Lecture 19 - Thu 2020 11 05 - Virtual
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Continue with GLOBAL BIFURCATION OF CYCLES.

 Note: please READ the summary below. The lecture actually starts below in Case
       2, but you should make sure you understand all of it. If you have doubts
       (any doubts) please ask at the office hours, or via piazza.  The summary
       includes examples.
 Note: The book has a lot of examples, arising from simple models for chemical,
       biological, and physical systems. You should read them. I will not cover
       them, else I will not be able to "reach" chaos.

Summary from last lecture:
  Arguments based on the existence of a "trapping hyper-tube" enclosing a stable
  limit cycle [which we assume is the case on at least one "side" of the
  bifurcation]. This leads to:

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Case 1 % --------------------------------------------------------------------- %
Flow around trapping tube persists on both "side" of the bifurcation. Thus: can
define a Poincare map, with the limit cycle then a fixed point.   Examining the
linearization, which has all the eigenvalues inside the unit disk on the stable
side, yields:

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Case 1.1:  Single real eigenvalue crosses out of the unit circle through 1.
           This is the ONLY possible case in 2-D, because there the Poincare
           map is an increasing scalar function.
"Generically" this leads to a                       Saddle-Node bifurcation
   of limit cycles.

EXAMPLE, in polar coordinates:   \dot{r} = r*((r^2-1)^2-d); \dot{\theta} = 1.
   For d < 0: no limit cycles, orbits spiral from 0 to infinity.
   For 0 < d < 1: two limit cycles, at r = \sqrt{1 \pm \sqrt{d}}. The smaller
      one is stable and the other is unstable. At d=1 a soft Hopf bifurcation
      happens and the smaller orbit vanishes into the critical point at the
      origin, which becomes a stable spiral.

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Case 1.2: Single real eigenvalue crosses out of the unit circle through -1.
          This bifurcation requires at least 3-D.
"Generically" this leads to a                     Period-Doubling bifurcation.
   Can be either"soft" (non-linearity stabilizes;
   new limit cycle is stable) or "hard" (non-linearity destabilizes).
Intuition [argument is very similar to the one for Hopf]. Nearby orbits switch
   sides [along the eigenvector that corresponds to the lambda ~ 1 eigenvalue]
   and either: move away from the limit cycle (lambda > 1), or towards it
   (lambda < 1). Nonlinear effects then balance this linear effect at some
   distance from the limit cycle, and produce another limit cycle. As usual,
   the distance from the new cycle to the old one grows like the square root
   of the deviation of the bifurcation parameter from the critical value.

EXAMPLE: we will see some examples later.
   These bifurcations are one important element in one "route to chaos".

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Case 1.3: Pair of complex conjugate eigenvalues crosses out of the unit circle.
          This bifurcation requires at least 3-D.
"Generically" this leads to a                Hopf bifurcation of limit cycles;
   either "soft" (non-linearity stabilizes)
   or "hard" (non-linearity destabilizes).  The new object is, generally, not
   a limit cycle. In fact, it is an orbit that travels on a "Torus = doughnut"
   enclosing the limit cycle, orbiting around the
   limit cycle as it moves along it.       Thus THESE ORBITS HAVE TWO PERIODS.

   NOTE: if you see the projection of the new object on a plane, what shows
         up is a "wobbling" loop.  So, another name:  WOBBLING BIFURCATION.

EXAMPLE: we will get back to this later.
Note: this bifurcation cannot occur in volume shrinking systems, because they
      require an orbit on a Torus [thus no shrinking].    Dissipative systems
are generally volume shrinking, so they do not occur there.   But they happen
in driven-dissipative systems, and they characterize another "route to chaos".

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Case 2 % --------------------------------------------------------------------- %
Flow around the tube destroyed; thus no Poincare map exists on one side of the
bifurcation.    This requires a critical point within the trapping hyper-tube.
Furthermore: this critical point must approach the limit cycle as the critical
bifurcation value is approached  [else one can select a smaller tube without a
critical point, and we are back to case 1].

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Case 2.1: Infinite period bifurcation.
   A critical point pops up right on the cycle, and then splits into two.
   Right at the bifurcation, the critical point on the limit cycle is
   asymptotically stable but not Liapunov.
   A bit beyond it, we get an excitable system.

   Right before the bifurcation, a critical slow down phenomena occurs along
   the limit cycle, whose period thus approaches infinity as the bifurcation
   is approached. Further:
   the approach to infinity is like .....  1/sqrt{|lambda-lambda_c|},
   where lambda is the bifurcation
   parameter, and lambda_c is the critical value.

EXAMPLE:  in polar coordinates
          \dot{r} = r(1-r^2); \dot{\theta} = 1 - a*cos(theta).
          Bifurcation happens at a = 1, where a critical point appears at
          r=1, theta = 0.
 
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2.2 Homoclinic bifurcation of limit cycles.
 A critical point approaches cycle. The only critical point with a flow pattern
 near it consistent with this is a saddle [we exclude "strange" critical points
 that require a degenerate linearization; "generic" assumption].

 As the saddle approaches the limit cycle, the period goes to infinity, and at
 the bifurcation the limit cycle becomes an homoclinic connection.

 This type of bifurcation can arise when a saddle point and a spiral point are
 nearby, and a Hopf bifurcation happens at the spiral point.    Then the limit
 cycle produced grows as the bifurcation parameter changes, and (if the saddle
 is close enough) it can collide with the saddle.

EXAMPLE, using "phase plane surgery" ................... #1
Computation of how the period approaches infinity. ..... #1

#1 See the scanned notes: "Homoclinic Bifurcation of a Limit Cycle"
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         TABLE OF PERIODS AND AMPLITUDES FOR THE VARIOUS BIFURCATIONS

Here d is the deviation of the bifurcation parameter from the critical value;
     d = |\lambda-\lambda_c|.

TYPE                AMPLITUDE      PERIOD
Hopf                O(sqrt(d))     O(1)
Saddle-Node         O(1)           O(1)
Infinite period     O(1)           O(1/sqrt(d))
Homoclinic          O(1)           log(d)
Period Doubling     O(1)           gets doubled
Wobling             O(1)           new O(1) period appears.
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