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Chaotic Dynamical Systems

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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

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Chaotic Dynamical Systems

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  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)

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