58
P. E. Kloeden and M. Yang
Fig. 4.2 Pullback attraction
as t 0 → −∞
Forward and pullback convergence coincide in an autonomous system, but are
independent of each other in a nonautonomous one. Consequently, two types of
nonautonomous attractors arise from these convergences, a forward attractor and a
pullback attractor.
4.3 Invariant Sets and Attractors of Processes
The nonautonomous case differs crucially from the autonomous one and requires the
introduction of new concepts The following definitions and results are taken from
the monographs [15, 17], see also the papers [3, 19, 24, 25].
Definition 4.2 A family A = {A n : n ∈ Z} of nonempty subsets of R
d is φinvariant if
φ
n, n 0 , A n 0
= A n , for all (n, n 0 ) ∈ Z
2
≥ ,
or, equivalently, if f n (A n ) = A n+1 for all n ∈ Z. It is said to be uniformly bounded
if there exists a bounded subset B of R
d such that A n ⊂ B for all n ∈ Z.
A uniformly bounded φ-invariant family is characterised by the bounded entire solutions.
Proposition 4.1 A uniformly bounded family A = {A n : n ∈ Z} is φ-invariant if
and only if for every pair n 0 ∈ Z and x 0 ∈ A n 0 there exists a bounded entire solution
χ such that χ n 0 = x 0 and χ n ∈ A n for all n ∈ Z.
Forward and pullback convergences can be used to define two distinct types of
nonautonomous attractors for a process φ. Define
dist R d (x, B) := inf
b∈B
− b
dist R d (A, B) := sup
a∈A
dist R d (a, B)
for nonempty subsets A, B of R
d .
Definition 4.3 A φ-invariant family A = {A n : n ∈ Z} of nonempty compact subsets of R
d is called a forward attractor if it forward attracts bounded subsets D of
R
d , i.e.,