1. Chaotic Dynamical Systems Experimental Approach Frank Wang

2. Striking the same key

3. Graphic Method Square root function f(x)=sqrt(x) Identity function y=x Vertical to the curve and horizontal to the line

4. Square Root Function

5. Logistic Difference Equation

6. Function Notation seed x0 orbit

7. lambda=2.5

8. lambda=3.1

9. lambda=3.8

10. lambda=3.8 histogram

11. Fixed Point and Periodic Point Fixed point: Periodic point:

12. Period-1

13. Period-2

14. Bifurcation Diagram

15. Period 3 Implies Chaos

16. Sarkovskii’s Theorem (1964)

17. Filled Julia Set Quadratic Map Filled Julia Set

18. Sonya Kovalevskaya Introduction of i to a dynamical system.

19. Kovalevskaya Top

20. C=0.33+0.45 i

21. C=0.5+0.5 i

22. C=0.33+0.57 i

23. C=0.33+0.573 i

24. C=-0.122+0.745 i

25. C= i

26. C=0.360284+0.100376 i

27. C=-0.75+0.1 i

28. Mandelbrot set and bifurcation Mandelbrot set

29. Period 3 window

30. Magnification of the Mandelbrot set

31. Period 7 bulb (2/7)

32. Period 8 bulb (3/8)

33. Period 9 bulb (4/9)

34. Period 13 bulb (6/13)

35. Julia set for (1+2 i) exp(z)

36. Julia set for 2.96 cos(z)

37. Julia set for (1+0.2 i) sin(z)