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 We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes. —Pierre Simon Laplace, A Philosophical Essay on Probabilities [39] [39]

8. Henri Poincaré, 1854 – 1912 Text In mathematics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long time, return to a state very close to the initial state. The Poincaré recurrence time is the length of time elapsed until the recurrence. The result applies to physical systems in which energy is conserved. The theorem is commonly discussed in the context of ergodic theory,dynamical systems and statistical mechanics.mathematicsonservnamical systemal mechanics.

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 Henri Poincaré 1854 – 1912 Isaac Newton 1643 – 1727

11. Iteration and a path to chaos: Laundry example: Wash once, clothes clean. Wash twice, cleaner still.

12. Iteration and a path to chaos: Some important operations don’t follow this pattern: Rabbit population next year.

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

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

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

16. What is the difference between chaos and randomness?

17. Notion of “deterministic chaos”

18. 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.

19. Linear Behavior

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

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

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

23. Logistic model Notions of: –birth rate –death rate –maximum carrying capacity k (upper limit of the population that the habitat will support, due to limited resources)

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

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

26. Nonlinear Behavior

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

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

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

30. 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”

31. Bifurcation Diagram

32. R 1 ≈ 3.0: period 2 R 2 ≈ 3.44949 period 4 R 3 ≈ 3.54409 period 8 R 4 ≈ 3.564407 period 16 R 5 ≈ 3.568759 period 32 R ∞ ≈ 3.569946 period ∞ (chaos) Period Doubling and Universals in Chaos (Mitchell Feigenbaum)

33. R 1 ≈ 3.0: period 2 R 2 ≈ 3.44949 period 4 R 3 ≈ 3.54409 period 8 R 4 ≈ 3.564407 period 16 R 5 ≈ 3.568759 period 32 R ∞ ≈ 3.569946 period ∞ (chaos) A similar “period doubling route” to chaos is seen in any “one-humped (unimodal) map.

34. Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 ≈ 3.0: period 2 R 2 ≈ 3.44949 period 4 R 3 ≈ 3.54409 period 8 R 4 ≈ 3.564407 period 16 R 5 ≈ 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) R 1 ≈ 3.0: period 2 R 2 ≈ 3.44949 period 4 R 3 ≈ 3.54409 period 8 R 4 ≈ 3.564407 period 16 R 5 ≈ 3.568759 period 32 R ∞ ≈ 3.569946 period ∞ (chaos) Rate at which distance between bifurcations is shrinking:

36. Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 ≈ 3.0: period 2 R 2 ≈ 3.44949 period 4 R 3 ≈ 3.54409 period 8 R 4 ≈ 3.564407 period 16 R 5 ≈ 3.568759 period 32 R ∞ ≈ 3.569946 period ∞ (chaos) Rate at which distance between bifurcations is shrinking: In other words, each new bifurcation appears about 4.6692016 times faster than the previous one.

37. Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 ≈ 3.0: period 2 R 2 ≈ 3.44949 period 4 R 3 ≈ 3.54409 period 8 R 4 ≈ 3.564407 period 16 R 5 ≈ 3.568759 period 32 R ∞ ≈ 3.569946 period ∞ (chaos) Rate at which distance between bifurcations is shrinking: In other words, each new bifurcation appears about 4.6692016 times faster than the previous one. This same rate of 4.6692016 occurs in any unimodal map.

38. Significance of dynamics and chaos for complex systems

39. Apparent random behavior from deterministic rules

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

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

42. 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

43. 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