4 Nonautonomous Attractors
63
Fig. 4.4 Solutions of x n+1
=
1
2 x n + e −n with different
initial conditions. The set {0}
is not invariant but looks
more and more more
invariant for later starting
times n 0 , i.e., asymptotically
invariant
-4
-3
-2
-1
0
1
2
3
4
0
5
10
15
20
25
However the definition of ω
∞
B does not require the system to be defined in the
distant past or for the attraction to be uniform in the initial time. Moreover, nothing
is said about invariance in this definition. For this reason (4.10), ω
∞
B will be called
the forward attracting set.
In the piecewise autonomous system, (4.10), ω
∞
B = [1 − λ, λ − 1] is positively
invariant for all time, but only invariant for negative time. The situation can be more
complicated (Fig. 4.4).
4.5.1 Asymptotic Invariance
Asymptotic invariance means that a set becomes more and more invariant the later
one starts, [10, 11, 16, 21].
Definition 4.6 A set A is said to be asymptotically positively invariant if for any
monotonic decreasing sequence ε p → 0 as p → ∞ there exists a monotonic increasing sequence N p → ∞ as p → ∞ such that
φ (n, n 0 , A) ⊂ B ε p (A) , n ≥ n 0 ,
for each n 0 ≥ N p , where
B ε p (A) := {x ∈ R
d
: dist R d (x, A) < ε}.
Definition 4.7 A set A is said to be asymptotically negatively invariant if for every
a ∈ A, ε > 0 and N > 0, there exist n ε and a ε ∈ A such that
ϕ (n ε , n ε − N , a ε ) − a < ε.