Lecture 26 - Tue 2024 12 10  Last lecture.
Two interesting dynamical systems. Cover one-or-both below.
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                                    Faraday instability and bouncing drops.
Bouncing drop system described in Lecture 23 [guest lecturer N. Liu].
Here I add a few, larger context, things:
1) Faraday discovered the instability in the mid-19 century. One of the
   first [or maybe the first] documented instances of a parametric
   instability.
   Describe the set up.
   Behavior can be understood in terms of Floquet theory.
   Describe what parametric forcing [as opposed to regular forcing] is,
      and how it has a richer behavior.
2) In the presence of dissipation, first instability is a sub-harmonic one
   [1/2 omega], which arises at some critical value of the forcing.
   Explain it as a resonance between the forcing and a wave with
   wave-number with corresponding wave-frequency equal to the subharmonic.
3) Drops can bounce indefinitely if the forcing is large enough.
4) Mention period doubling for bouncing [shows up for ping pong already].
5) Thresholds. If vertical forcing has acceleration gamma*sin(omega*t)
   then:
   gamma_B: threshold above which a drop can bounce [no merging].
   gamma_D: period doubling bifurcation for bouncing.
   gamma_F: Faraday instability.
   Then if  gamma_B, gamma_D < gamma < gamma_F
   drop is in resonance with waves. Walking happens at some
   gamma_D < gamma_W < gamma_F; provided vertical motion is still in the
   period double region.
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            The Manta ray attractor in a simple model for detonation waves.
Follow the talk [2020/12/01] in the pdf [posted in the web page].

Note that here we use alternative [A2] for Takens' theorem:
[A1] For a discrete system and a single signal {x_n}, use:
     (x_n, x_{n-1}, ...., x_{n-N}).
     Note the clear connection with the Lorenz map [discussed earlier].
[A2] Smooth signal b = b(t) where time derivatives are accessible, use:
     (b, b', b'', ...).
This last is the scheme we used for the "manta ray" attractor for pulsating
detonation waves in "The_Videos_and_the_Manta_ray_attractor.zip"
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EOF