Lecture 8 - Thu 2020 09 24 - Virtual
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Continue prior lecture: Flows in the circle and bifurcations.
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Start with Phase Plane
-1- Introduce system. x_t = f(x, y)  and y_t = g(x, y).
-2- Orbits/trajectories: curves in the plane.
-3- Assume f and g smooth. Therefore trajectories do not cross!
-4- Want to classify possible phase plane portraits. Strong restriction imposed
    by 3 and plane geometry: Jordan curve theorem leads to self-trapping of
    trajectories. Makes classification possible.
-5- Critical points; linearization ... delY_t = A delY
    [A 2x2 matrix and delY = [delx, dely] vector of perturbations to C.P.]
    Classification:
    Write characteristic
    polynomial in terms of:   \Delta = determinant = \lambda_1 * \lambda_2
                              \tau   = trace       = \lambda_1 + \lambda_2
    0 = det(A-\lambda) = \lambda^2 - tau*\lambda + \Delta
      = (\lambda - \tau/2)^2 + \Delta - \tau^2/4

    Draw regions in the \Delta-\tau plane [\tau vertical abscissa].
    Describe:
    saddle, nodes, spirals + improper nodes = star + degenerate nodes +
    centers.

    Saddles: contradiction?
             Trajectories should not cross, but they seem to. Uh?

    Stable and unstable nodes  AND  stable and unstable spirals.
    Transition set-ups [not structurally stable]. Do examples.
      Star: double eigenvalue with two eigenvectors.
      Degenerate node: double eigenvalue with single eigenvector.
      Center.
    Note that going across a transition is a bifurcation!

    Attracting and repelling critical points.
    Point out "degeneracy" when linear theory fails for \Delta = 0.
    Meta-theorem. When is linear picture not structurally stable.
                  Do pictures.

    STABILITY JARGON: Lyapunov stable;  Attracting;  AND  Asymptotically
       stable. Example of attracting but not Lyapunov stable.
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