(a) (b) (c) 4. (a) (b) (c) p.2/27

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2 (a) (b) (c) 4. (a) (b) (c) p.2/27

3 , 2000 p.3/27

4 , 2000 (?) p.3/27

5 , 2000 (?) p.3/27

6 , 2000 (?), 2002 p.3/27

7 (time series) ( 225 ) p.4/27

8 p.5/27

9 ?? p.5/27

10 ??? t ??? t p.6/27

11 ??? t ??? t p.6/27

12 ??? t ??? t p.6/27

13 Power 10 0 Power Frequency Frequency, p.7/27

14 x(n) x(3) x(2) x(1) n p.8/27

15 x(n) x(3) x(2) x(1) n x(n) x(n + 1) p.8/27

16 x(n) x(3) x(2) x(n+1) x(1) (x(1), x(2)) n x(n) p.8/27

17 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

18 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

19 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

20 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

21 y(t+1) 0.5 y(t+1) y(t) y(t) p.9/27

22 ??? t ??? t p.10/27

23 ??? t ??? t x(n + 1) = 4x(n)(1 x(n)) p.10/27

24 ??? y(t+1) t y(t) ??? y(t+1) t y(t) x(n + 1) = 4x(n)(1 x(n)) p.10/27

25 Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke p.11/27

26 Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke p.11/27

27 Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke p.11/27

28 Poincaré Hadamard Kalman Lorenz Rössler Li-Yorke? p.11/27

29 p.12/27

30 = p.12/27

31 = p.12/27

32 = p.12/27

33 = ( ) p.12/27

34 1. 2. ( ) (a) (b) (c) (d) p.13/27

35 y(t + 1) y(t + 2) y(t) y(t) KS p.14/27

36 x(n + 1) = ax(n)(1 x(n)) p.15/27

37 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

38 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

39 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

40 1. (Orbital Instability) 2. (Long-term unpredictability and short-term predictability) 3. (Self-similarity) 4. (Non-periodicity) 5. (Boundedness) p.19/27

41 Orbital instability ɛ(t) = ɛ(0)e λt ɛ(0) ɛ(0) : λ : p.20/27

42 x(n + 1) = 4x(n)(1 x(n)) x(0) = { x(t) t p.21/27

43 ɛ(0) ɛ(t) = ɛ(0)e λt p.22/27

44 ( ) D 0 = 0.63 D 0 = p.23/27

45 { x(n + 1) = 1 + y(n) ax(n) 2 y(n + 1) = bx(n) p.24/27

46 (folding) p.25/27

47 Power Frequency p.26/27

48 1. J. P. Eckmann and D. Ruelle: Ergodic theory of chaos and strange attractors, Reviews of Modern Physics, 57, 3, Part. 1, , P. Grassberger, T. Schreiber and C.Schaffrath, Nonlinear Time Sequence Analysis, International Journal of Bifurcation and Chaos, 1, 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, , ,, J79, 8, , p.27/27

(time series) ( 225 ) / / p.2/66

338 857 255 Tel : 48 858 3577, Fax : 48 858 3716 Email : tohru@ics.saitama-u.ac.jp URL : http://www.nls.ics.saitama-u.ac.jp/ tohru / / p.1/66 (time series) ( 225 ) / / p.2/66 / / p.3/66 ?? / / p.3/66 1.9.8.7.6???.5.4.3.2.1