Lecture 5 - Thu 2023 09 21
Continue with bifurcations; quick summary of what we have done:
--- What they are: qualitative change in phase portrait. Cannot map one
    into the other continuously.
--- We will start with 1-D; and CLASSIFY them BY DEGREE OF GENERALITY; in
    the order: (1) No assumptions; (2) Restrict to the case where solution
    persists across bifurcation; (3) Restric to the case where there is a
    left-right symmetry. All these bifurcations carry over to more than 1D.
--- For additional details see "baby bifurcation notes" in Web page.
    CRUCIAL PICTURE: \dot{x} = f(x, r). Picture of f "flapping" as r moves.
--- MOST GENERIC BEHAVIOR: SADDLE NODE BIFURCATION
    Creation, annihilation of zeros of f.
    Normal form: \dot{x} = r + x^2; only one case.
    Critical slow down phenomena.
--- Bifurcation diagram ~ "stack the phase portraits for each
                       parameter value in one single picture.
--- NORMAL FORMS: Can be justified using the inverse and implicit theorems.
    The math. statement is that there exist a transformation (local, valid
    only near the bifurcation point) reducing the ode to the normal form.
    Local means: Let the bifurcation for \dot{x} = f(x, r) happen at r = r_c,
    at the critical point x_c. Then the transformation reducing the system to
    normal form is valid in a neighborhood of (x_c, r_c).
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Now we ADD restrictions to the system, and look at LESS GENERIC SITUATIONS.
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                                                      TRANSCRITICAL BIFURCATION
Canonical or NORMAL FORM: \dot{x} = r*x - x^2.

"Simplest/generic" under "symmetry" assumption that a solution (critical
point) persists across the bifurcation [often true in physical systems].
IMPORTANT ............................... "Only one case" is still true.
Do bifurcation diagram.
Show what the "Picture of f "flapping" as r moves" means here.
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                                                          PITCHFORK BIFURCATION
Canonical or NORMAL FORM: \dot{x} = r*x \pm x^3.

"Simplest/generic" under left-right "symmetry" assumption.
This bifurcation corresponds to a "spontaneous symmetry breaking" (explain).
IMPORTANT: unless prior cases, sign changes cannot be mapped away.
           There are TWO cases: Subcritical (soft) and Supercritical (hard).
Do both bifurcation diagrams.
Point out difference in behavior across bifurcation:
          Continuous (subcritical/soft) versus abrupt (supercritical/hard).

Point out "Conservation of stability"; which applies to all three bifurcations
      which we have seen [as well as the ones that we will see later on]
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  WHAT HAPPENS WHEN THE BIFURCATION PARAMETER(s) IN THE EQUATION CHANGE SLOWLY?
  These were done in prior lectures
Hysteresis.
    Example: magnetization.
    Simple model: use s-shaped curve of equilibrium solutions f(x, r) = 0.
                  Two back-to-back saddle nodes.

Other examples we already saw:
    Light switch; rod buckling under pressure [remember the measuring tape].

We will see more later; but they are very common in nature. Other examples:
    Lasing transition: a laser only starts working after the pumping is large
           enough.
    Tipping points in climate. Geological evidence shows that many times in the
           history of Earth, some even triggered a dramatic change in the
           climate.
    Insect breaks; such as locust or bud worms infestations that can kill whole
           forests.
    Phase transitions.
They dramatically break the intuitive idea that if you change something just a
little bit, nothing much will happen.
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                                                           STRUCTURAL STABILITY
How stable are the bifurcations that we have seen to imperfect knowledge of the
system? Basically: do they persist if some perturbation is added to the system?

A very tricky question, because it is related to what kind of "perturbations"
are allowed. If "anything goes" there is very little to nothing that one can
say.
Hence we will limit ourselves to the case where the dynamical system may have
some "hidden" parameters [this includes imperfectly known system constants].
That is:
              \dot{x} = f(x, lambda, h)
where h are
the hidden parameters [there could be many]. I WILL NOT LOOK AT THIS TOPIC IN
ANY MATHEMATICAL DETAIL [Strogatz book has a bit more detail than what I will
do in the Lectures ... I encourage you to READ IT], and will only do an intuitive
treatment.

EXAMPLES:
Saddle-Node or turning point.  Structurally stable.
       Think of the "flapping f" model. Small changes to f will not destroy
       local minimums or local maximums, nor the fact that they are crossing
       zero.

Transcritical. For arbitrary perturbations it is not structurally stable.
       [but if the hypothesis that the critical point persists is enforced,
        then it is structurally stable].
       Show the two ways the transcritical bifurcation diagram can break.
       Case 1: there is no bifurcation at all, but just two branches getting
               close.
       Case 2: two back-to-back saddle nodes.

       Note that in case 2, a small "gap" in parameter values may exist
       that allows the system to go far away if there.
       Note that this "going far away" will happen SLOWLY [because of the
       critical slow down phenomena], and may occur in a very small region.
       In practice it may, or may not, be important.

Pitchfork. Same as transcritical: not structurally stable [unless the left
       right symmetry is enforced].
       EXAMPLE: bending of the measuring tape. Because the tape is not
                symmetric, it mostly only bends in one direction.
       IMPORTANT: the measuring tape shows that the concept of "Spontaneous
                SYMMETRY BREAKING" may fail if you loose this bifurcation.
       Show how the bifurcation diagram may break:
       Case 1: a saddle node plus a branch of solutions that gets close to
               the saddle, but it is isolated.
       Case 2: bifurcation point slides away from the turning point, so that
               the result is a transcritical near a saddle.
               This is what happens for the case of the measuring tape!
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                          Tracking bifurcation branches numerically and ISOLAS.

Using the math. that characterizes bifurcations
     Example: for a saddle, f(x_c, r_c) =  f_x(x_c, r_c) = 0.
a computer can track branches of steady states, and identify where they meet
a bifurcation, so the new branch of solutions can be identified. This way, at
least in principle, one can get the "full picture" of critical points.
EXCEPT ...
   that there may be branches of solutions that are disconnected from the rest,
   and not accessible in this way. These are called ISOLAS.
   Example from stability theory:
           Flow down a pipe; from Poiseuille to turbulence.
   Stability theory states that laminar flow is stable, but experiments show
   that this is not true; or do they?
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