Lecture 18 - Thu_2023 11 09 Bifurcations in the plane.
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Things I must not forget to say:
#1 Explain the difference between chaos and stochastic.
   Chaos is deterministic, albeit hard to compute with precision.
      Same initial data --> same answer.
      With a program in the *same* computer: same thing happens.
   Stochastic equations are not deterministic.
      Same initial data --> different answer answer.
      With a program in the *same* computer: same thing happens.
   If (small) noise is added to a chaotic system, noise will be amplified.
#2 Below, intuition for Hopf,
#3 Below, Landau hypothesis.
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                                                              HOPF BIFURCATION.
1) INTUITION: soft (nonlinear stabilize) and hard (nonlinear destabilize).
   Explain scaling intuitively: Why is it the cubic stuff that matters?
   [the math. reason is resonances

2) ANALYSIS   (This was mostly done in the prior lecture, so skip it here).

This follows naturally the small gain vdp expansion.
Consider         \ddot{x} + x = r x + N(x, \dot{x});     N = nonlinear terms.

  Switches from stable to unstable spiral as r crosses zero going up.

  Let r = sigma*epsilon, 0 < epsilon << 1, scale x = sqrt{epsilon} X

      Why sqrt{epsilon}? See intuitive explanation above. Or the math.: In
      an expansion the quadratic terms cause no resonance, so the linear
      terms have to balance the cubic terms to suppress resonances.

Then (for simplicity assume quadratic terms vanish, though not-needed).

   \ddot{X} + X = sigma epsilon X + epsilon C(X, \dot{X})

Derive:   X ~ A(tau) e^{i t} + c.c., with tau = epsilon t and

          2 i A_tau = (i sigma + gamma |A|^2) A.

Write A = rho e^{i phi} ==>  2 rho_tau = (sigma + gamma_I rho^2) rho
                             2 phi_tau = - gamma_R rho^2

gamma_I < 0  Soft Hopf \ Limit cycle amplitude ~ sqrt{|r|}  | START
gamma_R > 0  Hard Hopf /       """"" period    ~ O(1).      | TABLE!

3) LANDAU's HYPOTHESIS FOR TURBULENCE: sequence of "Hopf on a Hopf" --->
   quasi-periodic solutions. Explain; they "live" on Tori.
Problems: Cannot happen on dissipative systems [will explain later]; and:
          The "distance" between two nearby solutions grows linearly
          in time; breaks sensitive dependence on initial conditions.
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                                               BIFURCATIONS OF CRITICAL POINTS.
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).
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 of cycles. 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!

   Limit cycles amplitude ~ O(1)  | ADD TO
                period    ~ O(1)  | TABLE.

2) Infinite Period Bifurcation: Critical point pops-up on limit cycle; destroys it.
   \dot{r}     = r(1-r^2)      \ limit cycle exists for r > 1,
   \dot{theta} = r - sin theta / does not for r < 1  [note it re-appears at r = -1].

   Limit cycles amplitude ~ O(1)
                period    ~ O(1/sqrt{|r-r_c|})   [critical slowing down].

3) Homoclinic bifurcation.  Do the picture.

   Limit cycles amplitude ~ O(1)
                period    ~ O(1/log|r-r_c|)
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