Lecture 5 - Thu 2024 09 19
Continue with flows on the line: bifurcations.
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                                                       Recall from prior Lecture:
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                                                        SADDLE NODE BIFURCATION
Canonical or NORMAL form:                                    \dot{x} = r + x^2.
BIFURCATION DIAGRAM ~ "stack" the phase portraits for each
                       parameter value in one single picture.
Doable because the phase portrait for a 1-D system is just points in the line,
with arrows (flow direction) between them.
Point out "ONLY ONE CASE".                                      property: [#1C]
    [sign changes in r and x^2, or size changes in x^2, can be transformed out].
CRITICAL SLOW DOWN PHENOMENA [way to detect transition about to happen].
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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].
Note .............................. Only one case, property [#1C] is STILL TRUE.
Do BIFURCATION diagram <=========================================
Show what the "Picture of f "flapping" as r moves" means here.
Point out "Conservation of stability" here and prior one.
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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).
Note: unless prior ones; sign changes cannot be mapped away ....... [#1C] FAILS.
      TWO CASES: Subcritical (SOFT)/Supercritical (HARD).
Do BIFURCATION diagrams <========================================
Point out difference in behavior across bifurcation:
          Continuous (subcritical/soft) versus abrupt (supercritical/hard).
Again "Conservation of stability" principle holds. Applies to all three
bifurcations we have seen [as well as the ones that we will see later on]
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                                                             LECTURE STARTS HERE.
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                                                       EXAMPLES of BIFURCATIONS
Pitchfork
--- Over-damped bead in rotating loop [soft].
                Describe physics. Math details: read in book.
--- Simple model for column buckling.
                Describe physics only. Math details: in a problem set?
Example
    A-symmetrical column [more resistance to bending in one direction than
    the other]. Pitchfork becomes transcritical + saddle node.
    Show bifurcation diagram and physical example [measuring tape].
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  WHAT HAPPENS WHEN THE BIFURCATION PARAMETER(s) IN THE EQUATION CHANGE SLOWLY?
Hysteresis.
    Example: magnetization.
    Simple model: use s-shaped curve of equilibrium solutions f(x, r) = 0.
                  Two back-to-back saddle nodes.

Model for insect break: READ FROM THE BOOK.
  Describe qualitative picture of equilibrium slowly moving as two parameters
  change, going around a pitchfork [2-D surface with cusp singularity].
  Situation modeled: Forest dies periodically, ~15 years or so, due to the
    interaction between the insects (spruce budworm) and the birds].
    Fast dynamics: insects and birds.   Slow dynamics: forest growing.

  SEE PICTURE IN WEB PAGE [with this lecture summary] of how the surface of
  equilibrium solutions looks like.  The one I could not do at the lecture,
  with the (slow) path around the origin of the forest parameters, which is
  the cause for the repeated insect blooms followed by a forest crash.
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                                                  MORE EXAMPLES OF BIFURCATIONS
Light switch [see the notes "Model for switch" in this web page]
Rod buckling under pressure [remember the measuring tape]. Seen before
Lasing transition. Laser starts working after pumping reaches critical value.
Tipping points in climate. Geological evidence shows that many times in the
        history of Earth, some event triggered a dramatic change in climate.
Insect breaks. Locust or bud worms infestations that can kill whole forests.
Phase transitions.

Bifurcations dramatically break the intuitive idea that if you change something
just a little bit, nothing much will happen. They are quite common in nature.
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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?
[See the notes "Structural Stability" in this web page].

Very tricky question, as 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 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 [maybe many]. I WILL NOT LOOK AT THIS TOPIC IN MATH. 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.

EXAMPLE. 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.

EXAMPLE. Transcritical. For arbitrary perturbations not structurally stable.
--- but if the hypothesis that the critical point persists is enforced, then it
is structurally stable.
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.
In case 2, a small "gap" in parameter values may exist that allows the system
to go far away if the parameters place it there. However, 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.
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