Basins of Attraction

1. Basins of Attraction Dr. David Chan NCSSM TCM Conference February 1, 2002

3. Outline • Definitions and a Simple Example • Newton’s Method in the Real Plane • Newton’s Method in the Complex Plane • The Biology of a Species

4. Dynamical Systems A Dynamical System is a set of equations which model some changing phenomena. They often take the form of • Difference Equation(s) • Ordinary Differential Equation(s) • Partial Differential Equation(s)

5. Examples: • Precalculus - Population growth - Drug Dosage - Loans

6. More Examples: • Calculus -Function Iteration -Fixed points -Bifurcations -Periodic Orbits -Newton’s Method

7. Attractors An attractor is a point or a collection of points on which the system can limit. These often take the form of -Fixed Points -Periodic Orbits -Strange Attractors

8. Basins of Attraction The Basin of Attraction for an attractor is the set of points which limit on the attractor.

9. Example: Function iteration Two fixed points x=0 Has a basin of attraction of (-1,1). x=1 Has a basin of attraction of {-1,1}. Everything else goes to infinity!

10. Calculus—Newton’s Method • Used to find roots of a function by using tangent lines. • Formula:

12. Location of a horizontal tangent line.

13. Consider: Questions: • What is the basin of attraction for 0? • Are there other attractors other than the roots? • In what way(s) can Newton’s Method fail?

14. Question: • What is the basin of attraction for 0? • Answer: • There is a part of each ‘hump’ of sine which will give 0 as a root.

20. Question: • Are there other attractors other than the roots? • Answer: • There are periodic points.

21. Question: • In what way(s) can Newton’s Method fail? • Answer: • Move to the next hump at the same location.

22. Newton’s Method in the Complex Plane • Same method but involves using • complex arithmetic. • This is 2-dimensional. • has n different solutions. • And…

23. Z2 - 1

24. Z3 - 1

25. Z4 - 1

26. Z5 - 1

27. Z2 - 1

28. Z2.001 - 1

29. Z2.005 - 1

30. Z2.01 - 1

31. Z2.02 - 1

32. Z2.03 - 1

33. Z2.04 - 1

34. Z2.06 - 1

35. Z2.1 - 1

36. Z2.2 - 1

37. Z2.3 - 1

38. Z2.4 - 1

39. Z2.5 - 1

40. Z2.6 - 1

41. Z2.7 - 1

42. Z2.8 - 1

43. Z2.9 - 1

44. Z2.95 - 1

45. Z3 - 1

46. 1 x 1

47. 0.1 x 0.1

48. 0.01 x 0.01

49. 0.000001 x 0.000001

50. Newton’s Method: • Method fails at z=0. • Method fails at lots of points which • map to zero (eventually). • All these points have points of all • three colors near them.