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# Rで分かる力学系

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### Rで分かる力学系

1. 1. R 2009 3 1 ( ) R 2009 3 1 1 / 23
2. 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. 3. R? R ! ( ) R 2009 3 1 3 / 23
4. 4. = OK R? R ! ( ) R 2009 3 1 3 / 23
5. 5. = OK ⊂ R? R ! ( ) R 2009 3 1 3 / 23
6. 6. = OK ⊂ R? R ! ( ) R 2009 3 1 3 / 23
7. 7. = OK ⊂ R? R ( ) ! ( ) R 2009 3 1 3 / 23
8. 8. = OK ⊂ R? R ( ) ( ) ! ( ) R 2009 3 1 3 / 23
9. 9. = OK ⊂ R? R ( ) ( ) ///////////// ! ( ) R 2009 3 1 3 / 23
10. 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. 11. 1: x = −x ¨ ((x, x) ) ˙ : ⇔ ( ) R 2009 3 1 5 / 23
12. 12. 2: x = −x + c x ¨ ˙ (0, 0) : ( ) ⇒ ( ) R 2009 3 1 6 / 23
13. 13. 3: Van der Pol x = −x + c(1 − x2 ) x ¨ ˙ ⇒ ! ⇒ ( ) R 2009 3 1 7 / 23
14. 14. 4: Lorenz attractor ! ⇒ ( ) R 2009 3 1 8 / 23
15. 15. ( ) ( ) ( ) R 2009 3 1 9 / 23
16. 16. (bifurcation) ? 1 2 Saddle-Node Bifurcation Pitchfork Bifurcation 2 ( ) R 2009 3 1 10 / 23
17. 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. 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. 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. 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. 21. ! ( ) R 2009 3 1 15 / 23
22. 22. ¨ Rossler Attractor x = −y − z ˙ y = x + ay ˙ z = b + (c − x)z ˙ z xz ˙ ( ) R 2009 3 1 16 / 23
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. 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. 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. 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. 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. 28. ( ) R 2009 3 1 22 / 23
29. 29. 3 : , , ( ) R 2009 3 1 22 / 23
30. 30. 3 : , , : Saddle-Node Bifurcation Pitchfork Bifurcation ( ) R 2009 3 1 22 / 23
31. 31. 3 : , , : Saddle-Node Bifurcation Pitchfork Bifurcation ( ) R 2009 3 1 22 / 23
32. 32. 3 : , , : Saddle-Node Bifurcation Pitchfork Bifurcation R ( ) R 2009 3 1 22 / 23
33. 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