Notes on linearization around fixed points.


Polking et al treats this for 2 x 2 systems in 10.1.  We are just
getting a head start on the idea.


We are considering an autonomous equation of the form  y' = f(y).  Where
f(a) = 0, y' = 0, and y(t) = a is a fixed point for the differential equation.

Suppose we want to know what happens near y = a.  We construct an approximation
to the equation.  We assume that  the variable h = y-a is small.  We shift
the equation to be written in terms of h.  This looks like


          h' = (y-a)' = y' - a' = y' = f(y) = f(h+a) ~ f(a) + f'(a)h + error.


By Taylor's theorem, the error is small.  We call the equation

        

          h' = f(a) + f'(a)h  =  f'(a)h


the linearization about the fixed point y = a.

It is of course, a theorem, that under the right circumstances, solutions near
y(t) = a behave like the solutions of the linearization.  Of course, if 
f'(a) < 0, the solutions h ~ k expt(f'(a)t) approach 0 and we can deduce that
the solutions y(t) approach a as t goes to infinity. If f'(a) > 0, the 
solutions h(t) grow exponentially, and after awhile the linear approximation
is false.


Lets do an example.  Lets try our old friend the logistics equation

         y' = ry(1 - y/K).

This has fixed points y = 0 and y = K.  It is always easier to linearize
about y = 0.  The other fixed point is y = K.  

         Now h = y - 0 = y;

         The linearization is y' = ry.

The solutions to the linearization are y = k exp(rt).  They grow exponentially,
so y = 0 is an unstable fixed point.


At y = K, f'(K) =  r - 2 r y/K = -r.  Now h = y-K, and the  linearization is:

          h' = -rh.

We know that nearby solutions decay into the fixed point h = 0 or y = K. We
also know the decay rate:  y-K ~ k exp(-rt).  Hence y = K is a stable fixed 
point.


In general, when we are modeling with non-linear equations, we cannot expect
to solve them.  We may not even be able to graph them (try graphing something
that is 10 X 10!)   However, we can always look for fixed points and linearize
around the fixed point for clues as to behavior.

Now for some exercises:  Find all the fixed points of the equation
y' = y(y^2 - 9) and linearize the equation around each fixed point.

Find the fixed points of the equation y' = y cos(y). 
Explain which ones are stable, and verify your answer by finding the
linearization.