Lecture 12 - Thu 2024 10 17
Hamiltonian Systems; Gradient Systems; Liapunov Functions; Limit cycles.
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Left without doing from prior lecture:                      HAMILTONIAN SYSTEMS
Show that H is conserved for        \dot{q_n} = H_{p_n}   \dot{p_n} = - H_{q_n}
for n=1:N
Then  \dot{H} = Sum_n H_{q_n} \dot{q_n} + H_{p_n} \dot{p_n}
              = Sum_n H_{q_n} H_{p_n}   - H_{p_n} H_{q_n}  = 0.
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Gradient systems have the form   \dot{x_n} = -V_{x_n}          GRADIENT SYSTEMS
for some function (the potential)
defined in phase space.                    Note that \dot{V} = -(grad V)^2; [A]

Thus, if V has only isolated points where grad V = 0 (fixed points of system)
the solutions approach the local minimums of V.
    For a gradient system: local isolated maximums of V are repellers.
                           local isolated minimums of V are attractors.
These systems cannot have periodic solutions.
Physics: while Hamiltonian systems correspond to zero dissipation, gradient
         systems are the opposite: when driving-dissipation dominates the
         behavior.
Note: if V is strictly convex, then it has a single minimum, and [A] drives
      the solution towards this minimum.
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Physical systems with dissipation combine a Hamiltonian plus a gradient
         structure, with V strictly convex.

For an arbitrary system \dot{Y} = F(Y) in n-dimensions, the equation
for the evolution of the volume of a differential blob in phase space
is:
     \dot{V} = \dive(F) ......  Show this left to a problem set.

Use this to show that a Hamiltonian system preserves volume and physical
dissipation shrinks volume.
For a Hamiltonian \dive(F) = \sum (\dot{q_n})_{q_n} + (\dot{p_n})_{p_n}
      = \sum (H_{p_n})_{q_n} - (H_{q_n})_{p_n} = 0.
For a Gradient    \dive(F) = \sum (\dot{x_n})_{x_n} = - \sum (V{x_n})_x_n
                           = - Laplacian V  < 0 if V is strictly convex.
Note: obviously, if V has a saddle (not convex), near it it can expand volume!
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A Liapunov function is a function, L = L(x),                 LIAPUNOV FUNCTIONS.
defined on phase space such that (grad L) = 0 only at
critical points and \dot{L} < 0 elsewhere.  EXAMPLE: V for a gradient system.

Note that: the existence of a Liapunov function means that there are no
           periodic solutions.                                             [B]
Prove this by assuming a periodic solution and looking at the the
      variation of L over one period  DL = \int \dot{L} dt.
      Periodicity says DL = 0.  The Liapunov property says DL < 0.

Note: for proof do not strictly need \dot{L} < 0 away from critical points.
Enough to have: \dot{L} \leq 0, with the set where \dot{L} = 0 being
                transversal to the solutions [so on a periodic orbit there
                would only be a discrete set of such points].

Example: \ddot{x} + \dot{x}^3 + x = 0; take  L = 0.5*(\dot{x}^2 + x^2).
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Define: Isolated closed trajectory.                                LIMIT CYCLES.
Example: draw stable, unstable, and semi-stable limit cycles.
Example: in polar coordinates, let  \dot{\theta} = 1 and \dot{r} = r*f(r),
         f smooth.    Then r>0 fixed points of f yield limit cycles.
Example: van der Pol oscillator
         \ddot{x} + mu*(x^2-1)*\dot{x} + x = 0,  mu > 0.
         Decay for x large and growth for x small.
         Only one critical point.
         Later on we will see this yields a limit cycle. For now: numerical.
Draw what phase plane looks like for $mu$ small.
     Give hand-waving justification, and explain how averaging yields radius:
     Solution approximately a circle.
     Plug in approximate solution into equation  \dot{E} = mu*(1-x^2)*\dot{x}^2,
     where E = 0.5*(x^2+\dot{x}^2) = 0.5*r^2 to compute radius for which
     \dot{E} is approximately zero.
As $mu$ grows, sinusoidal character is lost, later on we will see what happens
     when $mu$ is large. For example; x = x(t) gets "jerky", and looks a bit
     like a square wave.
Explain physics behind the equation [electronic oscillator].
Analogous situations: Dad pushing swing with little kids; clocks that do tic-toc
... why?
Why are limit cycles a thing? It would seem that having an orbit that closes
exactly on itself is a very fickle thing, easy to destroy! Explain why not.
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                                                         van der Pol oscillator
For a somewhat detailed derivation of the van der Pol oscillator
equation, check [in [R] below]:
    Chapter V (Self-sustained oscillations)
    Part B (Forced oscillations in self-sustained systems)
The circuitry shown corresponds to a valve triode [there were no transistors
when van der Pol derived the equation], but the idea is the same if instead a
transistor triode is used. Note that valve components are no longer used for
consumer electronics, but they are still used for applications that require
either high power and/or small signal to noise ratio.

[R] Nonlinear vibrations in mechanical and electrical systems
    by Stoker, J. J., New York, Interscience Publishers, 1950 and 1963
You can find the book in the Library, as well as on-line in the
Internet Archive:
1963 version https://archive.org/details/nonlinearvibrati0000stok
1950 version https://archive.org/details/nonlinearvibrati00stok
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