ABSTRACT
Let us now look at difference systems of equations which can be brought to the form
x(n+ 1) = Λˆ ( I +
1 n Aˆ
) x(n) + g(n,x(n)) (7.1)
where Λˆ and Aˆ are constant coefficient matrices, g is convergently given for small x by
g(n,x) = ∑ k∈Nm
gk(n)xk (7.2)
with gk(n) analytic in n at infinity and
gk(n) = O(n−2) as n→∞, if m∑ j=1
kj ≤ 1 (7.3)
under nonresonance conditions: Let µ = (µ1, ..., µn) and a = (a1, ..., an) where e−µk are the eigenvalues of Λˆ and the ak are the eigenvalues of Aˆ. Then the nonresonance condition is
(k · µ = 0 mod 2pii with k ∈ Zm1)⇔ k = 0. (7.4)
The theory of these equations is remarkably similar to that of differential equations. We consider the solutions of (7.1) which are small as n becomes large.
