Lecture 21 - Thu 2020 11 12 - Virtual
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Floquet theory provides an "alternative" to linearizing a Poincare map to study
the stability of a limit cycle, and it is an important theory on its own - with
applications beyond stability of limit cycles.  While it is less intuitive than
the Poincare map, it is easy to set-up,  with no need to deal with the geometry
of a limit cycle [which can be quite complicated, and hard to ascertain in high
dimensions].
We will cover Floquet Theory at the end of the semester, if time permits.
However ... some movies:
1 Inverted pendulum ..... parametric stabilization.
2 Faraday Instability ... parametric destabilization
3 Quasi-periodicity.
4 Rotating Saddle [parametric stabilization]. SKIP
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NEW TOPIC: Beyond the plane; chaos in 3-D continuous systems and in maps.
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1) Show the iterations of the shear map pictures.
2) Rossler lecture [Matlab scripts illustrating the Rossler attractor].
3) Chaos wheel movie [Lorenz attractor].
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Lorenz equations.
--- Describe historical facts, Lorenz work. Simple model for convection:
    project dynamics on just 3 modes: Lorenz system [ode in 3-D].
--- Approximation is rather drastic in convection. Can show that, as more
    modes are kept, behavior changes [work by Howard, for example].
--- Is there a physical system for which the Lorenz equations are a good
    approximation?
    Can produce simple system where model is pretty good [Malkus and Howard].
    Describe early model with coffee cups glued to turn-table [worked 1st
    time they tried it, pretty much].
--- Describe what Malkus "wheel" does, and relate it to Lorenz attractor in
    3-D [Wings of a butterfly on which trajectory orbits, but jumps
         from one wing to the other at, seemingly, random intervals].
--- Describe fancier Malkus model [turns out, harder to get this to work!].
    Gave inspiration for many "chaos fountains".  [SHOWN IN A MOVIE EARLIER].
    Malkus anecdote of "chaos fountain" in Paris.
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Rossler attractor:
--- Simpler system to understand chaos [shown in movies].
    Describe attractor. Now one "apparent surface" where trajectories orbit
    around a center, and "randomly" leave for an excursion "outside" to come
    down elsewhere.
Describe key features of computations:
--- Cross-section. Very thin. Looks like a curve, but upon blow up [10^8 factor]
    some transversal structure starts to arise [folds].
--- Describe evolution of thing rectangle in a z-radial half-plane
             where x = r*cos(theta), y = r*sin(theta)
    as theta grows. "Horizontal" stretching [factor O(2)], exponential
    "vertical" compression, and FOLDING.
--- Describe Poincar\'e map after one 2*pi turn.

Attractor is now result of applying the Poincar\'e map infinitely many times.

Now split understanding of what the attractor looks like, and how it arises
in TWO STEPS.
(1) Vertical structure, z direction.
(2) Longitudinal structure, r direction.
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