Lecture 4 - Tue 2023 09 19
From Lecture 3:
--- Comments about numerical methods. Talk about the effect of errors,
    specially fixed point arithmetic. Note that "approximation" errors
    amount to replacing a dynamical system by a near one. Things that
    are structurally stable will persist. But fixed point errors, and
    convergence errors, etc. are a different story.       They "blur"
    transitions, for example.
--- Start with bifurcations. Quick summary is:
    --- Qualitative change in phase portrait. Cannot map one into the other.
    --- Will start with 1-D; and CLASSIFY them BY DEGREE OF GENERALITY.
    --- 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:
    Creation, annihilation of zeros <--- Saddle node bifurcation
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                                                        SADDLE NODE BIFURCATION
Canonical or NORMAL: \dot{x} = r + x^2.

Bifurcation diagram ~ "stack the phase portraits for each
                       parameter value in one single picture.
Point out "only one case"
   Sign changes in r and x^2, as well as size changes in
   x^2, can be transformed out].                                           [#A]
Critical slow down phenomena.
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                                                                   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.
     ^# For \dot{x} = f(x, r), with the bifurcation for r=r_c, at the critical
        point x_c, local means: valid in a neighborhood of (x_c, r_c).
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Now ADD restrictions to the system ==> 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].
Note ........................ Only one case, [#A] 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. <======= Next Lecture
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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 ..... [#A] 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. <============== Next Lecture
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                                                    EXAMPLES <==== Next Lecture
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?
                                                             <==== Next Lecture
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 only 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.
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