Lecture19 Tue 2024 11 12
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                           BIFURCATIONS OF CRITICAL POINTS (review done earlier)
We first discuss bifurcations where a critical point is stable on one side.
Explain: then, generically, there is a curve along which everything is
         attracted and where the bifurcation happens.
Another way to think of it: intersections of two curves [the null-clines].
        Note situation "the same" as 1-D, where the two "curves" for
        \dot{x} = f(x, mu)  are:  y = f(x, mu) and y = 0.

SADDLE NODE in 1-D ---> Saddle node in 2-D   \dot{y} = -y   \dot{x} = mu-x^2
    Do phase portraits for mu > 0, mu=0, mu < 0.

TRANSCRITICAL:  \dot{y} = -y and \dot{x} = mu*x - x^2 Saddle and node "exchange"

Example: \dot{x} = -a*x+y ................  nullcline: y = a*x
         \dot{y} = x^2/(1+x^2) - y .......  nullcline: y = x^2/(1+x^2)
Has both saddle-nodes and transcritical.

PITCHFORKS:  \dot{y} = -y and  \dot{x} = mu*x-x^3 (soft) \ add \dot{y} = -y
or                             \dot{x} = mu*x+x^3 (hard) /

Note analogy with Hopf: soft (nonlinear stabilize) and hard (nonlinear
                        destabilize). Explain scaling "intuitively"
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                                          BIFURCATIONS OF LIMIT CYCLES: GLOBAL.
Prior bifurcations involved the neighborhood of a critical point: local.

1) Saddle-Node bifurcation. Two cycles "collide" and annihilate each other.
   \dot{r}     = mu r + r^3 - r^5 = f(r)  <-- this eqn. undergoes saddle node
   \dot{theta} = omega

   Explain "puzzle": how can two curves approach each other *everywhere* and
   end up the same at the same value of bifurcation parameter. Why is this not
   extremely rare?  Answer: if one point coincides, then whole solution equal!

   Saddle-node corresponds to eigenvalue in Poincar\'e Map crosses through z=1.
   Do Poincar\'e Map for 2-D limit cycle.
   To TABLE .....................  amplitude ~ O(1)    period ~ O(1)

   Next two DO NOT FIT Poincar\'e pap picture. There   **********************
   is no Poincar\'e map as the "tube/strip" of nearby  **********************
   orbits tracking cycle shrinks to zero as the bifurcation is approached. **
   Requires a critical point on or approaching limit cycle.

2) Infinite Period Bifurcation: Crit. point pops-up on limit cycle; destroys it.
   \dot{r}     = r(1-r^2)    \ limit cycle exists for |a| > 1,
   \dot{theta} = a-sin theta / does not for |a| < 1.
   To TABLE: amplitude ~ O(1)  period ~ O(1/sqrt{|a-a_c|}) [critical slow down].

3) Homoclinic bifurcation.  Do the picture. Limit cycle cut by a saddle.
   To TABLE:    amplitude ~ O(1)    period    ~ O(1/log|r-r_c|)
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                                          Conditions for Hopf using two timing.
Do the example using the approach in the notes
       Hopf bifurcations using two timing and complex notation.
I only covered the idea, but did not go into the actual computations.
These are left for the students to read using the notes in the web page.
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