1. Announcements Topics: Work On:
sections 7.7 (improper integrals), (stability of dynamical systems)
* Read these sections and study solved examples in your textbook!
Work On:
Practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the link “SCHEDULE + HOMEWORK”)

2. Definite (Proper) Integrals
Assumptions: f is continuous on a finite interval [a,b].
= real number
proper integral
finite region

3. Improper Integrals
Why are the following integrals “improper”?

4. Improper Integrals Type I: Infinite Limits of Integration
Definition: Assume that the definite integral exists (i.e., is equal to a real number) for every Then we define the improper integral of f(x) on by provided that the limit on the right side exists.

5. Improper Integrals Type I: Infinite Limits of Integration
Illustration:
proper integral
finite region

6. Improper Integrals Type I: Infinite Limits of Integration
Examples:
Evaluate the following improper integrals.
(a) (b)

7. Improper Integrals Type I: Infinite Limits of Integration
When the limit exists, we say that the integral converges. When the limit does not exist, we say that the integral diverges.
Rule:
is convergent if and divergent if

8. Illustration
infinite area
finite area
diverges
converges

9. Improper Integrals Type I: Infinite Limits of Integration
More Examples:
Evaluate the following improper integrals.
(a) (b)

10. Equilibria
Definition: A point is called an equilibrium of the discrete-time dynamical system if Geometrically, the equilibria correspond to points where the updating function intersects the diagonal.
RECALL

11. Stability of Equilibria
An equilibrium is stable if solutions that start near the equilibrium move closer to the equilibrium. An equilibrium is unstable if solutions that start near the equilibrium move away from the equilibrium.

12. Checking Stability of Equilibria
To determine stability, we can use:
Cobwebbing
“Graphical Criteria” (if the the updating function is increasing at the equilibrium)
“Slope Criteria” i.e. the Stability Theorem (provided the slope at the equilibrium isn’t exactly -1 or 1)

13. Checking Stability of Equilibria
To determine stability, we can use:
Cobwebbing
“Graphical Criteria” (if the the updating function is increasing at the equilibrium)
“Slope Criteria” i.e. the Stability Theorem (provided the slope at the equilibrium isn’t exactly -1 or 1)

14. Checking Stability of Equilibria Using Cobwebbing

15. Checking Stability of Equilibria Using Cobwebbing

16. Checking Stability of Equilibria Using Cobwebbing

17. Checking Stability of Equilibria Using Cobwebbing

18. Checking Stability of Equilibria Using Cobwebbing

19. Checking Stability of Equilibria Using Cobwebbing

20. Checking Stability of Equilibria Using Cobwebbing

21. Checking Stability of Equilibria Using Cobwebbing

22. Checking Stability of Equilibria Using Cobwebbing

23. Checking Stability of Equilibria Using Cobwebbing

24. Checking Stability of Equilibria Using Cobwebbing

25. Checking Stability of Equilibria Using Cobwebbing

26. Checking Stability of Equilibria Using Cobwebbing

27. Checking Stability of Equilibria Using Cobwebbing

28. Checking Stability of Equilibria Using Cobwebbing

29. Checking Stability of Equilibria Using Cobwebbing

30. Checking Stability of Equilibria Using Cobwebbing

31. Checking Stability of Equilibria Using Cobwebbing

32. Checking Stability of Equilibria Using Cobwebbing

33. Checking Stability of Equilibria Using Cobwebbing

34. Checking Stability of Equilibria Using Cobwebbing

35. Checking Stability of Equilibria Using Cobwebbing

36. Checking Stability of Equilibria Using Cobwebbing

37. Checking Stability of Equilibria Using Cobwebbing

38. Checking Stability of Equilibria Using Cobwebbing

39. Checking Stability of Equilibria Using Cobwebbing

40. Checking Stability of Equilibria Using Cobwebbing

41. Checking Stability of Equilibria Using Cobwebbing

42. Checking Stability of Equilibria Using Cobwebbing

43. Checking Stability of Equilibria
To determine stability, we can use:
Cobwebbing
“Graphical Criteria” (if the the updating function is increasing at the equilibrium)
“Slope Criteria” i.e. the Stability Theorem (provided the slope at the equilibrium isn’t exactly -1 or 1)

44. Stability Theorem for DTDSs
An equilibrium is stable if the absolute value of the derivative of the updating function is < 1 at the equilibrium, i.e.,
Example:

45. Stability Theorem for DTDSs
An equilibrium is stable if the absolute value of the derivative of the updating function is < 1 at the equilibrium, i.e.,
Example:

46. Stability Theorem for DTDSs
An equilibrium is stable if the absolute value of the derivative of the updating function is < 1 at the equilibrium, i.e.,
Example:

47. Stability Theorem for DTDSs
An equilibrium is unstable if the absolute value of the derivative of the updating function is > 1 at the equilibrium, i.e.,
Example:

48. Stability Theorem for DTDSs
An equilibrium is unstable if the absolute value of the derivative of the updating function is > 1 at the equilibrium, i.e.,
Example:

49. Stability Theorem for DTDSs
If the slope of the updating function is exactly 1 or -1 at the equilibrium, i.e.,
then the equilibrium could be stable, unstable, or half-stable.
Example:

50. Stability Theorem for DTDSs
If the slope of the updating function is exactly 1 or -1 at the equilibrium, i.e.,
then the equilibrium could be stable, unstable, or half-stable.
Example:

51. Stability Theorem for DTDSs
Example: DTDS for a limited population

52. Stability Theorem for DTDSs
Example: DTDS for a limited population
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53. Stability Theorem for DTDSs
Example: logistic dynamical system

54. Stability Theorem for DTDSs
Example: logistic dynamical system

55. Checking Stability of Equilibria
To determine stability, we can use:
Cobwebbing
“Graphical Criteria” (if the the updating function is increasing at the equilibrium)
“Slope Criteria” i.e. the Stability Theorem (provided the slope at the equilibrium isn’t exactly -1 or 1)

56. Graphical Criterion for Stability of Equilibria for a DTDS with an Increasing Updating Function
An equilibrium is stable if the graph of the (increasing) updating function crosses the diagonal from above to below.
Example:

57. Graphical Criterion for Stability of Equilibria for a DTDS with an Increasing Updating Function
An equilibrium is unstable if the graph of the (increasing) updating function crosses the diagonal from below to above.
Example:

58. Graphical Criterion for Stability of Equilibria for a DTDS with an Increasing Updating Function
Example: DTDS for a limited population

59. Graphical Criterion for Stability of Equilibria for a DTDS with an Increasing Updating Function
Example: DTDS for a limited population
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