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2.4. Flows on Graßmann manifolds 47 1
c˜1
1 ˜1 .
andmodifyC toC = . with 1
c˜μ :=
If we define furthermore
1 c1μ
c˜m+n
if μ is such that R(aμ) = R(ak),
otherwise.
0
etζ(aj)v˜(aj)D(aj,C) = etζ(ak)∆0(t), ∆0(t) := et(ζ(aj)−ζ(ak))v˜(aj)D(aj,C)
R(aj )=R(ak )
and if it holds that
then W(·;WC˜1) is an almost-periodic solution of equation RDER with
lim (W (t; WC1 ) − W (t; WC˜1 )) = 0 t→∞
t>t0
and the convergence takes place at an exponential rate.
Remark 2.4.11. An important case for all applications is obtained if the di- chotomy condition
R (λm) < R (λm+1)
holds, i.e., there exists a gap in the real parts of the eigenvalues λ1, . . . , λm+n
where
of M.
i) If we denote
with
then
A
B := (vm+1,vm+2,...,vm+n)
A = (v˜m+1,...,v˜m+n) and if detA ̸= 0,
Wd = BA−1 (2.43)
∆(t) := R(aj )=R(ak )
|∆0(t)|≥δ>0 for t>t0,
is a solution of ARE which is called the dichotomic solution (see [Meda82],
Definition 2). Analogously, if R(λn) < R(λn+1) with det(v˜1, . . . , v˜n) ̸= 0
one defines the reverse dichotomic solution Wrd = B˜A˜−1 of ARE, where
A˜
B˜ =(v1,...,vn).
