Lecture 15 - Tue 2023 10 31
Trapping Regions, existence of periodic solutions, and Dulac's criterion.
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From prior two lectures; finish: Lyapunov, Limit Cycle, and Dulac.
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Interlude on discrete dynamical systems.
Consider the dynamical system
  x_0 = mu, with x_{n+1} = x_n^mu, where mu = sqrt{2}.
  What happens as n \to \infty?
There is a YouTube video that claims x_n goes to 2 = mu^{mu^{mu^{mu ...}}},
with the proof:
  Let y = mu^{mu^{mu^{mu ...}}}. Then y = mu^y, so must be y = 2.
Discussion: why is this proof wrong?
Deeper issue: when you see articles in magazines, internet, or even refereed
   journals, do not assume that they are correct. How to tell if they are.
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                                               Recall POINCARE-BENDIXON THEOREM.
Assume a C^1 dynamical system, defined in an open set Q in R^2.
Let X = X(t) be a solution of the system. Then, as t \to \infty, one of these
happens (nothing else is possible):
(1) X(t) approaches a critical point --- in particular: X may be a fixed point.
(2) X(t) approaches a periodic orbit --- in particular: X may be periodic.
(3) X(t) approaches a cycle graph.
(4) X(t) reaches the boundary of the set [if the set is unbounded, then X(t)
    may go to infinity].
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How to use Poincare Bendixon to show limit cycles exist.       TRAPPING REGIONS.
EXAMPLE #1: % ============================================ %
\dot{r} = r*(1-r^2) + mu*r^2*\cos\theta; \dot{\theta} = 1;  0 < mu < 1.

Find R_max so that, for r \geq R_max, \dot{r} < 0.
Find R_min so that, for r \leq R_min, \dot{r} > 0.
Since there are no fixed points in the annulus R_min < r < R_max, there must
      be a periodic solution.

System is:   \dot{x} = - y + [1-r^2 + mu*x]*x;
             \dot{y} =   x + [1-r^2 + mu*x]*y;

EXAMPLE #2: % ============================================ %
*READ* the glycolysis example in the book [example 7.3.2].

EXAMPLE #3: % ============================================ %
Sketch how the trapping region works for the van der Pol equation.
   The sketch shows how to show existence. Uniqueness follows by using
   an extension of Dulac. Then stability follows from Poincare Bendixon.

SEE NOTES NEXT TO THIS LECTURE IN THE WEB PAGE,
with details of the construction of a trapping region for van der Pol.

Note: The van der Pol equation is a particular example of a Lienard system,
      which also have a unique, stable, limit cycle [READ the book].
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                                            DULAC's criterion (see Lecture 13).
The trapping regions argument can tell you that there is a periodic solution
in the region. But how can you tell it is a limit cycle (i.e. isolated)? If you
can show that the trapping region has only one periodic solution (using Dulac
... explain how), then not only get that it is a limit cycle, but that it is
stable as well.

NOTE: in a few days I will post some notes showing the example of van der Pol,
      and how Dulac can be used there to show that the limit cycle is stable.
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                                                       RELAXATION OSCILLATIONS.
Start with this [topic developed in the next lecture].
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