Lecture 17 - Thu 2020 10 29 - Virtual
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Re: last lecture. Please read the notes on asymptotic versus convergent on the
    web page.


NEW TOPIC: BIFURCATIONS IN THE PLANE.

               First: bifurcations involving an isolated CP.

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1. Bifurcations of critical points involving loss of stability by a single
   eigenvalue: Saddle node, transcritical, pitchfork.
Typically this means that all directions but one remain "stable", so that all
the dynamics occurs along a 1-D curve   [leading order: line through unstable
eigenvector]. Thus: just take the 1-d cases and add a stable direction!

Examples:  
Show that all involve interactions of saddles and nodes. In particular:
      Explain why name: "saddle-node"
      Soft pitchfork: two nodes   and a saddle.
      Hard pitchfork: two saddles and a node.

Alternative way to understand them (in the plane): look at the geometry of
what the null-clines do.
For example: Nullclines crossing/uncrossing yields a saddle-node.
Adding restrictions [nullcline crossing cannot be destroyed, or symmetry]
yields the other cases.

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2. Bifurcation of a critical point involving a 2-eigenvalue stability loss.
   NEW case: Hopf bifurcation.
   Complex pair of eigenvalues crosses imaginary axis.

Bifurcation arises because of interplay between linearity and nonlinearity.
  Unstable linear terms and stabilizing nonlinearity produces stable
  limit cycle [soft Hopf]. Reverse produces unstable limit cycle [hard Hopf].
  Situation is "similar" to what happens for a pitchfork; except that the
  effect of the nonlinearity is given by an average as the orbit goes around.

Simple example/canonical equation:  \dot{r} = a*r-r^3  and \dot{theta} = 1. [A]

Explain why it has to be r^3; cannot be r^2:
Analysis explanation: with r^2 equations are not smooth at r = 0 [write them
   in terms of x and y.
Geometrical explanation: quadratic terms cancel at leading order in the radial
   direction. Their contribution to the vector field is the same at (x, y) and
   (-x, -y).
Perturbative explanation: the quadratic terms produce no resonances, hence no
   secular terms in the solution [will see this more clearly later].
  
From [A], note: (i) Super and sub-critical cases.  (ii) Bifurcation diagram.
   Amplitude of the limit cycle goes like sqrt{|a|}, where a is the departure
   from the bifurcation point at which linearized system has a center. This is
   generic: balance of linear and nonlinear terms happens at cubic level.
       Linear term epsilon*amplitude; where epsilon=imaginary part eigenvalues.
       Cubic nonlinearity amplitude^3
       Balance when amplitude ~ sqrt{epsilon}
    This can be justified with asymptotics for Hopf bifurcations.
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