Chaos Experiments for the Advanced Lab

1. Chaos Experiments for the Advanced Lab Robert DeSerio University of Florida Advance Lab Topical Conference Ann Arbor, Michigan -- July 2009

2. Advertisement Chaos is cool! Books Movies Video games Music (from 2005 tour)

3. “…it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible...”, Poincaré Jules Henri Poincaré (1854–1912) Aleksandr Lyapunov (1857–1918)

4. Edward Norton Lorenz (1917 – 2008) Lorenz attractor 1972 paper: "Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas? Les objets fractals, forme, hasard et dimension (1975) The Fractal Geometry of Nature (1982) Benoît Mandlebrot (1924-) Mandelbrot set

5. Three Chaos Experiments • Water drops • Example of fluid system • Chaotic pendulum • Example of mechanical system • Electronic Duffing oscillator • Example of electronic system

6. Analyses • Return maps , Poincaré sections, Attractors (strange and normal) • Plotting methods and concepts associated with chaotic sequences • Parameter fitting • Determine model parameters from data • Fractal dimensions and Lyapunov exponents • Determine attractor parameters • Simulations • For checking and hypothesizing

7. Drop chaos apparatus

9. data acquisition photogate signal goes to gate of counter. counter continually counts internal 20 MHz clock pulses, mode is set so rising edge of gate triggers buffered readings of count. software reads buffer data, ignores satellite drops, computes tn saves data to file.

10. raw data

11. 2-D Return Map t n+1 t n

12. 3-D Return map

13. apparatus

14. Chaos Interface

15. raw data q t

16. Electronic Duffing Oscillator

17. raw data q t

18. data acquisition/pendulum

19. data acquisition/elec. chaos

20. data • 200 per drive cycle • only explicit measurement --- • 50,000 cycles or n=10,000,000 q s • by differentiation of q • 2p/200 (n mod 200) • cyclic variable from 0 to 199/200 (2p)

21. phase space variables q, w, f Autonomous: System of first order differential equations without explicit time Equation of motion If angular acceleration is determined along with q, w, f, then the middle equation could be used in a linear regression to find k, m, G. G’, e. f is a cyclic variable, perfect for making Poincaré sections trivial to construct which then become two-dimensional and easy to display. Simulator uses the Runge-Kutte algorithm with these equations at 40 points per drive cycle. • definitions/initial conditions: • q --- position • w --- velocity • f --- drive phase Poincaré–Bendixson theorem: System at the minimum required dimensionality of three for chaos.

22. phase space 2-dimensional 3-dimensional w trajectory for driven harmonic oscillation q

23. driven H-O motion qvs. t graph 3-d phase space representation transient steady state motion (attractor)

24. Phase plot w q

25. low damping

26. high damping

27. asymmetric wells

28. Electronic chaos data 1.6 kHz (31 s of data) 70 Wresistance 170 Wresistance

29. fit

30. Simulator

31. comparison

32. Simulation w/o damping

33. fractal dimension

34. Lyapunov

35. exponential solutions

36. local behavior sq=0.002 rad sw=0.02 rad/s

37. delay map (Dt= period) Poincaré section

38. Conclusions • Chaos is interesting to students • My pendulum and circuit allow for nearly paradigmatic studies including • Creation of multiple Poincaré sections • Analysis of equation of motion, fractal dimension, and Lyapunov exponent • Drop chaos adds return map analysis • Chaos experiments should be an integral part of the advanced lab