Lecture 6 - Tue 2024 09 24
Continue with structural stability of bifurcations.
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                                                Structural Stability of Pitchfork
Not structurally stable [unless the left right symmetry is enforced].
Remind them of the possible cases, using "topology" arguments on Bif. Diagram.
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 (crooked or imperfect pitchfork)
        Example: measuring tape!
Measuring tape: 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.

MORE EXAMPLES of pitchforks: <------------------------- Covered in prior lectures
-- Over-damped bead in rotating loop [soft]. Already covered; read in the book.
   See the COMPANION MatLab SCRIPT TO THIS LECTURE [with the extras].
   Example involves neglecting inertia; 2nd order equation becomes 1st order.
   Paradox: how is this possible? 2nd order requires two IC, 1st order only one!
   Peak at the answer: in the phase plane, trajectories converge to a curve!
-- Simple model for column buckling. Done in prior lecture.
   Math details: in a problem set?
-- Model for insect break (spruce budworm): READ FROM THE BOOK.
   Qualitative picture: 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.
   Parameters: 1) Carrying capacity (of budworms) k varies with forest;
                   and depends on how many leaves are there in forest.
               2) Growth rate r (affected by birds)
                  predation rate (by birds).
   Forest assumed quasi-stationary [typical time scale ~10 years, long relative
   to worm life-cycle ~ months]. Predation rate by birds is taken as a function
   of worm density; and turns sharply up at some density were they become the
   food of choice.

   The bifurcation diagram involves two slowly changing parameters, k and r. The
   idea is that r grows with forest growth and k does not change very much.

   See the COMPANION PICTURES TO THIS LECTURE [with the extras]. They show the
   two-parameter bifurcation diagram in 3-D. The "path" followed by the forest
   in parameter space makes a counter-clock-wise loop around the center,
   crashing every time it hits the edge of the upper branch of solutions.
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                                      IMPORTANT NOTE about STRUCTURAL STABILITY
If we allow perturbations "beyond"                    \dot{x} = f(x, lambda, h),
things can get very complicated. For
example, in an equation such as             h*\ddot{x} + \dot{x} = f(x, lambda),
you cannot assume that just because
h is small you can drop the higher order term. In this particular example, the
behavior can be dramatically different if h > 0 or h < 0, even if small.
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                               Math. behind structural stability and normal forms
Having extra parameter h in \dot{x} = f(x, r, h) shows that, upon doing a Taylor
expansion, a perturbed normal form arises, with a constant being added. e.g.:
-- Saddle node: \dot{x} = r - x^2  ---> \dot{x} = c + r - x^2
                No change after renaming parameter; r --> c+r
-- Transcritical: \dot{x} = r x + x^2 ---> \dot{x} = c + r x + x^2
   Critical points
   change to          x = -0.5 r \pm 0.5 \sqrt{r^2-c}
            For c < 0; x_+ > x_-. Critical points never met, no bifurcation
            For c > 0; gap occurs for |r| < \sqrt{c}. Two saddle nodes.
-- Pitchfork left to the students to figure out.
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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?

Numerical methods and something learned from the "Computer exercises with a 1-D
map"
  Two types of errors: Truncation (e.g.: derivative approximated by finite
  difference) and Fixed Point Arithmetic. The errors amount to replacing a
  dynamical system by a near one. Thus, things that are structurally stable
  persist; other may not. In addition errors can "blur" transitions.
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Bifurcation Summary                  [not covered in class, for students to read]
For additional details see "baby bifurcation notes" in Web page.

--- What they are: qualitative change in phase portrait. Cannot map one
    into the other continuously.
    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.
--- Adding restrictions [solution remains through bifurcation, or symmetry]
    generates the cases: Transcritical and Pitchfork.
    Tow cases for Pitchfork: soft and hard.
--- 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).
--- STRUCTURAL STABILITY. Perturb equation, does bifurcation survive?
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                                                            FLOWS ON THE CIRCLE
Start with flows in the circle and EXAMPLE: \dot(theta) = r - sin(theta).
Pendulum with torque. Examine behavior as r grows from 0 to r > 1.
Compute time scale for critical slow down.
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