Lecture 8 - Thu 2023 10 03
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We now start the study of 2-D continuum dynamical systems. This have the form:
\dot{x} = f(x, y) and \dot{y} = g(x, y)
for some functions f and g [which we will, generally, assume nice and smooth].
We will be able to classify *ALL* the possible phase space portraits!!     [A1]
--- Introduce notion of of orbits/trajectories in the plane.
    Curves traced by the solution.
    Note: a solution gives ONE orbit, but an orbit corresponds to many slns.:
          If x = X(t), y = Y(t) is a solution, then x = X(t-t0), y = Y(t-t0)
          is also a solution, with the same orbit.
Note [A] is possible because solutions "do not cross", and solutions "trap"
     each other in the plane.
     What kind of spaghetti patterns can you get in a dinner plate if
     every spaghetti must be touching the plate along its full length?
     This is, actually, the question we will be answering. :-)
     Except that on a very large plate, with spaghetti of all lengths covering
     the entire plate; and one extra restriction: the pattern must be "smooth".

Important: the MatLab toolkit in the webpage has some phase plane tools. In
           particular: play with PHPLdemoB
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Quick description of the answer to [A]1, by hand-waving.                   [A2]
The Poincare-Bendixon theorem, which will be seeing later [no proof
though]. From [A2] we see that a key step in understanding the phase plane,
is knowing what happens near a critical point ---> next topic.

Just as in the phase line, the critical points (stationary solutions)
are crucial in "organizing" the phase portrait. However:
--- We need more than just critical points to understand what can happen.
--- 2-D stability of critical points is a bit more complicated than in 1-D.
--- We start by linearizing near a critical point.
STABILITY DEFINITIONS:
Attracting ............... trajectories in neighborhood converge to x*
Liapunov Stable .......... nearby trajectories stay there.
Asymptotically stable .... both attracting and Liapunov.
Do example of attracting but not Liapunov [in circle].
These are the ones in Strogatz's book, but you may encounter also:
Neutrally stable ......... another name for Liapunov.
Strongly stable .......... another name for asymptotically stable.

These definitions apply to critical points. Other kind of solutions will
require other concepts [e.g.: orbital stability for periodic solutions].
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We begin the study by looking at ........................... LINEAR 2-D SYSTEMS
That is:
        \dot{x} = a*x + b*y    and   \dot{y} = c*x + d*y,                    [B]

where a, b, c, and d are constants. Or in vector notation:  \dot{X} = A*X,
where X = [x; y] and  A = [a, b; c d] [MatLab notation here].

In the lecture: instead of X I will use x with a bar below to denote vector,
                and I will write A in the standard form.

WHY DO WE SAY [B] is LINEAR?   Explain.
NOTE: this is a REVIEW of 18.03, so I will go pretty fast here. If you have
      forgotten 18.03, take a look at the notes there.  **OR** you can take
      a look at the examples in "Lectures 7-8 pre.pdf" from M. Durey, posted
      in the web-site.
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                                                                       EXAMPLES
--- Harmonic oscillator, with solutions: x =   R1 cos(omega*t + theta_0)
                                         y = - R2 sin(omega*t + theta_0)
    where y = \dot{x}, omega = sqrt{k/m} for a mass-spring system,
    R2 = omega*R1, and R1 \geq 0 is an arbitrary constant.

    The orbits are ellipses, with principal axes R1 and R2.
--- A diagonal, with diagonal elements a, and -1.
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                         START WITH STUDY/REVIEW OF SOLUTIONS TO LINEAR SYSTEMS.
Recall:
General solution to \dot{X} = A X in terms of eigenvalues and eigenvectors.
Do an example.
Classification of the linear system phase portraits.
More examples.
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