Lecture 17 - Tue 2023 11 07  Two times scales; van der Pol for small gain
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                                                      Two times scales example.
We will consider the vdp equations for small gain:
                                     \ddot{x} + x + eps (x^2-1)\dot{x} = 0; [A]
where 0 < \epsilon \ll 1.
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                                           Bit of physics and Hopf bifurcation.
But first a bit of physics: 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,
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 epsilon > 0; the
nonlinear term always has the same sign. Thus the vdp equation only makes
sense for epsilon > 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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                                                          The two times example.
Do the example using the approach in the notes
       Hopf bifurcations using two timing and complex notation.
I will add to these notes a typed version of the hand-written
notes I used for the lecture.
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