Lecture 12 - Thu 2023 10 19
Index theory; Hamiltonian Systems; Liapunov Functions; Limit cycles.
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                                                                  INDEX THEORY
Linear stability: local info near C.P.  However: index theory ==> global info.
Define: index of a closed curve.  Unrelated to stability (time reversal
                                  does not change it).
Examples: compute index for a closed orbit; and near some critical points.

Properties: see page 15-17 Lectures 9-11 Matt Durey.

Define index of an isolated critical point.

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.
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.

Define index for an *isolated* critical point, using (01). Then:
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: show how index theory shows that the Rabbits and Sheeps system does
         not have any periodic orbits.
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                                                            HAMILTONIAN SYSTEMS.
Define Hamiltonian systems (see below). 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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                                                               GRADIENT SYSTEMS.
Gradient systems have the form   \dot{x_n} = -V_{x_n}
for some function (the potential) defined in phase space.

Note that \dot{V} = -(grad V)^2;

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.

Clearly this systems cannot have periodic solutions.
Physics: while Hamiltonian systems correspond to zero dissipation, gradient
         systems are the opposite: when driving-dissipation dominates the
         behavior.
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Physical systems with dissipation combine a Hamiltonian plus a gradient
         structure, with V strictly convex. [A]

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.
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                                                             LIAPUNOV FUNCTIONS.
A Liapunov function is a function, L = L(x), 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.                                             [A]
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 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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                                                                   LIMIT CYCLES.
Define: Isolated closed trajectory.
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.
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