From: Dave Rusin <rusin@math.niu.edu>
Subject: Re:  STABILITY ANALYSIS : Help Please
Date: Thu, 6 May 1999 07:36:24 -0500 (CDT)
Newsgroups: [missing]
To: lsouriau@club-internet.fr
Keywords: boundedness of terms of a linear recurrence

>Here is the PROBLEM :
>
>  (1) A(k) Wn+1(k) - B(k) Wn(k) + C(k) Wn-1(k) = D(k)
>       n coresponds to the current time step : t=n Dt.
>Initial conditions : W0(k) = W1(k) = 0
>
>I'd like to know what is the condition on factors A, B, C , D
>(function of  k, real number) so that Wn(k) is stable versus time ?
>(No divergence of Wn(k)  torward  + infinity )

Your problem may depend on  k, but the question concerns one  k  at a time.
So the question (for each  k)  seems to be,

 If  W0=W1=0  and  A Wn+1 - B Wn + C Wn-1 = D  for all n >=1, is {Wn} bounded?

That's a simple question of linear recurrence relations. If, for example,
A is nonzero [in your problem there may be particular values of  k  which
make  A(k)=0  even if  A(k)  is not identically zero]  then we let
F = sum(  Wn * X^n  )   be the formal power series; the given conditions
show  F = D / ( (1-X)(A X^2 - B X + C) ). Now use partial fractions to
write  F = a/(1-X) + b/(1-rX) + c/(1-sX)  [special cases if  A-B+C=0 or
B^2=4AC] to see  Wn = a + b r^n + c s^n, which stays bounded iff |r|<=1 and
|s|<=1 [special cases if b=0 or c=0, as well as if   r and s are purely
imaginary]

So apart from a few special cases the conclusion is, "W stays bounded iff
both roots of the polynomial  A X^2 - B X + C  have complex magnitude at
most equal to 1". This in turn simply means that the ratios (B/A, C/A) lie 
in a certain compact subset of the plane.

[deletia -- djr]