56
P. E. Kloeden and M. Yang
behaviour, so many of the concepts that have been developed and extensively investigated for autonomous dynamical systems are either too restrictive or no longer valid
or meaningful in the nonautonomous context [2, 4, 17].
A new feature of nonautonomous dynamical systems is that invariant sets are in
fact families of sets that are invariant in the sense that they mapped onto each other
as time evolves. Another important new feature is that limiting temporal behaviour
must now be characterised in two ways. For a discrete time autonomous dynamical
system, the elapsed time n − n 0 → ∞ if either n → ∞ with n 0 fixed or if n 0 → −∞
with n fixed. In the nonautonomous case the limits obtained may be different, if they
exist.
The former, called forward convergence, involves information about the future of
the system, whereas the latter, called pullback convergence, uses information from
the past. Two types of nonautonomous attractors arise from these convergences, a
forward and a pullback attractor. These consist of families of nonempty compact
subsets that are invariant in the above generalised sense and attract other sets (or
even families of sets) with respect to the corresponding convergence.
The results given below have appeared in published papers over the past twenty
years. They are developed systematically in monographs Kloeden and Rasmussen
[15] and Kloeden and Yang [17] as well as [19] which focuses on nonautonomous
difference equations. See also the monographs [2, 4].
Nonautonomous dynamical systems have many applications, especially in the biological sciences [14]. Aneta Stefanovska has been a pioneer in their use in cardiology
and physics, see [1, 5, 22, 23, 28, 29].
4.2 Nonautonomous Difference Equations
A nonautonomous difference equation on R
d has the form
x n+1 = f n (x n )
(4.1)
with mappings f n : R
d
→ R
d which may vary with time n. They are assumed to be
continuous here. Define
Z
2
≥ := {(n, n 0 ) ∈ Z
2
: n ≥ n 0 },
The nonautonomous difference equation (4.1) generates a solution mapping
φ : Z
2
≥ × R
d
→ R
d
through iteration, i.e.,
φ(n, n 0 , x 0 ) := f n−1 ◦ · · · ◦ f n 0 (x 0 )
P. E. Kloeden and M. Yang
behaviour, so many of the concepts that have been developed and extensively investigated for autonomous dynamical systems are either too restrictive or no longer valid
or meaningful in the nonautonomous context [2, 4, 17].
A new feature of nonautonomous dynamical systems is that invariant sets are in
fact families of sets that are invariant in the sense that they mapped onto each other
as time evolves. Another important new feature is that limiting temporal behaviour
must now be characterised in two ways. For a discrete time autonomous dynamical
system, the elapsed time n − n 0 → ∞ if either n → ∞ with n 0 fixed or if n 0 → −∞
with n fixed. In the nonautonomous case the limits obtained may be different, if they
exist.
The former, called forward convergence, involves information about the future of
the system, whereas the latter, called pullback convergence, uses information from
the past. Two types of nonautonomous attractors arise from these convergences, a
forward and a pullback attractor. These consist of families of nonempty compact
subsets that are invariant in the above generalised sense and attract other sets (or
even families of sets) with respect to the corresponding convergence.
The results given below have appeared in published papers over the past twenty
years. They are developed systematically in monographs Kloeden and Rasmussen
[15] and Kloeden and Yang [17] as well as [19] which focuses on nonautonomous
difference equations. See also the monographs [2, 4].
Nonautonomous dynamical systems have many applications, especially in the biological sciences [14]. Aneta Stefanovska has been a pioneer in their use in cardiology
and physics, see [1, 5, 22, 23, 28, 29].
4.2 Nonautonomous Difference Equations
A nonautonomous difference equation on R
d has the form
x n+1 = f n (x n )
(4.1)
with mappings f n : R
d
→ R
d which may vary with time n. They are assumed to be
continuous here. Define
Z
2
≥ := {(n, n 0 ) ∈ Z
2
: n ≥ n 0 },
The nonautonomous difference equation (4.1) generates a solution mapping
φ : Z
2
≥ × R
d
→ R
d
through iteration, i.e.,
φ(n, n 0 , x 0 ) := f n−1 ◦ · · · ◦ f n 0 (x 0 )