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38 Chapter 2. Global aspects
1 ≤ l ≤ m, 1 ≤ α ≤ n, and this is a solution of ARE as long as the denominator is not zero.
For the description of Γ, the set of solutions of ARE, we use the set of parameter matrices
G0ζ(ak) = {C ∈ Gζ(ak)| v˜(aν)D(aν,C) ̸= 0} ν∈J0(C)
and the set of indices
J(ak) =
If G0ζ(ak) ̸= ∅ then, according to formula (2.33) (to any matrix C ∈ G0ζ(ak))
there exists a corresponding solution,
WC = W(t,WC) =: (wlα(t,WC))1≤l≤m ,
1≤α≤n
to the equation ARE.
On the other hand it follows from the facts observed in (i) - (iii) that
any solution of equation ARE can be represented in the form (2.33), for some C ∈ G0ζ(ak),k ∈ {1,...,N}. To each family
Fk = {C ∈ Vn(C(m+n)×n)|J0(C) ⊂ J(ak)}
there exists the corresponding family Γk = {WC|C ∈ Fk} of solutions of equa- tion ARE and Γ = Nk=1 Γk. It was already observed in Section 2.1, Theorem
2.1.2 and Remark 2.1.4 that the number of solutions of equation ARE corre- sponds to the number of n-dimensional M-invariant subspaces of Cm+n being
0
complementary to the image of I . Alternatively, the number of elements
m
of Γ and the structure of Γ can easily be determined from (2.33) if we know #J0(C) and #J(ak),1 ≤ k ≤ N. Obviously Γ is either finite, if G0ζ(ak) ̸= ∅ and J(ak) = {k}, or otherwise uncountable.
Before going into a more detailed description of Γ we need another prepara- tory lemma showing that the determinants v˜(ak) and vlα(ak) as well as the determinants D(aν,C),C ∈ Gζ(ak), in (2.33) are not completely independent, which has great impact on the structure of the solutions of ARE and therefore clearly also on the phase portrait of RDE .
Lemma 2.4.2. (i) Let ak ∈ A and v˜(ak) = |v˜ν1k,...,v˜νnk| ̸= 0. If j ∈ {1, . . . , n + m} \ {ν1k , . . . , νnk }, then at least one of the determinants
| v˜ j , v˜ ν 2k , . . . , v˜ ν nk |
or
is non-zero.
J0(C). C ∈ G 0ζ ( a k )
|vj(l,α),vν2k(l,α),...,vνnk(l,α)|, 1≤l≤m, 1≤α≤n,