4 Nonautonomous Attractors
61
Fig. 4.3 Trajectories of the
piecewise autonomous
equation (4.10) with λ = 1.5
−1
5
2
0
10
−5
1
−10
15
−2
0
for some λ > 1, which corresponds to a switch between the two autonomous problems at n = 0. Its pullback attractor A of the resulting nonautonomous system has
component sets A n ≡ {0} for all n ∈ Z corresponding to the zero entire solution
(Fig. 4.3).
The pullback attractor is not a forward attractor in this example. In fact, this
example does not even have a bounded forward attractor. This is not surprising
since pullback attraction depends on the past behaviour of the system and not on its
behaviour in the distant future. See [18].
4.4.2 A Condition Ensuring Forward Convergence
The above counterexample shows that a φ-invariant family A = {A n , n ∈ Z} constructed in Theorem 4.2 need not be a forward attractor, even when the φ-positively
invariant family B = {B n , n ∈ Z} is a forward absorbing family. Another important
observation is that there should be no ω-limit points from inside the family B that
are not ω-limit points from inside the family A .
For each n 0 ∈ Z, the forward ω-limit set with respect to B is defined by
ω B (n 0 ) :=
m≥n 0
n≥m
φ(n, n 0 , B n 0 ).
Suppose that the B n are uniformly bounded in a compact set B, i.e., B n ⊂ B. Then
ω B (n 0 ) is nonempty and compact as the intersection of nonempty nested compact
subsets and
lim
n→∞
dist R d
φ(n, n 0 , B n 0 ), ω B (n 0 )
= 0 (fixed n 0 ).