2023 09 28 SUGGESTED READINGS
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Below a list of things that I have covered very lightly, or
not at all, with suggestions of where to read more about it.
The list is NOT ordered, by the way.

1) STRUCTURAL STABILITY.
  Covered in Strogatz' book. Also, see the notes posted in the web below
  the lecture summaries.
  Note that we used this idea when classifying the bifurcations, and
  neglected any situation that could be destroyed by a small perturbation
  [i.e.: in Taylor expansion, any terms not guaranteed to be zero, taken
         as non-zero].

2) IMPERFECT BIFURCATIONS AND CATASTROPHES.
   Related to structural stability (same references).
   What happens if there is an imperfection in the system, violating
   the assumptions. This is modeled by incorporating another parameter
   in the system, that measures departure from assumptions.
   Example:  \dot{x} = h + r*x - x^3
            [soft pitchfork with hidden parameter h=0].
   Note that the "Model for insect break" also involves a two-parameter
   model.

3) NORMAL FORMS. How to justify them and get them analytically?
   See the course notes: "Baby Normal Forms".

4) Toy model for column buckling; an example of a soft pitchfork.
   This maybe assigned in a problem set.

5) Model for insect break (spruce budworm).
   See Strogatz' book and paper references there.

6) Mathematical example with a hard pitchfork. Add 5-th order term to the
   normal form.
   \dot{x} = r*x + x^3 - x^5.
   LEFT TO YOU: do the bifurcation diagram for this example. You now have
                the tools to do it.

7) Hysteresis (bifurcation parameter changing slowly).
   This was done in the lectures; try it again on your own. What happens
   for \dot{x} = f(x, r) is such that the equilibrium solutions [i.e.:
   f(x, r) = 0] yield an s-shaped curve [two back-to-back saddle nodes]
   and r is varied back and forth, crossing the two bifurcation points?
   EXAMPLE: magnetization.
   Strogatz' book has several examples across its pages.

8) Population biology models.
   Example: Population growth/logistic equation (Explain).
    \dot{N} = k*N  with  k = r*(1-N/N_0).
   N_0 = carrying capacity of environment.
   A "derivation" is in Strogatz/ book.

   More complicated models of the form \dot{N} = f(N) have been
   proposed [see the "Allee effect" in Strogatz book; you can also
   "google" it]

   More generally, models that include more than one variable are used
   in various fields;
                        \dot{N} = F(N), where now N is a vector.
   Example:
   In epidemiology models like this are used, where the components of N
   may refer to different populations [sick people in initial stage, sick
   people in hospital, cured people, people never infected, and so on].
   PDE models are used when N is non-uniform ... but you can stay within
   ODE by splitting N by regions as well.

9) Imperfect bifurcation; example:
   Non-symmetric elastic blade under pressure (measuring tape)

   The symmetric blade yields a soft pitchfork:
                                                      \dot{x} = r*x - x^3 [A]
    The asymmetry introduced in the measuring tape
    (curvature) breaks the symmetry, but x=0
    (unbent) remains a solution.
    Simple model:                             \dot{x} = r*x + h*x^2 - x^3 [B]
    where h is a small number.

    Draw the bifurcation diagram and show that:
    The pitchfork breaks into a trans-critical and a nearby saddle-node,
        where the saddle-node is on the side opposite to the bias.
        I DREW THIS DIAGRAM IN THE BLACKBOARD.
    Note that conservation of stability preserved.

    TASK LEFT TO STUDENTS: verify that the bifurcation diagram I drew
                           corresponds to [B].
    Hint: the critical points follow from
         0 = r*x + h*x^2 - x^3 = (r + h*x - x^2)*x, so that
         x = 0   and   r = - h*x + x^2 = (x-h/2)^2 - h^2/4.

    Interpretation.
    Think of a measuring tape blade: If you slowly increase the pressure
    it will always bend towards the side where the tape is concave. But a
    small tap can make it bend the other way. Further: this "bent the
    other way" state exists and is stable *below* the pressure at which
    the tape bends in the biased direction. This is exactly what the broken
    bifurcation diagram I drew predicts.

    Note 1: you can get the same effect with the "Over-damped bead in
            rotating loop" if the loop is not perfectly round, so one
            side is favored.

10) Read from Strogatz' book:
    --- laser theshold example.
    --- population growth example [also covered in class].
    --- critical slow down phenomena for a saddle node bifurcation
        [also covered in class] Recall NORMAL FORM: \dot{x} = r + x^2.

11) EXAMPLE:  Model for a switch
    See the notes posted in the web below the lecture summaries.
    Justify the equations used here.
    Then write the a-dimensional form:
       time unit = nu/k; and length unit = a.
       Small parameter = m k/nu^2; neglect inertia.
       Non-dim numbers left: zeta/k; p/(a k); ell/a;
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