68
P. E. Kloeden and M. Yang
4.6.3 Skew Product Flows as Semi-dynamical Autonomous
Systems
A skew product flow (θ, ϕ) on P × R
d is an autonomous semi-dynamical system Π
on the extended state space X := P × R
d with the metric
dist X (( p 1 , x 1 ), ( p 2 , x 2 )) = d P ( p 1 , p 2 ) + +(x 1 − x 2 ,
where Π : T
+
× R
d
→ R
d is defined by
Π (t, (p 0 , x 0 )) = (θ t ( p 0 ), ϕ(t, p 0 , x 0 )) .
The initial condition and continuity properties of Π are straightforward. The
semi-group property follows from that of θ and the cocycle property of ϕ:
Π (s + t, (p 0 , x 0 )) = (θ s+t ( p 0 ), ϕ(s + t, p 0 , x 0 ))
= (θ s ◦ θ t ( p 0 ), ϕ(s, θ t ( p 0 ), ϕ(t, p 0 , x 0 )))
= Π (s, (θ t ( p 0 ), ϕ(t, p 0 , x 0 ))) = Π (s, Π (t, (p 0 , x 0 ))) .
This representation as an autonomous semi-dynamical system is useful since it provides insights into how one could define invariant sets and attractors for nonautonomous systems.
Proposition 4.4 Suppose that A is a uniform attractor (i.e., uniformly attracting
in both the forward and pullback senses) of a skew-product system (θ, ϕ) and that
p∈P A p is precompact in R
d .
Then the union A :=
p∈P { p} × A p is the global attractor of the autonomous
semidynamical system Π .
Without uniform attraction as in Proposition 4.4 a pullback attractor need not give a
global attractor, but the following result does hold.
Proposition 4.5 If A is a pullback attractor for a skew-product system (θ, ϕ) and
p∈P A p is precompact in R
d , then A :=
p∈P { p} × A p is the maximal invariant
compact set of the autonomous semidynamical system Π .
The set A here need not be the global attractor of Π . In the opposite direction, the
global attractor of the associated autonomous semidynamical system always forms
a pullback attractor of the skew-product system.
Proposition 4.6 If the autonomous semidynamical system Π has a global attractor
A =
p∈P
{ p} × A p ,
then A = {A p : p ∈ P} is a pullback attractor for the skew-product system (θ, ϕ).
P. E. Kloeden and M. Yang
4.6.3 Skew Product Flows as Semi-dynamical Autonomous
Systems
A skew product flow (θ, ϕ) on P × R
d is an autonomous semi-dynamical system Π
on the extended state space X := P × R
d with the metric
dist X (( p 1 , x 1 ), ( p 2 , x 2 )) = d P ( p 1 , p 2 ) + +(x 1 − x 2 ,
where Π : T
+
× R
d
→ R
d is defined by
Π (t, (p 0 , x 0 )) = (θ t ( p 0 ), ϕ(t, p 0 , x 0 )) .
The initial condition and continuity properties of Π are straightforward. The
semi-group property follows from that of θ and the cocycle property of ϕ:
Π (s + t, (p 0 , x 0 )) = (θ s+t ( p 0 ), ϕ(s + t, p 0 , x 0 ))
= (θ s ◦ θ t ( p 0 ), ϕ(s, θ t ( p 0 ), ϕ(t, p 0 , x 0 )))
= Π (s, (θ t ( p 0 ), ϕ(t, p 0 , x 0 ))) = Π (s, Π (t, (p 0 , x 0 ))) .
This representation as an autonomous semi-dynamical system is useful since it provides insights into how one could define invariant sets and attractors for nonautonomous systems.
Proposition 4.4 Suppose that A is a uniform attractor (i.e., uniformly attracting
in both the forward and pullback senses) of a skew-product system (θ, ϕ) and that
p∈P A p is precompact in R
d .
Then the union A :=
p∈P { p} × A p is the global attractor of the autonomous
semidynamical system Π .
Without uniform attraction as in Proposition 4.4 a pullback attractor need not give a
global attractor, but the following result does hold.
Proposition 4.5 If A is a pullback attractor for a skew-product system (θ, ϕ) and
p∈P A p is precompact in R
d , then A :=
p∈P { p} × A p is the maximal invariant
compact set of the autonomous semidynamical system Π .
The set A here need not be the global attractor of Π . In the opposite direction, the
global attractor of the associated autonomous semidynamical system always forms
a pullback attractor of the skew-product system.
Proposition 4.6 If the autonomous semidynamical system Π has a global attractor
A =
p∈P
{ p} × A p ,
then A = {A p : p ∈ P} is a pullback attractor for the skew-product system (θ, ϕ).