62
P. E. Kloeden and M. Yang
Since A n 0 ⊂ B n 0 and A n = φ(n, n 0 , A n 0 ) ⊂ φ(n, n 0 , B n 0 )
lim
n→∞
dist R d (A n , ω B (n 0 )) = 0 (fixed n 0 ).
(4.11)
Moreover, ω B (n 0 ) ⊂ ω B (n
0 ) ⊂ B for n 0 ≤ n
0 . Hence the set
ω
∞
B :=
n 0 ∈Z
ω B (n 0 )
is a nonempty and compact subset of B. From (4.11) it is clear that
lim
n→∞
dist R d
A n , ω
∞
B
= 0.
(4.12)
The ω-limit points for dynamics starting inside the family of sets A are defined by
ω
∞
A :=
n 0 ∈Z
n≥n 0
A n =
n 0 ∈Z
n≥n 0
φ(n, n 0 , A n 0 ) ⊂ B,
which is nonempty and compact as a family of nested compact sets. Obviously, ω
∞
A
⊂ ω
∞
B ⊂ B. The example below shows that the inclusions may be strict.
The following result was proved in [13].
Theorem 4.3 A is forward attracting from within B if and only if ω
∞
A = ω
∞
B .
If B is forward absorbing, then A will then be a forward attractor.
4.5 Forward Attracting Sets
Forward attraction (4.4) is very different conceptually from pullback attraction (4.5)
in that it is about what happens in the distant future and not in actual time, i.e., current
time. Pullback convergence, is in this sense the natural generalisation of convergence
in autonomous systems, which depends only on the elapsed time since starting, so
its limit sets exist, in fact, in actual time. Moreover, as seen in system (4.7) above,
forward attractors need not be unique.
A curious feature of forward attractors in the sense of Definition 4.3 is that they
require the entire past history of the system to be known. Indeed, its construction
in Theorem 4.2 is based on pullback convergence, although forward convergence
is about the distant future and should be independent of the past. In fact, forward
convergence should not even require the system to be defined in the past.
The future limiting dynamics in (4.10) is contained in the omega limit set ω
∞
B . It
includes what Haraux [9] and Vishik [30] called a uniform attractor, i.e., a compact
set which attracts the forward dynamics of the system uniformly in the initial time
and is minimal in the sense that it is contained in all sets with this property.
P. E. Kloeden and M. Yang
Since A n 0 ⊂ B n 0 and A n = φ(n, n 0 , A n 0 ) ⊂ φ(n, n 0 , B n 0 )
lim
n→∞
dist R d (A n , ω B (n 0 )) = 0 (fixed n 0 ).
(4.11)
Moreover, ω B (n 0 ) ⊂ ω B (n
0 ) ⊂ B for n 0 ≤ n
0 . Hence the set
ω
∞
B :=
n 0 ∈Z
ω B (n 0 )
is a nonempty and compact subset of B. From (4.11) it is clear that
lim
n→∞
dist R d
A n , ω
∞
B
= 0.
(4.12)
The ω-limit points for dynamics starting inside the family of sets A are defined by
ω
∞
A :=
n 0 ∈Z
n≥n 0
A n =
n 0 ∈Z
n≥n 0
φ(n, n 0 , A n 0 ) ⊂ B,
which is nonempty and compact as a family of nested compact sets. Obviously, ω
∞
A
⊂ ω
∞
B ⊂ B. The example below shows that the inclusions may be strict.
The following result was proved in [13].
Theorem 4.3 A is forward attracting from within B if and only if ω
∞
A = ω
∞
B .
If B is forward absorbing, then A will then be a forward attractor.
4.5 Forward Attracting Sets
Forward attraction (4.4) is very different conceptually from pullback attraction (4.5)
in that it is about what happens in the distant future and not in actual time, i.e., current
time. Pullback convergence, is in this sense the natural generalisation of convergence
in autonomous systems, which depends only on the elapsed time since starting, so
its limit sets exist, in fact, in actual time. Moreover, as seen in system (4.7) above,
forward attractors need not be unique.
A curious feature of forward attractors in the sense of Definition 4.3 is that they
require the entire past history of the system to be known. Indeed, its construction
in Theorem 4.2 is based on pullback convergence, although forward convergence
is about the distant future and should be independent of the past. In fact, forward
convergence should not even require the system to be defined in the past.
The future limiting dynamics in (4.10) is contained in the omega limit set ω
∞
B . It
includes what Haraux [9] and Vishik [30] called a uniform attractor, i.e., a compact
set which attracts the forward dynamics of the system uniformly in the initial time
and is minimal in the sense that it is contained in all sets with this property.