60
P. E. Kloeden and M. Yang
has a forward attractor A
(r ) with component subsets
A
(r )
n =
r [−1, 1],
n ≤ 0,
1
2 n r [−1, 1],
n > 0.
(4.8)
These forward attractors are not pullback attractors.
For a long time there was no counterpart of Theorem 4.1 for nonautonomous
forward attractors. In fact, such construction (4.6) was shown by Kloeden and Lorenz
[12, 13] to hold within any positively invariant family, but provides only a candidate
for a forward attractor; other conditions must also hold. A key observation for the
construction of a forward attractor is provided by the next theorem [13]. It is based
on the following important property of forward attractors [13], which requires the
following definition of a forward absorbing family. See also [12].
Definition 4.5 A family B = {B n : n ∈ Z} of nonempty compact subsets of R
d is
called forward absorbing if for every bounded subset D of R
d and n ∈ Z there exists
an N (n, D) ∈ N such that
φ (n, n 0 , D) ⊆ B n for all n ≥ n 0 + N (n, D).
Proposition 4.2 A uniformly bounded forward attractor A = {A n : n ∈ Z} in R
d
has a φ-positively invariant family B = {B n : n ∈ Z} of nonempty compact subsets
with A n ⊂ B n for n ∈ Z, which is forward absorbing.
Theorem 4.2 Suppose that a process φ on R
d has a φ-positively invariant family
B = {B n : n ∈ Z} of nonempty compact subsets of R
d .
Then φ has a maximal φ-invariant family A = {A n : n ∈ Z} in B of nonempty
compact subsets determined by
A n =
n 0 ≤n
φ
n, n 0 , B n 0
for each n ∈ Z.
(4.9)
In view of Proposition 4.2, the components sets of any forward attractor can
be constructed in this way. Note that nothing is assumed here about the dynamics
outside of the family B.
4.4.1 A Counterexample
Consider the piecewise autonomous equation
x n+1 =
λ n x n
1 + |x n |
,
λ n :=
λ,
n ≥ 0,
λ
−1
,
n < 0
(4.10)
P. E. Kloeden and M. Yang
has a forward attractor A
(r ) with component subsets
A
(r )
n =
r [−1, 1],
n ≤ 0,
1
2 n r [−1, 1],
n > 0.
(4.8)
These forward attractors are not pullback attractors.
For a long time there was no counterpart of Theorem 4.1 for nonautonomous
forward attractors. In fact, such construction (4.6) was shown by Kloeden and Lorenz
[12, 13] to hold within any positively invariant family, but provides only a candidate
for a forward attractor; other conditions must also hold. A key observation for the
construction of a forward attractor is provided by the next theorem [13]. It is based
on the following important property of forward attractors [13], which requires the
following definition of a forward absorbing family. See also [12].
Definition 4.5 A family B = {B n : n ∈ Z} of nonempty compact subsets of R
d is
called forward absorbing if for every bounded subset D of R
d and n ∈ Z there exists
an N (n, D) ∈ N such that
φ (n, n 0 , D) ⊆ B n for all n ≥ n 0 + N (n, D).
Proposition 4.2 A uniformly bounded forward attractor A = {A n : n ∈ Z} in R
d
has a φ-positively invariant family B = {B n : n ∈ Z} of nonempty compact subsets
with A n ⊂ B n for n ∈ Z, which is forward absorbing.
Theorem 4.2 Suppose that a process φ on R
d has a φ-positively invariant family
B = {B n : n ∈ Z} of nonempty compact subsets of R
d .
Then φ has a maximal φ-invariant family A = {A n : n ∈ Z} in B of nonempty
compact subsets determined by
A n =
n 0 ≤n
φ
n, n 0 , B n 0
for each n ∈ Z.
(4.9)
In view of Proposition 4.2, the components sets of any forward attractor can
be constructed in this way. Note that nothing is assumed here about the dynamics
outside of the family B.
4.4.1 A Counterexample
Consider the piecewise autonomous equation
x n+1 =
λ n x n
1 + |x n |
,
λ n :=
λ,
n ≥ 0,
λ
−1
,
n < 0
(4.10)