Lecture 6 - Tue 2023 09 26

From a topic covered in 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.
%
% ----------------------------------------------------------------------------- %
                                EXAMPLES of physical systems with bifurcations:
Pitchfork
--- Over-damped bead in rotating loop [soft].
                Describe physics. Math details: read in book.
    Equations, maybe just write them [check derivation in book, simple mechanics].
    Write viscous limit and do details of pitchfork.
    See the COMPANION MatLab SCRIPT TO THIS LECTURE [with the extras].
    What exactly does "large dissipation" mean? DIMENSIONAL ANALYSIS
    PARADOX: show them how, in phase plane, trajectories converge to a curve.
 

--- Simple model for column buckling.
    Describe physics only. Math details: in a problem set?

Model for insect break (spruce budworm): 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.

    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 is assumed quasi-stationary [typical time is ~10 years, long
    relative to worm life-cycle (measured in months). Predation rate by
    birds is taken as just a function of worm density; and turns sharply
    up at some density were they become the food of choice.

    The bifurcation diagram in this case 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.
%
% ----------------------------------------------------------------------------- %
Note on 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} + x = f(x, lambda);
    you cannot assume that just because h is small you can drop the higher
    order term. In particular, note that the behavior can be dramatically
    different if h > 0 or h < 0, even if small.
%
% ----------------------------------------------------------------------------- %
                                             FLOWS ON THE CIRCLE [Next Lecture]
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.
%
% ----------------------------------------------------------------------------- %
EOF