Tsukuba.R#4

1. R
2009 3 1
( ) R 2009 3 1 1 / 23

2. : ( )
!
, @tkf, id:tkf41, (id:artk )
:
4 1
:
http://arataka.wordpress.com
: Python, C/C++, PHP, Javascript
R
: /
( ) R 2009 3 1 2 / 23

3. R?
R
!
( ) R 2009 3 1 3 / 23

4. = OK
R?
R
!
( ) R 2009 3 1 3 / 23

5. = OK
⊂
R?
R
!
( ) R 2009 3 1 3 / 23

6. = OK
⊂
R?
R
!
( ) R 2009 3 1 3 / 23

7. = OK
⊂
R?
R
( )
!
( ) R 2009 3 1 3 / 23

8. = OK
⊂
R?
R
( )
( )
!
( ) R 2009 3 1 3 / 23

9. = OK
⊂
R?
R
( )
( )
/////////////
!
( ) R 2009 3 1 3 / 23

10. 3
x = −x
¨
( ) x = −x + c x
¨ ˙
!) x = −x + c(1 − x2 ) x
Van der Pol ( ¨ ˙
Lorenz attractor ( !) x = σ(y − x)
˙
y = x(ρ − z) − y
˙
z = xy − βz
˙
( ) R 2009 3 1 4 / 23

11. 1:
x = −x
¨
((x, x) )
˙
: ⇔
( ) R 2009 3 1 5 / 23

12. 2:
x = −x + c x
¨ ˙
(0, 0)
:
( )
⇒
( ) R 2009 3 1 6 / 23

13. 3: Van der Pol
x = −x + c(1 − x2 ) x
¨ ˙
⇒
!
⇒
( ) R 2009 3 1 7 / 23

14. 4: Lorenz attractor
!
⇒
( ) R 2009 3 1 8 / 23

15. ( )
( )
( ) R 2009 3 1 9 / 23

16. (bifurcation) ?
1 2
Saddle-Node Bifurcation
Pitchfork Bifurcation
2
( ) R 2009 3 1 10 / 23

17. 1: Saddle-Node Bifurcation (1/2)
: x = x2 − r
˙
x=0
˙
! x = x2 − rx = (x + r)(x − r)
˙
√
x=± r( )
r>0
y y y
y = x^2 - r
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 2009 3 1 11 / 23

18. 1: Saddle-Node Bifurcation (2/2)
: x = x2 − r
˙
(a) r < 0: ( )
(b) r = 0:
x<0 x=0
( )
x>0
(c) r > 0: √
x < √r x=0
( )
x> r
y y y
y = x^2 - r
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 2009 3 1 12 / 23

19. 2: Pitchfork Bifurcation (1/2)
: x = −x3 + rx
˙
x=0
˙
x
! x = −x3 + rx = − 3 (x + r)(x − r)
˙
√
x=± r( )
r>0
x=0
y y y
y = x^3 + rx
dx/dt < 0
dx/dt < 0
dx/dt < 0
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 2009 3 1 13 / 23

20. 2: Pitchfork Bifurcation (2/2)
: x = −x3 + 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 y y
y = x^3 + rx
dx/dt < 0
dx/dt < 0
dx/dt < 0
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 2009 3 1 14 / 23

21. !
( ) R 2009 3 1 15 / 23

22. ¨
Rossler Attractor
x = −y − z
˙
y = x + ay
˙
z = b + (c − x)z
˙
z xz
˙
( ) R 2009 3 1 16 / 23

23. ¨
Rossler Attractor
(a) c = 4 (b) c = 6 (c) c = 8.5
(d) c = 8.7 (e) c = 9 (f) c = 12
(g) c = 12.8 (h) c = 13 (i) c = 18
( ) R 2009 3 1 17 / 23

24. Chua’s Circuit
x = c1 (y − x − g(x))
˙
y = c2 (x − y + z)
˙
z = −c3 y
˙
m0 −m1
g(x) = m1 x + (|x + 1| − |x − 1|)
2
g(x)
( ) R 2009 3 1 18 / 23

25. Chua’s Circuit
(a) c3 = 50 (b) c3 = 35 (c) c3 = 33.8
(d) c3 = 33.6 (e) c3 = 33 (f) c3 = 25.58
( ) R 2009 3 1 19 / 23

26. CTRNN (3 nodes)
n
τi xi = −xi + wi j tanh(x j + b j )
˙
j=1
n=3
tanh
( ) R 2009 3 1 20 / 23

27. CTRNN (3 nodes)
τ3 = 1.0 τ3 = 2.0 τ3 = 3.0
τ2 = 1.0
τ2 = 1.9
τ2 = 2.0
τ2 = 4.0
( ) R 2009 3 1 21 / 23

28. ( ) R 2009 3 1 22 / 23

29. 3 :
, ,
( ) R 2009 3 1 22 / 23

30. 3 :
, ,
:
Saddle-Node Bifurcation
Pitchfork Bifurcation
( ) R 2009 3 1 22 / 23

31. 3 :
, ,
:
Saddle-Node Bifurcation
Pitchfork Bifurcation
( ) R 2009 3 1 22 / 23

32. 3 :
, ,
:
Saddle-Node Bifurcation
Pitchfork Bifurcation
R
( ) R 2009 3 1 22 / 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, 469-509 (1995)
( ) R 2009 3 1 23 / 23