
x(n) x(3) x(2) x(n+1) x(1) (x(1), x(2)) n x(n) p.8/27
x(n) x(3) x(2) x(n+1) (x(2), x(3)) x(1) (x(1), x(2)) n x(n) p.8/27
x(n) x(3) x(2) x(n+1) (x(2), x(3)) x(1) (x(1), x(2)) n x(n) p.8/27
x(n) x(3) x(2) x(n+1) (x(2), x(3)) x(1) (x(1), x(2)) n x(n) p.8/27
x(n) x(3) x(2) x(n+1) (x(2), x(3)) x(1) (x(1), x(2)) n x(n) p.8/27
1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 y(t+1) 0.5 y(t+1) 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y(t) 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y(t) p.9/27
1 0.9 0.8 0.7 0.6??? 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400 450 500 1 t 0.9 0.8 0.7 0.6??? 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400 450 500 t p.10/27
1 0.9 0.8 0.7 0.6??? 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400 450 500 1 t 0.9 0.8 0.7 0.6??? 0.5 0.4 0.3 0.2 0.1 0 0 50 100 150 200 250 300 350 400 450 500 t x(n + 1) = 4x(n)(1 x(n)) p.10/27
1 1 0.9 0.9 0.8 0.8 0.7 0.7??? 0.6 0.5 0.4 y(t+1) 0.6 0.5 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 50 100 150 200 250 300 350 400 450 500 1 t 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y(t) 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6??? 0.5 0.4 y(t+1) 0.5 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 50 100 150 200 250 300 350 400 450 500 t 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 y(t) x(n + 1) = 4x(n)(1 x(n)) p.10/27
Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke p.11/27
Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke p.11/27
Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke p.11/27
Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke? p.11/27
p.12/27
= p.12/27
= p.12/27
= p.12/27
= ( ) p.12/27
1. 2. ( ) 3. 4. (a) (b) (c) (d) p.13/27
y(t + 1) y(t + 2) y(t) y(t) KS p.14/27
1. 1 2. 3. 2 x(n + 1) = ax(n)(1 x(n)) p.15/27
1. 2. x(n + 1) = fµ(x(n)), x(n) R k (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)) (automonous system) ẋ(t) = f(x(t)) (nonautomonous system) ẋ(t) = f(x(t), t) (Input Output System) ẋ(t) = f(x(t), u(t)) p.16/27
1. x(n + 1) = fµ(x(n)), x(n) R k x(0) (n ) k x(n) 2. (a) (fixed point) (b) (limit cycle) (c) (torus) (d) (chaos) p.17/27
k R k /Z k R/Z (k 2) 0 1 k n λ λ λ λ i < 0 1 = 0 i = 0 i > 0 (i=1,.., m 1) (i=1,..., k) λ (i=1,..., n) i < 0 λ λ (i=2,...,n) i < 0 m = 0 λ (i=k+1,.., n) i < 0 (i=m+1,.., n) p.18/27
1. (Orbital Instability) 2. (Long-term unpredictability and short-term predictability) 3. (Self-similarity) 4. (Non-periodicity) 5. (Boundedness) p.19/27
Orbital instability ɛ(t) = ɛ(0)e λt ɛ(0) ɛ(0) : λ : p.20/27
x(n + 1) = 4x(n)(1 x(n)) x(0) = { 0.1 0.1 + 10 8 1 0.9 0.8 0.7 0.6 x(t) 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 30 35 40 t p.21/27
ɛ(0) ɛ(t) = ɛ(0)e λt p.22/27
( ) D 0 = 0.63 D 0 = 1.585 p.23/27
{ x(n + 1) = 1 + y(n) ax(n) 2 y(n + 1) = bx(n) 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.22 0.2 0.18 0.16 0.14 0.12 0.195 0.19 0.185 0.18 0.175 0.17 0.165 0.16 0.4 1.5 1 0.5 0 0.5 1 1.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0.66 0.68 0.7 0.72 0.74 0.76 0.78 p.24/27
(folding) p.25/27
10 2 10 1 Power 10 0 10 1 10 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Frequency p.26/27
1. J. P. Eckmann and D. Ruelle: Ergodic theory of chaos and strange attractors, Reviews of Modern Physics, 57, 3, Part. 1, 617 656, 1985. 2. P. Grassberger, T. Schreiber and C.Schaffrath, Nonlinear Time Sequence Analysis, International Journal of Bifurcation and Chaos, 1, 3, 521 547, 1991. 3. H. D. I. Abarbanel, R.Brown, J.J. Sidorowich and L. S. Tsimring, The analysis of observed chaotic data in physical systems, Reviews of Modern Physics, 65, 4, 1331 1392, 1993. 4.,, J79, 8, 814 819, 1996. p.27/27