Lecture 14 - Thu 2023 10 26
Limit Cycles. Dulac's criterion. Poincare Bendixon. Trapping Regions
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From prior two lectures; finish: Lyapunov, and start with Limit Cycles
Extras on limit cycles:
-1- What does "isolated mean"
-2- Classification by stability in 2-D and 3-D.
-3- Introduce notion of Poincare Map, and linearization at limit cycle,
    to analyze the stability in 3-D. Note the close analogy with the critical
    points in 2-D [e.g.: saddle if 0 < |lambda_1| < 1 < |lambda_2|, etc] ...
    except that a negative eigenvalue means the perturbed orbit cork-screws
    around limit cycle with n+0.5 turns per cycle around the limit cycle.

Introduce the van der Pol oscillator and explain the physics behind it: and
RLC circuit with a feed-back loop via a triod [in the original 1920] or a
transistor later on. Analog to a swing with a dad pushing a kid ... or the
way a grand-father pendulum is kept going --- or why mechanical watches and
clocks do "tick-tock".
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