Lecture 7 - Tue 2020 09 22 - Virtual
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From Lecture 6
Do recap of 1-D bifurcation at the end of the lecture.

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Bifurcations for MULTI-D (example) involving critical points.
Note that (in general) we are only interested in stable solutions.  Thus the
bifurcation must involve at least one critical point that is stable. This is
the crucial assumption that makes the phenomena 1-D. The reason is that then
the bifurcation will involve a critical point that loses stability, and
generically this means
(a) one eigenvalue of the linearized problem that crosses from negative to
    positive [with one associated eigenvector] ... 1-D situation.
or
(b) A pair of complex eigenvalues that cross from stable to unstable. This
    will produce a 2-D situation and a bifurcation (Hopf) that we will see
    later.

SKETCH OF THE ALGEBRA.

System is:                         \dot{Y} = F(Y, r), Y and F vectors.
To find critical points must solve       0 = F(Y, r).

Numerically, given a solution (Y0, r0), find a solution for
slightly different r:
                       0 = F(Y0 + dY, r0 + dr) = (F_Y)_0*dY + (F_r)_0*dr + HOT

Implicit function guarantees a single dY-small solution exists for dr small
enough, provided (F_Y)_0 is not singular. Can find it by, say, Newton's method.
Then no bifurcation.

What happens if A=(F_Y)_0 is singular?
The generic case is that A has a single zero eigenvalue, with multiplicity
one. In addition, all the other eigenvalues must have negative real parts
[else situation would be unstable through-out, as any eigenvalue with positive
 real part would remain so for dr small].

This leads to a saddle-node or turning point bifurcation, where the solution
exists only for dr > 0 (or only for dr < 0), and there are two such solutions.

How do you find it? You use the "standard" expand in Taylor series for dY and
dr small, and keep dominant terms only. It turns out that this leads to the
scaling:
         dY = O(epsilon) with dr = epsilon^2 or dr = -epsilon^2
and epsilon small
         [the expansion can be justified using the implicit function theorem].
Then we seek a solution to the dynamical system of the form

       Y = Y0 + epsilon*Y1(T) + epsilon*Y2(t, T) + ...  [T = epsilon*t]

which is solved by plugging into the equation and solving each order separately
[first O(epsilon), then O(epsilon^2), etc].
..... Note T is has the scaling for the critical slowing down.
Note also that, when expanding, we only need to worry about the components
of the solution along the eigenvector that corresponds to the vanishing
eigenvalue, because all the other components decay exponentially.
Hence the 1-D behavior.

Generic assumptions:
--- single zero eigenvalue with 1-D eigenspace.
--- When solving at second order, need generic condition for a certain
    number [which has to be non-zero].
    --- In 1-D this is the second derivative.
    --- In n-D this is an object constructed from the second
        derivative tensor and the left and right eigenvectors.

Give pointers as to how a similar expansion works for the transcritical
bifurcation in the general n-D case: \dot{y} = F(y, r). Same for pitchfork.
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Flows in the circle and bifurcations.
--- Example: damping controlled; torsion driven pendulum:
 \dot{theta} = r - sin(theta).
 (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}.   [done in a prior lecture].

--- Simple models in applications occur for:
    Phase-locking + Oscillating neurons [heart, fire-flies] + Josephson
          junctions ==>  See \S 4.6 Strogatz Book.
    The eqn. for the phase difference analogous to driven pendulum. Pendulum
    going around generates oscillating voltage of very high frequency.
--- Explain mechanics of how an equation involving only phases can be justified:
    (i)   Assume strongly stable oscillations [limit cycles].
    (ii)  Then can forget the "amplitude" and just study the system by the
          dynamics of its phase. For a single oscillator this leads to
          \dot{theta} = w (constant, if theta defined properly).
    (iii) For two oscillators then \dot{theta_j} = w_j + F_j(theta_1, theta_2)
    (iv)  Example of PHASE LOCKING: two oscillators coupled, where the coupling
          depends only on the phase difference, then
          \dot{\phi} = r + f(phi),
          where phi = theta_1-theta_2, r = w_1-w_2, and f = F_1-F_2.
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