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2.4. Flows on Graßmann manifolds 45
Obviously, this implies that (2.40) with C0 instead of C and the summation over R(aj) = R(ak) attracts at an exponential rate W(t,WC0) which also yields the assertion about the stable set.
This theorem shows that in the case when an ARE yields infinitely many solutions they can also attract many solutions of RDE, but, in contrast to The- orem 2.4.6 the set of attraction also depends on the initial data.
In the two subsequent theorems we describe the periodic and almost- periodic solutions of RDER which clearly correspond to the set of periodic and almost-periodic functions defined by the representation formula (2.14) or (2.33). Using this fact we get the assertions of the two following theorems. We omit the proofs since they are the same as the proofs of Theorems 2.4.6, 2.4.8.
Theorem 2.4.9. Let C ∈ Gp(ak) where
Gp(ak) := {C ∈ Vn(C(m+n)×n)|D(ak,C) ̸= 0 and D(aj,C) = 0
if either R(aj) ̸= R(ak) or if I(aj),I(ak) incommensurable} and with det Q0 (C ) ̸= 0. Then W (·; WC ) is a periodic solution of equation RDER
with
where the sums run over all indices j such that R(aj) = R(ak) and I(aj),I(ak) are commensurable. Then
- The solution (2.41) may have real poles.
- Every periodic solution of equation RDER is either of the form (2.41) for some k ∈ {1,...,N} and for some matrix in Gp(ak), or it can be written in the form (2.42) (see Theorem 2.4.10).
- The set of all periodic solutions of equation RDER is either empty or uncountable.
Now let C1 ∈ G˜p(ak) where
G˜p(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) and I(aj),I(ak) incommensurable} 1 1
eitI(aj)vlα(aj)D(aj,C) j
wlα(t,WC) = eitI(aj)v˜(aj)D(aj,C) , (2.41) j
c1 c˜1
1 . ˜1 .
andmodifyC = . to C = . by
11 cm+n c˜m+n
