Lecture 10 - Tue 2024 10 08 --- Phase plane examples and conservative systems.
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A phase plane system  \dot{x} = f(x, y)  and  \dot{y} = g(x, y) is  CONSERVATIVE
if there is a function E = E(x, y) [the "energy"] defined in the
phase plane [or in some sub-region of interest] such that:
         (1)   E is defined on an open set, twice continuously differentiable.
         (2)   $\grad E \neq 0$ (except at C.P.)
         (3)   The solutions satisfy 0 = dE/dt
               [by the chain rule this is the same as f*E_x + g*E_y = 0].

Via the implicit function theorem, (2) implies that through each regular point
(i.e.: not critical) in the phase plane there is a unique level curve for E.
Then from (3) the curve has to be an orbit, because (f, g) is orthogonal to
\grad E.
........ i.e.: THE ORBITS ARE LEVEL CURVES FOR E, and ANY LEVEL CURVE FOR E
               IS AN ORBIT while away from critical points.

For good behavior we require the following properties related to C.P.
(4) The fixed points are isolated.
(5) grad E \neq 0 at non-critical points.... already included in (2)
(6) grad E = 0 at critical points .......... already included in (2)
(7) The hessian of E is non-degenerate at the C.P., so surface nearby
    is either a saddle or a local max./min. This also means that the
    critical point will be either a saddle or a center.

"Energies" without these properties are useful, but you need to analyze them
separately. See the example in [E10.1] below.

Note:   CONSERVATIVE SYSTEMS CANNOT HAVE sinks/attractors or sources/repellers.
Reason: all the points near such a CP belong to an orbit that converges to the
        CP as either t \to \infty or t \to -\infty. This would make E constant
(equal to value at CP) near the CP, thus violating (2).
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                             Simple standard tricks to get conserved quantities:
1) Multiply equation(s) by something and add, to get exact differential.
   Example (particle in a potential V = V(x))   m \ddot{x} + V^\prime(x) = 0.
   Multiply by \dot{x} and use
   the chain rule to get:        dE/dt = 0,   where   E = 0.5*m*\dot{x}^2 + V.

   In this case E is the mechanical energy.
   Note: local maximums of V produce saddles  in E (thus saddle critical points).
         local minimums of V produce minimums in E (thus center critical points).

   We saw an example of this in lecture 9 (1-D mass with double well potential).

2) In 2-D look at the equation   dy/dx = \dot{y}/\dot{x} = g(x, y)/f(x, y).
   This is an equation involving
   only x and y. If you can solve it (e.g. by separation of variables), the
   solution will have an integration constant. Writing this integration constant
   in terms of x and y produces a conserved quantity.
   Note that dy/dx is the equation for the ORBITS [i.e.: the curves, no time].

   This approach also works for the particle in a potential example, which we
   can write as  \dot{x} = v   and   \dot{v} = -V^\prime/m
   Then  dv/dx = -v^\prime/(m*v), or  m*v*dv = - V^\prime dx, which leads to
         0.5*m*v^2 + V = constant = E.
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                                                                         EXAMPLES
These examples are developed in some detail in Durey's Lectures_9_11_pre.pdf
I will only sketch the main points in the lecture; and I may not cover all
the examples below, but you SHOULD read Lectures_9_11_pre.pdf --- which does
include more examples!

Bottom of page 9.  EXAMPLE: The importance of being ISOLATED ............ [E10.1]
     \dot{x} = x*y  and \dot{y} = -x^2  has E = x^2+y^2 conserved.
     Do not get closed orbits near min./max.
     E fails: (4) The fixed points are isolated.
              (6) grad E = 0 at critical points.
Top of page 6.     EXAMPLE: Rabbits and Sheep ........................... [E10.2]
     \dot{x} = 3*x - x^2 - 2*xy (rabbits)  Note: logistic terms + competition.
     \dot{y} = 2*y - y^2 - x*y  (sheep)    Rabbits have edge on growth;
                                           Sheep have edge on competition.     
    Interested only in x, y non-negative quadrant. Show quadrant invariant.           
    Compute fixed points and Jacobian, and classify CP.
    Use Poincare Bendix to argue stable/unstable manifolds of saddle must
        go to the nodes or infinity. Exclude periodic [index theory, later].
    Model too simplistic: one species always dies "competitive exclusion".

Bottom of page 1/top of page 2 .......................................... [E10.3]
Example: \dot{x} = x+e^{-y};    C.P. (-1, 0);  A = | 1  -1|
         \dot{y} = -y;                             | 0  -1|      
Eigenvalues/vectors: {lambda =  1; v = (1, 0)} and {lambda = -1;   v = (1, 2)}
Nullclines: 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.
Nullclines split plane into 4 regions [\dot{x} and \dot{y} positive/negative].
     Start by drawing orbits from \dot{x} = 0 nullcline "out".
     Draw stable manifolds for the saddle (cannot cross nulcline \dot{x} = 0).
     Use the direction field to draw phase portrait.

Page 12. Nonlinear pendulum; Use energy to draw phase plane ............. [E10.4]
This was done in prior lecture.

Page 13. Dissipative nonlinear pendulum ................................. [E10.5]
For small dissipation, use the equation for the energy [which is
dissipated slowly] to construct phase diagram.
Here the energy gives us a first peek at Liapunov functions. <===========

Pages 10-11. Reversible systems ......................................... [E10.6]
                                THIS TO NEXT LECTURE
   Define reversible.
   Prove: linear centers stay centers if they are on the symmetry axis.
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