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for n ≥ 1 is a discrete-time cocycle mapping. It is clear that the mapping x 0 →
ϕ (n, s, x 0 ) is continuous due to the continuity of the composition of continuous
functions. One can also show that the mappings s → ϕ (n, s, x 0 ) are continuous
on P, see [15].
4.6.2 Attractors of Skew Product Flows
Then definition of invariance and pullback and attractors for attractors generalise
to skew product flows. Essentially, θ
n
( p 0 ) replaces the actual time n + n 0 and p 0
replaces the initial time n 0 .
Definition 4.9 A family A = {A p : p ∈ P} of nonempty subsets of R
d is called
ϕ-invariant for a skew-product system (θ, ϕ) on P × X if
ϕ(n, p, A p ) = A θ n ( p) for all n ∈ Z
+
, p ∈ P.
It is called ϕ- positively invariant if
ϕ(n, p, A p ) ⊆ A θ n ( p) for all n ∈ Z
+
, p ∈ P.
Definition 4.10 A family A = {A p : p ∈ P} of nonempty compact subsets of R
d
is called pullback attractor of a skew-product system (θ, ϕ) on P × R
d if it is ϕinvariant and pullback attracts bounded sets, i.e.,
dist R d
ϕ( j, θ − j ( p), D), A p
= 0 for j → ∞
(4.13)
for all p ∈ P and all bounded subsets D of R
d .
It is called a forward attractor if it is ϕ-invariant and forward attracts bounded
sets, i.e.,
dist R d
ϕ( j, p, D), A θ j ( p)
= 0 for j → ∞.
(4.14)
As with processes, the existence of a pullback for skew-product systems ensured by
that of a pullback absorbing system.
Definition 4.11 A family B = {B p : p ∈ P} of nonempty compact subsets of R
d
is called a pullback absorbing family for a skew-product system (θ, ϕ) on P × R
d
if for each p ∈ P and every bounded subset D of R
d there exists an N p,D ∈ Z
+ such
that
ϕ
j, θ − j ( p), D
⊆ B p for all j ≥ N p,D , p ∈ P.
The following result generalises the theorem for autonomous semidynamical systems and Theorem 4.1 for processes. The proof is similar in the latter case, essentially