1. Complexity: Ch. 2 Complexity in Systems 1

2. Dynamical Systems Merely means systems that evolve with time not intrinsically interesting in our context What is interesting are certain non- linear dynamical systems So lets work our way up to those Complexity in Systems 2

3. What’s coming in this chapter Dynamical systems Nonlinear dynamical systems Nonlinear chaotic dynamical systems Order in nonlinear chaotic dynamical systems Complexity in Systems 3

4. Dynamical systems Complexity in Systems 4

5. Newtonian Mechanics A triumph of reductionism Led to Laplace’s description of a predictable, clockwork universe Famous 19 th century complacency that physics was essentially complete except for some details Complexity in Systems 5

6. Worms in the apple of certainty Quantum mechanics and Heisenberg’s Principle not sure this is relevant Discovery of presumably well- posed problems that are pathologically dependent on initial conditions Complexity in Systems 6

7. On initial conditions The state of a dynamical system, such as the Solar System, depends on the position and velocity of all of its components at some particular time, called initial conditions, and the application of its equations of motion (i.e., Newton’s laws, updated). Complexity in Systems 7

8. The logistic map The logistic map is the name of a class of dynamical systems that play a large role in complexity theory. Think of it as the logistic model of population growth Complexity in Systems 8

9. The logistic map “Logistic” may come from the French word meaning “to house” but that’s only a guess. “Map” just means function. Complexity in Systems 9

10. The logistic population model Complexity in Systems 10

11. Show us pictures! Complexity in Systems 11

12. Same R, different starting value Complexity in Systems 12

13. Larger R Complexity in Systems 13

14. Slightly larger R Complexity in Systems 14

15. R = 4: SDIC and chaos Complexity in Systems 15

16. Bifurcation map Complexity in Systems 16

17. First bifurcation Complexity in Systems 17

18. Bifurcation map Complexity in Systems 18

19. Onset of chaos (period doubling) Complexity in Systems 19

20. Mitchell Feigenbaum Complexity in Systems 20

21. Feigenbaum’s Constant Complexity in Systems 21 4.6692016

22. Example: logistic map Complexity in Systems 22

23. Complexity in Systems 23 δ = 4.669 201 609 102 990 671 853 203 821 578

24. Takeaway Simple, deterministic systems can generate apparent random behavior Long term prediction for such systems may be impossible in principle Such systems may show surprising regularities: period-doubling and Feigenbaum’s Constant Complexity in Systems 24