R¤ÇÊ¬¤«¤ëÎÏ³Ø·Ï - ¡ÁÊ¬´ô¤ÎÍÍ»Ò¤ò²Ä»ë²½¤·¤Æ¤ß¤ë¡Á

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1 .... R ( ) R / 23

2 : ( id:tkf41, (id:artk ) : 4 1 : : Python, C/C++, PHP, Javascript R : / ( ) R / 23

3 R? R! ( ) R / 23

4 = OK R? R! ( ) R / 23

5 = OK R? R! ( ) R / 23

6 = OK R? R! ( ) R / 23

7 = OK R? R ( )! ( ) R / 23

8 = OK R? R ( ) ( )! ( ) R / 23

9 = OK R? R ( ) ( ) /! ( ) R / 23

10 3 ẍ = x ( ) ẍ = x + cẋ Van der Pol (!) ẍ = x + c(1 x 2 )ẋ Lorenz attractor (!) ẋ = σ(y x) ẏ = x(ρ z) y ż = xy βz ( ) R / 23

11 1: ẍ = x ((x, ẋ) ) : ( ) R / 23

12 2: ẍ = x + cẋ (0, 0) : ( ) ( ) R / 23

13 3: Van der Pol ẍ = x + c(1 x 2 )ẋ! ( ) R / 23

14 4: Lorenz attractor! ( ) R / 23

15 ( ) ( ) ( ) R / 23

16 (bifurcation)? 1 2 Saddle-Node Bifurcation Pitchfork Bifurcation 2 ( ) R / 23

17 1: Saddle-Node Bifurcation (1/2) : ẋ = x 2 r ẋ = 0! ẋ = x 2 rx = (x + r)(x r) x = ± r ( r > 0 ) y y = x^2 r y y dx/dt < 0 x x x dx/dt > 0 dx/dt > 0 dx/dt > 0 dx/dt > 0 (a) r < 0 (b) r = 0 (c) r > 0 ( ) R / 23

18 1: Saddle-Node Bifurcation (2/2) : ẋ = x 2 r (a) r < 0: ( ) (b) r = 0: x < 0 x = 0 x > 0 ( ) (c) r > 0: x < r x = 0 x > r ( ) y y = x^2 r y y dx/dt < 0 x x x dx/dt > 0 dx/dt > 0 dx/dt > 0 dx/dt > 0 (a) r < 0 (b) r = 0 (c) r > 0 ( ) R / 23

19 2: Pitchfork Bifurcation (1/2) : ẋ = x 3 + rx ẋ = 0! ẋ = x 3 + rx = x (x + r)(x r) 3 x = ± r ( r > 0 ) x = 0 y = x^3 + rx y y y dx/dt < 0 dx/dt < 0 dx/dt < 0 dx/dt < 0 dx/dt > 0 x dx/dt > 0 x dx/dt > 0 dx/dt > 0 x (a) r < 0 (b) r = 0 (c) r > 0 ( ) R / 23

20 2: Pitchfork Bifurcation (2/2) : ẋ = x 3 + rx (a) r < 0: x = 0 (b) r = 0: x = 0 (c) r > 0: x < 0 x = r x > 0 x = + r x = 0 ( ) y = x^3 + rx y y y dx/dt < 0 dx/dt < 0 dx/dt < 0 dx/dt < 0 dx/dt > 0 x dx/dt > 0 x dx/dt > 0 dx/dt > 0 x (a) r < 0 (b) r = 0 (c) r > 0 ( ) R / 23

21 ! ( ) R / 23

22 Rössler Attractor ẋ = y z ẏ = x + ay ż = b + (c x)z ż xz ( ) R / 23

23 (g) c = 12.8 (h) c = 13 (i) c = 18 ( ) R / 23 Rössler Attractor (a) c = 4 (b) c = 6 (c) c = 8.5 (d) c = 8.7 (e) c = 9 (f) c = 12

24 Chua s Circuit ẋ = c 1 (y x g(x)) ẏ = c 2 (x y + z) ż = c 3 y g(x) = m 1 x + m 0 m 1 2 ( x + 1 x 1 ) g(x) ( ) R / 23

25 Chua s Circuit (a) c 3 = 50 (b) c 3 = 35 (c) c 3 = 33.8 (d) c 3 = 33.6 (e) c 3 = 33 (f) c 3 = ( ) R / 23

26 CTRNN (3 nodes) τ i ẋ i = x i + n w i j tanh(x j + b j ) j=1 n = 3 tanh ( ) R / 23

27 CTRNN (3 nodes) τ 3 = 1.0 τ 3 = 2.0 τ 3 = 3.0 τ 2 = 1.0 τ 2 = 1.9 τ 2 = 2.0 τ 2 = 4.0 ( ) R / 23

28 ( ) R / 23

29 3 :,, ( ) R / 23

30 3 :,, : Saddle-Node Bifurcation Pitchfork Bifurcation ( ) R / 23

31 3 :,, : Saddle-Node Bifurcation Pitchfork Bifurcation ( ) R / 23

32 3 :,, : Saddle-Node Bifurcation Pitchfork Bifurcation R ( ) R / 23

33 Kathleen T. Alligood Tim Sauer James A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer, 1997 Steven H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Pr, 2001 Randall D. Beer, On the Dynamics of Small Continuous-Time Recurrent Neural Networks, Adaptive Behavior, Vol. 3, No. 4, (1995) ( ) R / 23

ver.1 / c /(13)

ver.1 / c /(13) 1 -- 11 1 c 2010 1/(13) 1 -- 11 -- 1 1--1 1--1--1 2009 3 t R x R n 1 ẋ = f(t, x) f = ( f 1,, f n ) f x(t) = ϕ(x 0, t) x(0) = x 0 n f f t 1--1--2 2009 3 q = (q 1,..., q m ), p = (p 1,..., p m ) x = (q,