Dynamics, Chaos, and Prediction

1. Dynamics, Chaos, and Prediction

2. Aristotle, 384 – 322 BC

3. Nicolaus Copernicus, 1473 – 1543

4. Galileo Galilei, 1564 – 1642

5. Johannes Kepler, 1571 – 1630

6. Isaac Newton, 1643 – 1727

7. Pierre- Simon Laplace, 1749 – 1827

8. Henri Poincaré, 1854 – 1912

9. Werner Heisenberg, 1901 – 1976

10. Dynamical Systems Theory: • The general study of how systems change over time • Calculus • Differential equations • Discrete maps • Algebraic topology • Vocabulary of change • The dynamics of a system: the manner in which the system changes • Dynamical systems theory gives us a vocabulary and set of tools for describing dynamics • Chaos: • One particular type of dynamics of a system • Defined as “sensitive dependence on initial conditions” • Poincaré: Many-body problem in the solar system Isaac Newton 1643 – 1727 Henri Poincaré 1854 – 1912

11. Dr. Ian Malcolm “You've never heard of Chaos theory? Non-linear equations? Strange attractors?”

12. Dr. Ian Malcolm “You've never heard of Chaos theory? Non-linear equations? Strange attractors?”

13. Chaos in Nature • Heart activity (EKG) • Computer networks • Population growth and dynamics • Financial data • Dripping faucets • Electrical circuits • Solar system orbits • Weather and climate (the “butterfly effect”) • Brain activity (EEG)

14. What is the difference between chaos and randomness?

15. What is the difference between chaos and randomness? Notion of “deterministic chaos”

16. A simple example of deterministic chaos:Exponential versus logistic models for population growth Exponential model: Each year each pair of parents mates, creates four offspring, and then parents die.

17. Linear Behavior

18. Linear Behavior: The whole is the sum of the parts

19. Linear Behavior: The whole is the sum of the parts Linear: No interaction among the offspring, except pair-wise mating.

20. Linear Behavior: The whole is the sum of the parts Linear: No interaction among the offspring, except pair-wise mating. More realistic: Introduce limits to population growth.

21. Logistic model • Notions of: • birth rate • death rate • maximum carrying capacityk (upper limit of the population that the habitat will support, due to limited resources)

22. Logistic model • Notions of: • birth rate • death rate • maximum carrying capacityk (upper limit of the population that the habitat will support due to limited resources) interactions between offspring make this model nonlinear

23. Logistic model • Notions of: • birth rate • death rate • maximum carrying capacityk (upper limit of the population that the habitat will support due to limited resources) interactions between offspring make this model nonlinear

24. Nonlinear Behavior

25. Nonlinear behavior of logistic model birth rate 2, death rate 0.4, k=32 (keep the same on the two islands)

26. Nonlinear behavior of logistic model Nonlinear: The whole is different than the sum of the parts birth rate 2, death rate 0.4, k=32 (keep the same on the two islands)

27. Logistic map aaa Lord Robert May b. 1936 Mitchell Feigenbaum b. 1944

28. LogisticMap.nlogo 1. R = 2 2. R = 2.5 3. R = 2.8 4. R = 3.1 5. R = 3.49 6. R = 3.56 7. R = 4, look at sensitive dependence on initial conditions Notion of period doubling Notion of “attractors”

29. Bifurcation Diagram

30. Period Doubling and Universals in Chaos(Mitchell Feigenbaum) R1 ≈ 3.0: period 2 R2 ≈ 3.44949 period 4 R3 ≈ 3.54409 period 8 R4 ≈ 3.564407 period 16 R5 ≈ 3.568759 period 32 R∞ ≈ 3.569946 period ∞ (chaos)

31. Period Doubling and Universals in Chaos(Mitchell Feigenbaum) R1 ≈ 3.0: period 2 R2 ≈ 3.44949 period 4 R3 ≈ 3.54409 period 8 R4 ≈ 3.564407 period 16 R5 ≈ 3.568759 period 32 R∞ ≈ 3.569946 period ∞ (chaos) A similar “period doubling route” to chaos is seen in any “one-humped (unimodal) map.

32. Period Doubling and Universals in Chaos(Mitchell Feigenbaum) R1 ≈ 3.0: period 2 R2 ≈ 3.44949 period 4 R3 ≈ 3.54409 period 8 R4 ≈ 3.564407 period 16 R5 ≈ 3.568759 period 32 R∞ ≈ 3.569946 period ∞ (chaos) Rate at which distance between bifurcations is shrinking:

33. Period Doubling and Universals in Chaos(Mitchell Feigenbaum) R1 ≈ 3.0: period 2 R2 ≈ 3.44949 period 4 R3 ≈ 3.54409 period 8 R4 ≈ 3.564407 period 16 R5 ≈ 3.568759 period 32 R∞ ≈ 3.569946 period ∞ (chaos) Rate at which distance between bifurcations is shrinking:

34. Period Doubling and Universals in Chaos(Mitchell Feigenbaum) In other words, each new bifurcation appears about 4.6692016 times faster than the previous one. R1 ≈ 3.0: period 2 R2 ≈ 3.44949 period 4 R3 ≈ 3.54409 period 8 R4 ≈ 3.564407 period 16 R5 ≈ 3.568759 period 32 R∞ ≈ 3.569946 period ∞ (chaos) Rate at which distance between bifurcations is shrinking:

35. Period Doubling and Universals in Chaos(Mitchell Feigenbaum) In other words, each new bifurcation appears about 4.6692016 times faster than the previous one. R1 ≈ 3.0: period 2 R2 ≈ 3.44949 period 4 R3 ≈ 3.54409 period 8 R4 ≈ 3.564407 period 16 R5 ≈ 3.568759 period 32 R∞ ≈ 3.569946 period ∞ (chaos) Rate at which distance between bifurcations is shrinking: This same rate of 4.6692016 occurs in any unimodal map.

36. Significance of dynamics and chaos for complex systems

37. Significance of dynamics and chaos for complex systems • Apparent random behavior from deterministic rules

38. Significance of dynamics and chaos for complex systems • Apparent random behavior from deterministic rules • Complexity from simple rules

39. Significance of dynamics and chaos for complex systems • Apparent random behavior from deterministic rules • Complexity from simple rules • Vocabulary of complex behavior

40. Significance of dynamics and chaos for complex systems • Apparent random behavior from deterministic rules • Complexity from simple rules • Vocabulary of complex behavior • Limits to detailed prediction

41. Significance of dynamics and chaos for complex systems • Apparent random behavior from deterministic rules • Complexity from simple rules • Vocabulary of complex behavior • Limits to detailed prediction • Universality