64
P. E. Kloeden and M. Yang
Some additional assumptions about the future behaviour of the system are needed
to ensure asymptotical positive or negative invariance.
Assumption 4.1 There exists a φ-positive invariant compact subset B in R
d such
that for any bounded subset D of R
d and every n 0 ≥ N
∗ there exists a N D ≥ 0 for
which
φ(n, n 0 , x 0 ) ∈ B ∀n ≥ n 0 + N D , x 0 ∈ D.
Assumption 4.2 The process is Lipschitz continuous in initial conditions in B on
finite time integer intervals [n 0 , n 0 + N ] uniformly in n 0 ≥ N
∗ , i.e., there exists a
constant L B > 0 independent of n 0 ≥ N
∗ such that
ϕ(n, τ , x 0 ) − ϕ(n, τ , y 0 ) ≤ x 0 − y 0 e
L B (n−τ )
≤ x 0 − y 0 e
L B N
for all x 0 , y 0 ∈ B and N
∗
≤ τ < n ≤ τ + N .
Theorem 4.4 Let Assumption 4.1 hold, then ω
∞
B is asymptotically positively invariant. If, in addition, Assumptions 4.2 holds, then ω
∞
B is also asymptotically negatively
invariant.
Assumption 4.2 holds for a nonautonomous difference equation if the functions
the f n : R
d
→ R
d are Lipschitz continuous on B uniformly in n ≥ N
∗ , i.e., only in
the distant future.
4.6 Skew Product Flows
The formulation of a skew product flow is more complicated than that of a process, but
it includes more information about how the system changes in time [11]. It consists
of an autonomous dynamical system θ (full group) on a base space P, which is the
source of the nonautonomity in a cocycle mapping ϕ acting on the state space R
d .
The autonomous dynamical system here is often called the driving system.
Definition 4.8 A discrete time skew product flow (θ, ϕ) on P × R
d consists of a
discrete time autonomous dynamical system θ = {θ n } n∈Z acting on a metric space
(P, d P ), which is called the base space, i.e.,
(i) θ 0 ( p) = p,
(ii) θ n+m ( p) = θ n ◦ θ m ( p),
(iii) (n, p) → θ n ( p) continuous
for all p ∈ P and s, t ∈ Z, and a cocycle mapping ϕ : Z
+
× P × R
d
→ R
d acting
on a metric space (R
d
, d
d
R ), which is called the state space, i.e.,
P. E. Kloeden and M. Yang
Some additional assumptions about the future behaviour of the system are needed
to ensure asymptotical positive or negative invariance.
Assumption 4.1 There exists a φ-positive invariant compact subset B in R
d such
that for any bounded subset D of R
d and every n 0 ≥ N
∗ there exists a N D ≥ 0 for
which
φ(n, n 0 , x 0 ) ∈ B ∀n ≥ n 0 + N D , x 0 ∈ D.
Assumption 4.2 The process is Lipschitz continuous in initial conditions in B on
finite time integer intervals [n 0 , n 0 + N ] uniformly in n 0 ≥ N
∗ , i.e., there exists a
constant L B > 0 independent of n 0 ≥ N
∗ such that
ϕ(n, τ , x 0 ) − ϕ(n, τ , y 0 ) ≤ x 0 − y 0 e
L B (n−τ )
≤ x 0 − y 0 e
L B N
for all x 0 , y 0 ∈ B and N
∗
≤ τ < n ≤ τ + N .
Theorem 4.4 Let Assumption 4.1 hold, then ω
∞
B is asymptotically positively invariant. If, in addition, Assumptions 4.2 holds, then ω
∞
B is also asymptotically negatively
invariant.
Assumption 4.2 holds for a nonautonomous difference equation if the functions
the f n : R
d
→ R
d are Lipschitz continuous on B uniformly in n ≥ N
∗ , i.e., only in
the distant future.
4.6 Skew Product Flows
The formulation of a skew product flow is more complicated than that of a process, but
it includes more information about how the system changes in time [11]. It consists
of an autonomous dynamical system θ (full group) on a base space P, which is the
source of the nonautonomity in a cocycle mapping ϕ acting on the state space R
d .
The autonomous dynamical system here is often called the driving system.
Definition 4.8 A discrete time skew product flow (θ, ϕ) on P × R
d consists of a
discrete time autonomous dynamical system θ = {θ n } n∈Z acting on a metric space
(P, d P ), which is called the base space, i.e.,
(i) θ 0 ( p) = p,
(ii) θ n+m ( p) = θ n ◦ θ m ( p),
(iii) (n, p) → θ n ( p) continuous
for all p ∈ P and s, t ∈ Z, and a cocycle mapping ϕ : Z
+
× P × R
d
→ R
d acting
on a metric space (R
d
, d
d
R ), which is called the state space, i.e.,