127
-1 5 -12 -9 -6 -3
0
3
6
12 15
a) 15
15
12
12
9
6
3
3
>0
0
- 3
- 3
-6
-6
-9
- 9
-12
-12
-15
-15
-15 -12 -9 -6 -3
0
9
12 15
x
-15 - 12 - 9 - 6 - 3 0
9
12 15
b) 15
15
\ ~
12
9
6
3
/
.I
>0
0
-3
-3
- 6
-6
- 9
- 9
- 12
. \ 2
-1 5
-15
- \5 -12 -9 -6 -3
0
9
\ 2
\5
X
Figure 20: PDF of the Lorenz model in the X- Y plane, low-pass filtered to remove oscillations around a regime centroid (a) from the unforced model (b) with a constant fa.
This behaviour can be understood in terms of the heterogeneity of finitetime singular values on the attractor. In particular, at the regime centroids
(PDF maxima for the state vector time averaged over the fast oscillation
time scale) the singular values are rather small, corresponding to the fact
that the attractor is rather stable in this part of phase space. On the other
hand, in other parts of the attractor, particularly near the origin, the singular values are particularly large, corresponding to a very unstable saddle
-1 5 -12 -9 -6 -3
0
3
6
12 15
a) 15
15
12
12
9
6
3
3
>0
0
- 3
- 3
-6
-6
-9
- 9
-12
-12
-15
-15
-15 -12 -9 -6 -3
0
9
12 15
x
-15 - 12 - 9 - 6 - 3 0
9
12 15
b) 15
15
\ ~
12
9
6
3
/
.I
>0
0
-3
-3
- 6
-6
- 9
- 9
- 12
. \ 2
-1 5
-15
- \5 -12 -9 -6 -3
0
9
\ 2
\5
X
Figure 20: PDF of the Lorenz model in the X- Y plane, low-pass filtered to remove oscillations around a regime centroid (a) from the unforced model (b) with a constant fa.
This behaviour can be understood in terms of the heterogeneity of finitetime singular values on the attractor. In particular, at the regime centroids
(PDF maxima for the state vector time averaged over the fast oscillation
time scale) the singular values are rather small, corresponding to the fact
that the attractor is rather stable in this part of phase space. On the other
hand, in other parts of the attractor, particularly near the origin, the singular values are particularly large, corresponding to a very unstable saddle