Lecture 14 - Thu 2024 10 24
Phase Plane: Dulac's criterion. Trapping regions. Dissipation. Limit cycle Stab.
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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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One more way to rule out periodic orbits:                     DULAC's CRITERION.
\dot{x} = f(x) valid in a simply connected region of R^2
There is a scalar g(x) defined there such that div(g(x)\dot{x}) = div(g*f)
    never vanishes in the region [hence it is either positive or negative].
Then no periodic solutions can exist in the region.

Proof: Use Gauss theorem  \int_Q \div(g*\dot{x}) dA = \int_C g*\dot{x}.\hat{n} ds
where C = periodic orbit, Q = interior of C, \hat{n} = unit normal to C.
The left in this equation is not zero, while the right is zero. Contradiction.

EXAMPLE:  \dot{x} = x*(2-x-y)   and   \dot{y} = y*(4*x-x^2-3)
          Then use g = 1/(x*y) on first quadrant.

Note: Dulac's criterion can be "boosted" so as to show that a given periodic
      solution is unique [extend it to regions with one hole and apply it to
      the annulus between two periodic solutions, to reach a contradiction];
      but we will not go into this.

When we get to the limit cycle for van der Pol, we will see an application of
this.
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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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                                     Examples of dissipation in physical systems.
In lecture 12 I argued that, generically, macroscopic mechanical systems are a
Hamiltonian-Gradient mix
    \dot{q_n} =   H_{p_n} - Phi_{q_n} \ [A]
    \dot{p_n} = - H_{q_n} - Phi_{p_n} /

Simple example: \ddot{x} + \nu\dot{x} + V'(x) = 0.
Writing this as a system:   \dot{x} = v   and \dot{v} = -V'(x) -\nu v.

This has the form in [A] with     H = 0.5 v^2 + V  and Phi = 0.5 \nu v^2.

Writing a system in the form in [A] can be nontrivial, the form is non-unique,
and the physical meaning of H and Phi can be unclear [hence we will not do
much with such systems, other than the volume calculation we did in lecture
13]. Example:

Consider now non-linear dissipation   \ddot{x} + \nu(x) \dot{x} + V'(x) = 0,
where now nu(x) > 0 is a function.
How do we turn this into a Hamiltonian-Gradient mix? Turns out not obvious!
First: introduce mu = mu(x) such that d mu/dx = nu.
Then, write the equation as         d/dt (\dot{x} + \mu(x)) + V'(x) = 0.
Now, let p = \dot{x} + \mu(x)
         q = x
Then     \dot{q} = p - mu(q) =  H_p - Phi_q \  where H   = 0.5*p^2+V
         \dot{p} = -V'(q)    = -H_q - Phi_p /  
and Phi = Phi(q) is defined by d Phi/dq = mu(q).
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                                                       RELAXATION OSCILLATIONS.
Start with this [topic developed in the next lecture].
1) Introduce question: why does a chalk make noise? Why does pulling a chair
   along the floor make noise? Why is the motion sometimes very jerky? Why do
   sound amplifiers do annoying noises when the gain is too high?

To begin answering these questions we will look at the van der Pol limit cycle
when mu is large:
                   \ddot{x} - mu (1-x^2)\dot{x} + x = 0.

Introduce z by x = \dot{z} [defined up to a constant].
Substitute into equation and integrate:
               \ddot{z} - mu (1- (1/3)\dot{z}^2)\dot{z} + z = 0;
select constant in z to make rhs zero here.

Rayleigh equation. Give examples of terms like (1- (1/3)\dot{z}^2)\dot{z}
in situations where the velocity of something is driven towards some
preferred value: active fluids and traffic flow.

Continue next lecture.
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