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 - Loans - Drug Dosage - Population growth

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

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

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

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. Z 2 - 1

24. Z 3 - 1

25. Z 4 - 1

26. Z 5 - 1

27. Z 2 - 1

28. Z 2.001 - 1

29. Z 2.005 - 1

30. Z 2.01 - 1

31. Z 2.02 - 1

32. Z 2.03 - 1

33. Z 2.04 - 1

34. Z 2.06 - 1

35. Z 2.1 - 1

36. Z 2.2 - 1

37. Z 2.3 - 1

38. Z 2.4 - 1

39. Z 2.5 - 1

40. Z 2.6 - 1

41. Z 2.7 - 1

42. Z 2.8 - 1

43. Z 2.9 - 1

44. Z 2.95 - 1

45. Z 3 - 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.

51. Precalculus: Biology One species-Adult & Children

52. Simplified Equations

53. Questions: 1. What happens if r>0 and a>0. This models competition.

56. Questions: 2. What happens if r<0? Everything dies out!

57. Questions: 3.What happens if r>0 and a<0? This models cannibalism.