Lecture 4 - Thu 2020 09 10 - Virtual
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Bifurcations: Recap prior lectures
--- Example: tape buckling [column buckling, other bucklings, clickers].
    Other examples: lasers, turbulence, etc.  ........... 

--- Define it mathematically. Qualitative change in phase
    portrait. Cannot map one into the other.

Bifurcations: We start with 1-D; and CLASSIFY them BY DEGREE OF GENERALITY.
............  For additional details see "baby bifurcation notes" in Web page.

--- \dot{x} = f(x, r). Picture of f "flapping" as r moves.
    Most generic behavior:
    Creation, annihilation of zeros <--- Saddle node bifurcation
    
--- SADDLE NODE BIFURCATION
    [also known as turning point, fold, blue sky, ...]
    Canonical or NORMAL: \dot{x} = r + x^2.
    Bifurcation diagram.
    Critical slow down phenomena.
    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

Note: "Normal forms" can be justified using the Inverse and implicit theorems.
      The mathematical statement is that there exist a transformation (local,
valid only near the bifurcation point^1) reducing the ode to the normal form.
     ^1 If the ode is \dot{x} = f(x, r), and the bifurcation occurs for r=r_c
        at the critical point x_c, then local means: valid in a neighborhood
        of (x_c, r_c).

To get more bifurcations, we ADD restrictions to the system [this means less
generic situations]:

--- TRANSCRITICAL BIFURCATION: NORMAL FORM: \dot{x} = r*x - x^2.
    Justified as "simplest" under "symmetry" assumption that a solution
    (critical point) persists across the bifurcation [often true in physical
    systems].
    Note ........................ Point out #A is still true.
    Do bifurcation diagram.
    Point out "Conservation of stability" here and prior one.

--- PITCHFORK BIFURCATION: NORMAL FORM: \dot{x} = r*x \pm x^3.
    Justified as "simplest" under left-right "symmetry" assumption.
    This bifurcation corresponds to a spontaneous symmetry breaking.
    Note: unless prior ones; here change of sign cannot be mapped away.
          #A fails
    Two cases: Subcritical (soft)/Supercritical (hard).
  
    Do bifurcation diagram.
    Point out difference in behavior across bifurcation:
          Continuous (subcritical/soft) versus abrupt (supercritical/hard).
    Again "Conservation of stability" principle holds.

EXAMPLES:
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.
Transcritical:
    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].

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.

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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