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Understanding Linear Systems: Eigenvalues, Eigenvectors, and Equilibria

Explore eigenvalues, eigenvectors, and equilibria in linear systems of equations, with a focus on small oscillations and stability analysis. Learn how to linearize nonlinear equations and interpret phase plane diagrams. Dive into intriguing examples from ecosystem modeling to love affairs mathematics.

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Understanding Linear Systems: Eigenvalues, Eigenvectors, and Equilibria

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Presentation Transcript

  1. Recall:Finding eigvals and eigvecs

  2. Recall: Newton’s 2nd Law for Small Oscillations Equilibrium: F=0 ~0

  3. Systems of 1st-order, linear, homogeneous equations How we solve it (the basic idea). Why it matters. How we solve it (details, examples).

  4. Solution: the basic idea

  5. General solution

  6. General solution

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

  8. Conversion to 1st order

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

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

  11. 2-eqns: ecosystem modeling reproduction getting eaten eating starvation

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

  13. Ecosystem modeling

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

  15. The linearized system Phase plane diagram

  16. Linear, homogeneous systems

  17. Solution

  18. Interpreting σ

  19. Interpreting σ

  20. General solution

  21. N=2 case yesterday

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

  23. Strange Attractor Need N>3

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

  25. 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)

  26. 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)

  27. Example: Out of touch with feelings

  28. Limit cycle J R

  29. Example: Birds of a feather

  30. 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

  31. 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 …?)

  32. 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 …?)

  33. Example: Birds of a feather

  34. Example: Birds of a feather

  35. J R

  36. J R

  37. J R

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

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