1. Recall: Pendulum

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

3. Recall: Finding eigvals and eigvecs

4. Nonlinear systems: the qualitative theory Day 8: Mon Sep 20 1.How we solve it (the basic idea). 2.Why it matters. 3.How we solve it (details, examples). Systems of 1st-order, linear, homogeneous equations

5. Solution: the basic idea

6. General solution

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

9. Conversion to 1 st order

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

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

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

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

14. Equilibria

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

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

25. General solution

26. N=2 case Recall

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

28. Need N>3

29. b. repellora. attractor c. saddle Interpreting two σ’s both real

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

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

32. The mathematics of love affairs R(t)= Romeo’s affection for Juliet J(t) = Juliet’s affection for Romeo Response to own feelings (><0) Response to other person (><0) Strogatz, S., 1988, Math. Magazine 61, 35.

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 R J

36. Example: Birds of a feather

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

38. Example: Birds of a feather R J

39. Decaying case: a>b R J

40. Saddle: a<b R J

41. R J

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

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