1 / 37

Chaotic Dynamical Systems

Chaotic Dynamical Systems. Experimental Approach Frank Wang. Striking the same key. Graphic Method. Square root function f(x)=sqrt(x) Identity function y=x Vertical to the curve and horizontal to the line. Square Root Function. Logistic Difference Equation. Function Notation. seed x0

carl
Download Presentation

Chaotic Dynamical Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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)

More Related