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Recall: Pendulum

Recall: Pendulum. Unstable Pendulum. Exponential growth dominates. Equilibrium is unstable. Recall: Finding eigvals and eigvecs. Nonlinear systems: the qualitative theory Day 8: Mon Sep 20. Systems of 1st-order, linear, homogeneous equations. How we solve it (the basic idea).

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Recall: Pendulum

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  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.

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