Lecture 11 - Thu 2024 10 10  Continue with phase plane.
Reversible systems; Index theory; Hamiltonian Systems.
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Here we continue with the topics in Lecture 10. As in there, you can find extra
details in either Strogatz' book, or in Durey's Lectures_9_11_pre.pdf (posted in
the Web page).    While I may not cover everything in Lectures_9_11_pre.pdf; you
SHOULD read Lectures_9_11_pre.pdf (and the book).
Note: below, stuff like "Pages 10-11 LD" means pages in Lectures_9_11_pre.pdf
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Pages 10-11 LD.                                              REVERSIBLE SYSTEMS
Define reversible. Motivate by example of mass in potential:
                   \ddot{x} + V^\prime(x) = 0, (or pendulum).
Prove: LINEAR CENTERS STAY CENTERS if they are on the symmetry axis of a
       reversible system.
Point out that reversible and conservative are NOT the same:

Example of a conservative system that is not reversible: left to a problem set.

Example of a reversible system that is not conservative:
  Given by drawing phase plane.
  --- Place two saddles along the x-axis, at some x = \pm a.
  --- Place two nodes   along the y-axis, at some y = \pm b.
      Node at y = b unstable and node at y = -b stable.
  --- Connect repelling main axis of node at y = b with attracting direction
      of the saddles, and make the repelling secondary axis the y axis.
  --- Reflect across x-axis and reverse arrows.

More examples [to be read by students]
Bottom page 10 LD. Reversible system  \dot{x} =  y-y^3  and  \dot{y} = -x-y^2.
Bottom page 11 LD. Reversible, but not Conservative system.
     \dot{x} = -2*cos(x)-cos(y)   and   \dot{y} = -2*cos(y)-cos(x).
     Reversible across line x = -y.    x to -y; y to -x; t to -t;
     Note system has a symmetry across x = y;  but it does not flip time
     [so, not related to reversibility ... need the time reversal for the
      argument that linear centers are also nonlinear centers].
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We have seen that a system can be reversible but not conservative.         SKIP
Can a system be conservative, but not reversible?
    Yes: look at  \dot{x} = E_y   and   \dot{y} = -E_x                      [A]
    and  select E
         with level curves near a local minimum that have no symmetry.

[A] belongs to a very important class of conservative systems:
    "Hamiltonian systems".  See below.
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pages 15-17 LD (and book)                                          INDEX THEORY
Linear stability: local info near C.P.    Index theory: GLOBAL info.
Define: index of a closed curve.  Unrelated to stability (time reversal
                                  does not change it).
Example. Compute index for a closed orbit.
Example. Compute index of curve enclosing: saddle, node, center, spiral.

Important property: moving curve does not change index [as long as motion
          does not bring curve across a CP, where index is not defined].    
Use this to: define index of an *isolated* critical point.

Properties (prove them).
01) Curve deformed continuously *without* crossing a fixed point .... I = const.
02) Field is changed continuously *without* vanishing on curve ...... I = const.
03) If curve does not include any fixed points ...................... I = 0.
04) Reversing field direction (t \to -t) does not change the index.
    Thus: index and stability are not related.   [Proof: this adds pi to angles]
05) If closed curve is actually an orbit (i.e.: periodic solution) .. I = 1.
06) Note (03) and (05) ==> a closed orbit must include critical points.
07) Introduce curve surgery and algebra of curves. Then     I_{A+B} = I_A + I_B.
08) Let C be a closed curve without fixed points, enclosing the fixed points
    P_1 ... P_N.    Then I_C = \sum_n I_n;    where I_n = index P_n.
    Note that this generalizes (03).                        Proof: use (07).

EXAMPLE: Index theory shows the Rabbit-Sheep system has no periodic orbits.
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                                                            HAMILTONIAN SYSTEMS
Define Hamiltonian systems. Motivate with masses in a potential.
   Assume an even dimension system, with
   phase space parameterized by  q = (q_1, ..., q_N) and p = (p_1, ..., p_N).
   Let H = H(q, p). Then
                         \dot{q_n} = H_{p_n}   \dot{p_n} = - H_{q_n}
   is a classical Hamiltonian system, where H is the Hamiltonian (= Energy),
   and it is a conserved quantity [SHOW IT]
All the equations of non-dissipative classical mechanics have this form.
I doubt we will have time to look at Hamiltonian systems, unfortunately.
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