140 Antonio Navarra
Singular spectra of A (solid) and exp(A) (dashed)
1.5
1.5
1.5
1
' .
\
......... -..... .
. .... " .... ,' .........
0.5
0.5
"\
O
O
O
O
50
100
O
50
100
O
50
100
Eigenvalues of A
0.4
0.4
0.4
0.2
~~
0.2
0.2
00
:~oo
O 00 §
O
O (t)
O
00
-0.2 8 O O -0.2
-0.2
O O
-0.4
-0.4
-0.4 L-~~~----.J
-0.8 -0.6 -0.4 -0.2
-0.8 -0.6 -0.4 -0.2
-0.45 -0.4 -0.35 -0.3
Amplification factor of optimal m odes (solid) and normal modes (dashed)
1 .5 .--~-~----.
1.5
1. 5.--~------,
0.5
O'----~--_ · ' --'- " :.:J "
O
10
20
30
Time Steps
Fig. 8.2 Normal modes and finite time instabilities. The figure illustrates the difference
between the asymptotic problem of stability (normal modes of exponential shape in time)
and the finite time behaviour (polynomial growth in time). It is also shown that when the
operator tends to become self-adjoint the differences disappear. The bottom panel also
indicate how basic states that are stable regarding the asymptotic shape are in fact capable of
growth over finite times.