4 Nonautonomous Attractors
57
for all n > n 0 with n 0 ∈ Z, and each x 0 ∈ R
d with the initial value φ(n 0 , n 0 , x 0 ) :=
x 0 .
These solution mappings of nonautonomous difference equations (4.1) generate an abstract discrete time nonautonomous dynamical system formulated as a 2parameter semigroup or process [7, 8] on the state state space R
d and time set Z.
Definition 4.1 A (discrete time) process on the state space R
d is a mapping φ :
Z
2
≥ × R
d
→ R
d which satisfies the following initial value, 2-parameter evolution
and continuity properties:
(i) φ(n 0 , n 0 , x 0 ) = x 0 for all n 0 ∈ Z and x 0 ∈ R
d ,
(ii) φ(n 2 , n 0 , x 0 ) = φ (n 2 , n 1 , φ(n 1 , n 0 , x 0 )) for all n 0 ≤ n 1 ≤ n 2 in Z and x 0 ∈ R
d ,
(iii) the mapping x 0 → φ(n, n 0 , x 0 ) of R
d into itself is continuous for all (n, n 0 ) ∈
Z
2
≥ .
Entire solutions, which are defined for all n ∈ Z, play an important role in dynamical systems. An entire solution solution χ
∗ of a process satisfies
φ(n, n 0 , χ
∗
n 0
) = χ
∗
n for all (n, n 0 ) ∈ Z
2
≥ .
The obvious idea of convergence of other solutions to an entire solution χ
∗ reads
φ (n, n 0 , x 0 ) − χ
∗
n
→ 0 as n → ∞ (n 0 fixed).
(4.2)
It involves a moving target and is called forward convergence (Fig. 4.1).
To obtain convergence to the point χ
∗
n for a particular time n one has to start
progressively earlier, i.e.,
φ (n, n 0 , x 0 ) − χ
∗
n
→ 0 as n 0 → −∞ (n fixed).
(4.3)
It is called pullback convergence.
Note that pullback convergence does not involve the system running backwards in
time, rather it runs forwards from an ever earlier starting time. Pullback convergence
has long been used under other names, e.g., to construct entire solutions [20], probability measures, e.g., [6], or by Kolmogorov under the name absolute probability
(Fig. 4.2).
Fig. 4.1 Forward
convergence as t → ∞
57
for all n > n 0 with n 0 ∈ Z, and each x 0 ∈ R
d with the initial value φ(n 0 , n 0 , x 0 ) :=
x 0 .
These solution mappings of nonautonomous difference equations (4.1) generate an abstract discrete time nonautonomous dynamical system formulated as a 2parameter semigroup or process [7, 8] on the state state space R
d and time set Z.
Definition 4.1 A (discrete time) process on the state space R
d is a mapping φ :
Z
2
≥ × R
d
→ R
d which satisfies the following initial value, 2-parameter evolution
and continuity properties:
(i) φ(n 0 , n 0 , x 0 ) = x 0 for all n 0 ∈ Z and x 0 ∈ R
d ,
(ii) φ(n 2 , n 0 , x 0 ) = φ (n 2 , n 1 , φ(n 1 , n 0 , x 0 )) for all n 0 ≤ n 1 ≤ n 2 in Z and x 0 ∈ R
d ,
(iii) the mapping x 0 → φ(n, n 0 , x 0 ) of R
d into itself is continuous for all (n, n 0 ) ∈
Z
2
≥ .
Entire solutions, which are defined for all n ∈ Z, play an important role in dynamical systems. An entire solution solution χ
∗ of a process satisfies
φ(n, n 0 , χ
∗
n 0
) = χ
∗
n for all (n, n 0 ) ∈ Z
2
≥ .
The obvious idea of convergence of other solutions to an entire solution χ
∗ reads
φ (n, n 0 , x 0 ) − χ
∗
n
→ 0 as n → ∞ (n 0 fixed).
(4.2)
It involves a moving target and is called forward convergence (Fig. 4.1).
To obtain convergence to the point χ
∗
n for a particular time n one has to start
progressively earlier, i.e.,
φ (n, n 0 , x 0 ) − χ
∗
n
→ 0 as n 0 → −∞ (n fixed).
(4.3)
It is called pullback convergence.
Note that pullback convergence does not involve the system running backwards in
time, rather it runs forwards from an ever earlier starting time. Pullback convergence
has long been used under other names, e.g., to construct entire solutions [20], probability measures, e.g., [6], or by Kolmogorov under the name absolute probability
(Fig. 4.2).
Fig. 4.1 Forward
convergence as t → ∞