Lecture 9 - Thu 2024 10 03
Finish 2-D Linear systems. Start with (nonlinear, generic) 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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Finish example with repeated eigenvalue \neq 0, and a single eigenvector.
\dot{x} = lambda*x;  and  \dot{y} = lambda*y + x.
Solution:  x = x0 e^{lambda t}   and   y = (y0 + t x0) e^{lambda t}
Do the case lambda < 0 [lambda > 0 is the same, revert arrows].
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               STABILITY DEFINITIONS [Illustrate with drawings and give examples]
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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EXAMPLE ["Romeo and Juliet"]      \dot{R} = -a*J   and   \dot{J} = b*R,  [RJ]
                                  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}. Note that [RJ]
is just the harmonic oscillator!

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 [see Lectures_1_2_pre.pdf by
Matt Durey in the Lectures web page] Forward and Backwards Euler, Improved
Euler, Runge-Kutta, etc., can all 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.)
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EXAMPLE: center becomes spiral due to nonlinear terms.
         \dot{x} = -y + a*x*(x^2+y^2)  and  \dot{y} = x + a*y*(x^2+y^2)
         Linearization at the origin gives a center.
         Go to polar, using  x*\dot{x} + y*\dot{y} = r*\dot{r}
                             x*\dot{y} - y*\dot{x} = r^2*\dot{theta}
         Then  \dot{r} = a*r^3   and   \dot{theta} = 1.
               Spirals for r \neq 0.

EXAMPLE: det(A) = 0 degeneracy --->                     *** Done in lecture 8.
         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. Done in prior lectures.

EXAMPLE: Even worse A = 0 can turn into very complicated patterns.
         Done in prior lectures: points that are half saddle and half node.
         We will see more complicated examples later

EXAMPLE: \dot{x} = -x*(1-x)*(1+x) = -x + x^3  and  \dot{y} = -2*y.
         Yields: Saddle, node, and saddle on real axis at x = -1, 0, 1.
         Special property: nullclines are also "special" solutions. UNUSUAL!
         This example is on top of page 4 in the notes:
         Lectures_9_11_pre.pdf Matt Durey notes.
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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 equations are:          \ddot{x} + V^\prime = 0;  V = -0.5*x^2 + 0.25*x^4.
                  This example is on the bottom of page 8 in the notes:
                  Lectures_9_11_pre.pdf Matt Durey notes.

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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