1. Numerical Solutions and chaotic behavior 1 Numerical Solutions and Chaos

2. A Taste of Code def derivs(y, tsoln): th, a, pt, pa = y tdot = (pt - (1 + cos(a)) * pa)/(2.0 - cos(a)**2) adot = pa - tdot * (1.0 + cos(a)) ptdot = -g*(2.0 * sin(th) + sin(th + a)) padot = -tdot * (tdot + adot) * sin(a) -g * sin(th + a) return np.array((tdot, adot, ptdot, padot)) Numerical Solutions and Chaos 2

3. A Boy’s First Computer Numerical Solutions and Chaos 3

4. Neolithic Computing Bendix G15 $60,000 450 tubes 2160 words of memory punched paper tape i/o optional dent in the front panel 10 char/sec typewriter Numerical Solutions and Chaos 4

5. Richard Hamming (1915-1998) The purpose of computing is insight, not numbers. Numerical Solutions and Chaos 5

6. Begin with g = 0 (no gravity) Numerical Solutions and Chaos 6

7. numerical results (g-free) Numerical Solutions and Chaos 7 timeE  0.011.000.001.00 0.00 100.081.0039.86-1.781.000.69 200.031.0077.052.191.000.03 800.151.00310.92-1.861.000.67 900.081.00348.052.221.000.06 1000.11.00388.12-2.231.000.08

8. tip motion (g-free) Numerical Solutions and Chaos 8

9. Now turn on gravity Numerical Solutions and Chaos 9

10. numerical results (g active) timeE  0.011.001.570.00 10.041.001.68-0.69-1.71-0.59 20.041.000.741.05-3.05-0.84 80.011.002.06-39.790.400.32 90.081.00-0.96-45.441.320.84 100.061.00-0.41-46.791.25-0.45 Numerical Solutions and Chaos 10

11. tip motion (g active) Numerical Solutions and Chaos 11

12. Chaos Numerical Solutions and Chaos 12

13. Edward Lorenz (1917-2008) Meteorologist who contributed to chaos theory. Encountered pathological sensitivity to initial conditions in early weather prediction simulations. Published Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas? Numerical Solutions and Chaos 13

14. Lorenz’ Informal Definition Chaos: When the present determines the future, but the approximate present does not approximately determine the future. Numerical Solutions and Chaos 14

15. Some properties of chaos Numerical Solutions and Chaos 15

16. Numerical Chaos Numerical Solutions and Chaos 16

17. using numbers We generated two numerical solutions to the double pendulum with initial conditions that differed by one part in 1000 (0.1%). For each solution we computed the location of the tip of the pendulum, and then computed the distance, as a function of time, between the two tips. Numerical Solutions and Chaos 17

18. Tip offsets versus time Numerical Solutions and Chaos 18

19. Tip offsets versus time (log) Numerical Solutions and Chaos 19