From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Asymptotic behaviour of differential equations.
Date: 13 Jul 1998 23:46:36 GMT

In article <6nmomt$b2p3@hkpa05.polyu.edu.hk>,
en - LEE Hon Chor <enhclee@hkp.hk> wrote:

>I am investigating the asymptotic behaviour of a system of
>differential equations such that when z (the dependent variable) ->
>infinity, the system reduces to a system of linear o.d.e..
[...]
>So my thought is that
>the form of the asymptotic solution of the above system is
>just the usual one for a system of linear o.d.e. with known
>initial conditions.

It's not necessarily true that the solutions to the sets of equations

	y1' = a y1 + b y2 + f1(y1,y2)
	y2' = c y1 + d y2 + f2(y1,y2)
and
	y1' = a y1 + b y2
	y2' = c y1 + d y2

tend to agree as  t-> infinity, just because  f=(f1,f2)->(0,0) as
||(y1,y2)|| -> infinity, which is what I think you're trying to say.

As stated, you don't even know that  (y1, y2) is growing in 
the solution to either problem, so that the behaviour of  f  far from the
origin is irrelevant. Consider for example the system

	y1' =    -y2 + y1/(y1^2+y2^2)
	y2' = y1     + y2/(y1^2+y2^2)

starting at (y1,y2)=(1,0); the solution is  y1=sqrt(1+2t)*cos(t), 
y2=sqrt(1+2t)*sin(t) while the solution for the corresponding linearized
problem is simply  (cos(t), sin(t)). The fact that  f->0 for large
y_i is irrelevant here.
	
dave