Lecture 6 - Thu 2020 09 17 - Virtual
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Continue and finish with prior lecture; from structural stability for
trans-critical onward.

Finishing details.
--- To students: read in the book the example for a laser theshold.
--- To students: read the population growth example in the book.
--- Critical slow down phenomena for saddle node bifurcation
    [NORMAL FORM: \dot{x} = r + x^2].

Example of a critical slow down phenomena for saddle node bifurcation IN THE
CIRCLE. Then
        \dot{theta} = f(theta, r),  f periodic of period 2*pi.

Then assume f is positive for r > r_0, develops a zero at theta_0 for r=r_0,
and goes below zero in a segment around x_0 for r < r_0.
Note: solution periodic for r > r_0, with period  T = \int_0^2*pi dtheta/f.
      Does not contradict "no periodic in 1-D" because this applies to the
      line! In the circle theta+2*pi ~ theta and you can get periodic.
Note: as r to r_0, T grows to infinity.
Note: for r > r_0 solutions approach the stable critical point.
Note: for r = r_0 there is a semi-stable critical point and it takes
      infinite time to go around the circle.

Time scale characterizing the critical slow down. How does T grow as
epsilon^2 = r-r_0 vanishes [this defines epsilon].
For epsilon small T is dominated by values of theta near the minimum, where
(generically) f is quadratic. Thus, modulo constants:

T ~ \int dtheta/(epsilon^2 + theta^2) ~ 1/epsilon = 1/sqrt{r-r_0}.

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STRUCTURAL STABILITY (SS) summary, with "preturbed" normal forms.

Saddle node normal form:                 \dot{x} = h+r + x^2              SS

Trans-critical normal form:              \dot{x} = h + r*x - x^2.     Not SS
If we KNOW x=0 stays as solution, then   \dot{x} = (r+h)*x - x^2.         SS

Pitchfork normal form:                   \dot{x} = h + r*x - x^3.     Not SS
If we KNOW x = 0 stays as solution,
      only symmetry is broken, then      \dot{x} = r*x + h*x^2 - x^3. Not SS

If h preserves the symmetry, then:      \ dot{x} = (r+h)*x - x^3.         SS

All this follows by the technique of expanding and only keeping dominant
terms [and then normalizing so all the coefficients are \pm 1].
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