Recall: Pendulum

1. Recall: Pendulum

2. Unstable Pendulum Exponential growth dominates. Equilibrium is unstable.

3. Recall:Finding eigvals and eigvecs

4. Nonlinear systems: the qualitative theoryDay 8: Mon Sep 20 Systems of 1st-order, linear, homogeneous equations How we solve it (the basic idea). Why it matters. How we solve it (details, examples).

5. Solution: the basic idea

6. General solution

7. General solution

8. Systems of 1st-order, linear, homogeneous equations 1. 3. 2. Why important? Higher order equations can be converted to 1st order equations. A nonlinear equation can be linearized. Method extends to inhomogenous equations.

9. Conversion to 1st order

10. Another example Any higher order equation can be converted to a set of 1st order equations.

11. Nonlinear systems: qualitative solution e.g. Lorentz: 3 eqnschaos phase plane diagram • Stability of equilibria is a • linear problem • qualitative description • of solutions

12. 2-eqns: ecosystem modeling reproduction getting eaten eating starvation

13. Ecosystem modeling reproduction getting eaten eating starvation Reproduction rate reduced OR: Starvation rate reduced

14. Equilibria

15. Equilibria

16. Linearizing about an equilibrium 2nd-order (quadratic) nonlinearity

17. Linearizing about an equilibrium 2nd-order (quadratic) nonlinearity small really small small

18. The linearized system cancel

19. The linearized system Phase plane diagram

20. The “other” equilibrium Section 6 Problem 4 ?

21. Linear, homogeneous systems

22. Solution

23. Interpreting σ

24. Interpreting σ

25. General solution

26. N=2 case Recall

27. Interpreting two σ’s a. attractor (stable) b. repellor (unstable) c. saddle (unstable) d. limit cycle (neutral) e. unstable spiral f. stable spiral

28. Strange Attractor Need N>3

29. Interpreting two σ’sboth real a. attractor b. repellor c. saddle

30. Interpreting two σ’s:complex conjugate pair d. limit cycle e. unstable spiral f. stable spiral

31. Interpreting two σ’s a. attractor b. repellor c. saddle d. limit cycle e. unstable spiral f. stable spiral

32. The mathematics of love affairs Strogatz, S., 1988, Math. Magazine61, 35. R(t)=Romeo’s affection for Juliet J(t) = Juliet’s affection for Romeo Response to own feelings (><0) Response to other person (><0)

33. The mathematics of love affairs(S. Strogatz) R(t)=Romeo’s affection for Juliet J(t) = Juliet’s affection for Romeo Response to own feelings (><0) Response to other person (><0)

34. Example: Out of touch with feelings

35. Limit cycle J R

36. Example: Birds of a feather

37. Example: Birds of a feather both real positive if b>a negative if b<a negative b<a: both negative (romance fizzles) b>a: one positive, one negative (saddle …?) c. saddle decay eigvec growth eigvec

38. Example: Birds of a feather J R

39. Decaying case: a>b J R

40. Saddle: a<b J R

41. J R

42. Homework Sec. 6, p. 89 #4: Sketch the full phase diagram: ? ? #6: Optional

43. Why a saddle is unstable J R No matter where you start, things eventually blow up.