Lecture 16 - Thu 2024 10 31
Two times scales for van der Pol for small gain; Hopf Bifurcation
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                                                      Two times scales example.
van der Pol equations for small gain:
                                     \ddot{x} + x + mu (x^2-1)\dot{x} = 0; [A]
where mu = eps nu,
0 < eps \ll 1, and nu = \pm 1.
Remind students of the motivation done at the end of Lecture 15, then
cover section 2.1 of the notes "Weakly Nonlinear Things: Oscillators".

After done, point "weird" behavior here: some sort of bifurcation occurs here
as mu crosses zero. There a limit cycle of amplitude ~ 2 [circle of radius 2]
switches from unstable (mu < 0) to stable (mu > 0), with a center at mu = 0.

Intuitively this is NOT what you expect from, say, an electric circuit as you
crank up the gain. What should happen is that there should be no oscillation
if the gain is too small, then at some critical gain oscillations should start
... but they should be small! Same thing happens when you push a swing; the
amplitude grows from zero as you push harder.            WHAT IS GOING ON HERE?
Explanation: [A] involves some scaling and non-dim that hides the physics,
             as explained next.
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                            van der Pol, a bit of physics, and Hopf bifurcation.
The equation represents the balance of several effects
a) Inductance-Capacitance (oscillations).
   This alone would give                                      \ddot{x} + x = 0;
   where a time scale such that the frequency is "one" has
   been selected [we use this time scale everywhere].
b) Resistance [Ohm's law]. This alone would give      \ddot{x} + R \dot{x} = 0;
   where 1/R > 0 is the typical time of decay for
   the current in the circuit in units of the frequency.
c) A (nonlinear) feed-back loop that pumps energy
   into the system. This alone would give  \ddot{x} = A \dot{x} - B x^2 \dot{x};
   where A and B are positive constants.
   Here we assume "small" amplitude and keep only the first nonlinear
   correction. There is no x \dot{x} term because the feedback does not depend
   on the sign of x.

Adding it all yields:     \ddot{x} + (R-A)\dot{x} + B x^2 \dot{x} + x = 0.

Then we assume that A is slightly bigger than R, with \epsilon = A - R,
nu = 1, and select a scale for x in which B = epsilon.

WHY THIS SCALE?  We will see next that this is the scale at which the
oscillations arise, i.e.: for [A] the limit cycle has size O(1).

NOTE THAT this last transformation can only be done when nu = 1; the
nonlinear term always has the same sign. Thus the vdp equation only makes
sense for mu > 0.  It is easier to see the physics by writing the
equation in the form [choose scale for x so that B = 1]

          \ddot{x} - \epsilon \dot{x} + x^2 \dot{x} + x = 0.  [B]

Then what happens is: for epsilon < 0, the circuit does not oscillate,
and the solution remains at x \equiv 0. For epsilon > 0, a limit cycle
of size sqrt{epsilon} appears, and we can reduce [B] to [A] by scaling x.

THIS IS AN EXAMPLE OF A --------------> HOPF BIFURCATION
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                                                   Hopf bifurcation Normal form.
The arguments above show that, if we do not scale
the nonlinear term to be in-sink with the pumping, the equation for the radius
of the limit cycle in van der Pol takes the form [mu small here]

  da/dtau = mu*a - a^3, so that for mu > 0 there is an equilibrium radius
                        of size \sqrt{\mu}, while there is no limit cycle
                        for mu < 0 [the unstable limit cycle for mu < 0 in
                        vdp is "fake". The scaling of the nonlinear term by
                        \sqrt{(A-R)/B} cannot be done for A < R!!!

  This is the normal form for a supercritical (soft) Hopf bifurcation
  For a subcritical (hard) Hopf the normal form is da/dtau = mu*a + a^3.

  Do bifurcation diagrams and explain difference. Note analogy with pitchfork.
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                                                                   WHAT IS NEXT?
Getting periods for periodic solutions with high accuracy [Pincar\'e-Lindstedt].
Explain generic set-up under which Hopf happens, and intuition.
   [Two complex eigenvalues crossing imaginary axis balancing
    some nonlinearity. Explain why dominant non-linearity is cubic].
Two-timing calculation to get Hopf in generic system. How do you tell from
    it if nonlinearity stabilizes/destabilizes; including degenerate case.
Bifurcations in the plane [local and global].
Stability of limit cycles in 2-D and 3-D using Poincare map.
Higher order bifurcations and torii.
Landau's hypothesis for chaos.
Start with 3-D, and chaos.
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