Lecture 7 - Thu 2023 09 28
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                                                           FLOWS ON THE CIRCLE.
GENERIC CASE: \dot{theta} = f(theta); theta = angle; ** f PERIODIC **

Draw "circle-phase diagram" for "typical" example [graphic technique].
Note that; if f never vanishes ... get PERIODIC solution.
     Does not contradict "no periodic in 1-D" because this applies to the
     line! In the circle theta+2*pi ~ theta and you can get periodic.
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                                                                        EXAMPLE
Over-damped pendulum with torque. \dot(theta) = r - sin(theta).
Examine behavior as r grows from 0 to r > 1.
Compute time scale for critical slow down.
--- Saddle Node bifurcations at (r =  1, theta =  pi/2),
                            and (r = -1, theta = -pi/2).
    Note the critical slow down phenomena for r slightly above 1
    or slightly below -1 [ghost of a critical point].
    Solutions are periodic, but period is very long!

 (i)   QUALITATIVE DESCRIPTION: up and down equilibriums move towards each
       other, and vanish in a saddle-node bifurcation.
 (ii)  CRITICAL SLOWING DOWN means here: Wobbling behavior. Pendulum moves
       fast on one side, slows down to a crawl on the other.
 (iii) Calculate PERIOD DEPENDENCE on how close to critical the parameter
       is: Period T = O(1/sqrt{r-r_c}.   [next topic].
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The CRITICAL SLOW DOWN phenomena for a saddle node bifurcation IN THE CIRCLE.
Equation:
          \dot{theta} = f(theta, r),  f periodic of period 2*pi.

Assume: f is positive for r > r_0, develops a zero at theta_0 for r=r_0,
        and goes below zero in a segment around x_0 for r < r_0.
        At r=r_0 minimum of f at theta_0 is generic [second derivative > 0].
We can "normalize" f_r(x0, r0) = 1, so that r-r0 = distance from minimum
        of f to the theta axis.

Note: solution periodic for r > r_0, with period  T = \int_0^2*pi dtheta/f.
Note: as r to r_0, T grows to infinity.
Note: for r < r_0 solutions approach the stable critical point.
Note: for r = r_0 there is a semi-stable critical point and it takes
      infinite time to go around the circle.

Time scale characterizing the critical slow down. How does T grow as
epsilon^2 = r-r_0 vanishes [this defines epsilon].
For epsilon small T is dominated by values of theta near the minimum, where
(generically) f is quadratic. Thus, modulo constants:

T ~ \int dtheta/(epsilon^2 + theta^2) ~ 1/epsilon = 1/sqrt{r-r_0}.
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POTENTIALS IN THE CIRCLE
For the line, and \dot{x} = f(x), you can always find a "potential" so that
                  \dot{x} = - V'(x).
Note then that  \dot{V} = - (V'(x))^2 ... flow tries to minimize V.

On the circle, \dot{theta} = f(theta), can have a "pseudo-potential" such
that f = V',
but generally NOT a "true potential" (V periodic in theta).

QUESTION TO STUDENTS: what do you need from f to guarantee a periodic V?

Note that, if a true potential exists, then solutions go to the minimums
of V, and you cannot have a periodic solution (a periodic continuous function
always has a minimum).
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Back to the "torsion driven pendulum" equation: \dot{theta} = r-sin(theta).
--- This equation also shows in simple models for:
    #1 Josephson junctions ==>  See \S 4.6 Strogatz Book.
    #2 Phase-locking

#1 The equation for the phase difference across the junction behaves like a
   driven pendulum. The pendulum going around is associated with generating
   an oscillating voltage of very high frequency.

#2 Assume two oscillators which are very stable. The oscillations may occur
   in a high dimension dynamical system, but the orbits are attracted to a
   periodic solution there. We will see examples of this later ... the whole
   field of electronics is based on being able to do stuff like this. The
   same phenomena happens in biology [e.g. fireflies, cicadas, heart].
Given this:

(1) Each oscillator can be characterized by a "phase", where is it in its
    orbit. Parameterize the orbit so that the speed at which the solution
    moves along the orbit is a constant, and the orbit length is 2*pi.
    Then single oscillator reduces to:
    \dot{theta} = omega (a constant).                          DO DRAWING!

(2) Now couple two oscillators (or more). If the coupling is not too strong,
    the orbits will not change much [remember, very stable stuff]. Then get
    equations:
                \dot{theta_j} = omega_j + F_j(theta_1, theta_2);  j=1, 2.

(3) PHASE LOCKING: imagine that the coupling between two oscillators depends
    only on the phase difference. Then
      \dot{theta_1} = omega_1 + F_1(theta_1 - theta_2);
      \dot{theta_2} = omega_2 + F_2(theta_1 - theta_2);
    Hence
      \dot{\phi} = r + f(phi),   where phi = theta_1-theta_2,
                                         r = omega_1-omega_2,
                                     and f = F_1-F_2.
    A saddle node bifurcation, with a stable critical point showing up, leads
    to phi = constant ===> phase locking.

    This is not the only way phase locking can happen, there other ways.
    Note that phase locking is a very important phenomenon:
      heart beat requires muscle cells to contract in sync.
      cicadas synchronize their singing.
      fireflies (south-east Asia) can do the same.
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