4 Nonautonomous Attractors
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with n and θ −n( p) changed to n 0 and n 0 − n, respectively, but additional complications due to the fact that the pullback absorbing family is no longer assumed to be
ϕ-positively invariant here.
Theorem 4.5 Let (P, d P ) be a complete metric space and suppose that a skewproduct system (θ, ϕ) on P × R
d has a pullback absorbing set family B = {B p :
p ∈ P}. Then there exists a pullback attractor A = {A p : p ∈ P} with component
sets determined by
A p =
n≥0
j≥n
ϕ
j, θ − j ( p), B θ − j ( p)
.
(4.15)
This pullback attractor is unique if its component sets are uniformly bounded.
The pullback attractor of a skew-product system (θ, ϕ) has some nice properties
when its component subsets are contained in a common compact subset or if the
state space P of the driving system is compact. See [31].
Proposition 4.3 Suppose that A(P) :=
p∈P A p is compact for a pullback attractor
A = {A p : p ∈ P}. Then the set-valued mapping p → A p is upper semi-continuous
in the sense that
dist R d
A q , A p
→ 0 as q → p.
On the other hand, if P is compact and the set-valued mapping p → A p is upper
semi-continuous, then A(P) is compact.
Pullback attractors are in general not forward attractors. However, when the state
space P of the driving system is compact, then one has the following partial forward
convergence result for the pullback attractor [31].
Theorem 4.6 In addition to the assumptions of Theorem 4.5, suppose that P is
compact and suppose that the pullback absorbing family B is uniformly bounded by
a compact subset C of X . Then
lim
n→∞
sup
p∈P
dist R d (ϕ(n, p, D), A(P)) = 0
(4.16)
for every bounded subset D of X , where A(P) :=
p∈P A p , which is compact.
If the pullback attractor here consists of singleton sets corresponding to a periodic
entire trajectory, then A(P) represents the limit cycle and the convergence (4.16)
corresponds to orbital stability.