Lecture 9 - Thu 2023 10 05
Finish 2-D Linear systems. Start with phase plane.
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Note: the materiel here is covered both in Strogatz's book, as well as in
"Lectures 9-11 pre.pdf" from M. Durey, posted in the web-site.
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Do example with repeated eigenvalue \neq 0, and a single eigenvector.
\dot{x} = a*x;  and  \dot{y} = a*y + x; with a < 0 [a > 0 similar].
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Do a "Romeo and Juliet" example:      \dot{R} = -a*J   and   \dot{J} = b*R,
   a, b, positive constants.
Romeo is a bit of a jerk, and says Juliet is "clingy" if she expresses love
for him, and he becomes concerned when she rejects him. Neither does he have
a "center" for his love (it is all Juliet driven). Juliet does not have a
center either, and simply responds to Romeo. Not surprisingly, the result is
a never ending cycle of love and hate, with frequency sqrt{a*b}.

This is a structurally unstable situation/model. If either Juliet or Romeo
develops a bit of a "center", things will (literally) spiral out of control.

You can find more examples in the "Lectures 7-8 pre.pdf" from M. Durey, posted
in the web-site [as well as in the book].
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*** TOPIC COVERED IN Lecture 8 ***                            BEGIN PHASE PLANE
Objective: get phase portrait. Behavior is controlled by these main elements:
1) Fixed points and their type and stability.
2) Periodic solutions, which correspond to closed orbits.
3) Cycle graphs [we will define these later].
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                                                                       NUMERICS
The numerical schemes mentioned for 1-D problems [In the notes, not the
lectures: Forward and Backwards Euler, Improved Euler, Runge-Kutta, etc]
can be extended to n-D.
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                                                                        EXAMPLE
Example: \dot{x} = x+e^{-y};  C.P. (-1, 0);  lambda =  1;   v = (1, 0);
         \dot{y} = -y;                       lambda = -1;   v = (1, 2);
                                             A = | 1  -1|
                                                 | 0  -1|
Draw nullclines; note that nullclines are NOT orbits in general.
     In this case, one of them (y = 0) yields 3 orbits.
Note y \to 0 as t \to \infty, so \dot{x} ~ x + 1 (i.e. x ~ e^t for t large).
     y \to \pm \infty as t \to -\infty.
Draw now the stable and unstable manifolds for the saddle (note stable
     manifolds cannot cross nulcline \dot{x} = 0).
Now use the direction field to draw phase portrait.
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*** TOPIC COVERED IN PRIOR Lectures ***                EXISTENCE and UNIQUENESS
In drawing phase portraits, it is important to know solutions do not cross.
Quote IVP existence and uniqueness theorem for \dot{x} = f(x), with simplified
hypothesis: f continuous with bounded derivatives in neighborhood of I. Data.
Note:       OK for derivative to have discontinuities!
           [but here we will assume cont. partial derivatives].

===> A closed trajectory splits phase portrait in two [inside and outside].
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*** TOPIC COVERED IN PRIOR Lectures ***      PHASE PORTRAIT NEAR CRITICAL POINT
When does the linearization give the correct
phase portrait near a critical point?
ANSWER: when small perturbations of the linear system's matrix do not change
        the linear phase portrait (i.e.: the "meta-theorem" applies.)

EXAMPLE: center becomes spiral due to nonlinear terms.    *** DO this lecture
EXAMPLE: det(A) = 0 degeneracy --->                       *** DO this lecture
         saddle or node due to nonlinear terms.
         Look at \dot{x} = x^3 or \dot{x} = -x^3 and \dot{y} = -y.
         Mixes also possible: \dot{x} = \pm x^2
EXAMPLE: Stars can become nodes or spirals.
EXAMPLE: Even worse A = 0 can turn into very complicated patterns.
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                                                          CONSERVATIVE SYSTEMS.
First, do example with potential mechanical systems,
and get conservation of energy.

General definition: E defined on open sets, with $\grad E \neq 0$,
except at C.P. --- explain why.

Show: Conservative systems cannot have sinks/attractors or sources/repellers.
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                                  Example: 1-D mass with double well potential.
Generally: local max. of V are saddles and local min. are centers [non-linear!]
Graphical illustration of periodic motion [roll back and forth on graph of V].
Draw phase portrait.

The standard tricks to get conserved quantities:
1) Multiply equation(s) by something and add, to get exact differential.
2) In 2-D look at dy/dx = \dot{y}/\dot{x}. If you can solve this by separation
   of variables, the integration constant is a conserved quantity.
   Note that: dy/dx is the equation for the ORBITS [i.e.: the curves, no time].
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