Page 54 - Textos de Matemática Vol. 36

44 Chapter 2. Global aspects
In the next theorems we step up to a more complicated situation if there exist more terms with R(aj) = R(ak). First we discuss a situation with infinitely many solutions of ARE as possible attractors.
Theorem 2.4.8. If C ∈ G0ζ(ak), then WC := (wlα)1≤l≤m , where 1≤α≤n
vlα(aj)D(aj,C)
v˜(aj)D(aj,C) (2.40)

is a solution of equation ARE. Let C0 ∈ G˜0ζ(ak) with
G˜0ζ(ak) := {C ∈ Vn(C(m+n)×n)|D(ak,C) ̸= 0, and D(aj,C) = 0
if R(aj) > R(ak), or if R(aj) = R(ak) with ζ(aj) ̸= ζ(ak), and v˜(aj)D(aj,C) ̸= 0}.
ζ(aj)=ζ(ak)
Then there exists a solution W0 of equation ARE and t0 > 0 with
lim W(t,WC0)=W0 t→∞
t>t0
and the convergence takes place at an exponential rate. W0 is obtained from the right– hand side of (2.40) by replacing C with C0 and summing up only over those indices with
R(aj) = R(ak).
The stable set Ms(S0) ∩ Gn0 (Cm+n) in Gn0 (Cm+n), where S0 = Im(V C),
C ∈ G0ζ(ak), is given as
Ms(S0) ∩ Gn0 (Cm+n) = {Im(V C0)|C0 ∈ G˜0ζ(ak)}.
Proof. The assertion about the solution of ARE follows from Theorem 2.4.4, iii). From the definition of the set of C0-matrices and the general representation formula (2.33) we infer that
wlα =
j∈J0(C) j∈J0(C)
R(aj )≤R(ak ) wlα(t,WC0)=
etζ(aj )vlα(aj )D(aj , C0)
etζ(aj)v˜(aj)D(aj,C0) , v˜(aj)D(aj,C0) ̸= 0.
where
R(aj )≤R(ak )
R(aj )=R(ak )