Cheban D.N. Global Attractors Of Non-autonomous Dissipative Dynamical Systems
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Triangular maps 477
where H(f) =
{f
m
| m ∈ Z
+
} and the over bar denotes the closure in C(Z
+
×E, E),
and f
m
(n, u) = f(n + m, u) for all n ∈ Z
+
and u ∈ E. Denote by (C(Z
+
×
E, E), Z
+
, σ) the dynamical system of translations. Here σ(m, g) := g
m
for all
m ∈ Z
+
and g ∈ C(Z
+
×E, E).
Let Ω be the hull H(f) of a given function f ∈ C(Z
+
× E, E) and denote the
restriction of (C(Z
+
× E, E), R, σ) on Ω by (Ω, Z
+
, σ). Let F : Ω × E → E be a
continuous map defined by F (g, u) = g(0, u) for g ∈ Ω and u ∈ E. Then equation
(14.35) can be rewritten in such form:
u
n+1
= F (σ
n
ω, u
n
),
where ω := g and σ
n
ω := g
n
.
Definition 14.13 The function f ∈ C(Z
+
×E, E) is said to be periodic (almost
periodic, recurrent), if f ∈ C(Z
+
×E, E) is a periodic (almost periodic, recurrent)
point of the dynamical system of translations (C(Z
+
×E, E), Z
+
, σ).
If the function f ∈ C(Z
+
× E, E) is periodic (almost periodic, recurrent), then
the set Ω := H(f) is the compact minimal set of the dynamical system (C(Z
+
×
E, E), Z
+
, σ) consisting from periodic (almost periodic, recurrent) points.
14.6 Pseudo recurrent solutions
Definition 14.14 Recall that an autonomous dynamical system (Ω, Z
+
, σ) is said
to be pseudo recurrent, if the following conditions are fulfilled:
a) Ω is compact;
b) (Ω, Z
+
, σ) is transitive, i.e. there exists a point ω
0
∈ Ω such that Ω =
{σ
n
ω
0
| n ∈ Z
+
};
c) every point ω ∈ Ω is stable in the sense of Poisson, i.e.
N
ω
= {{n
k
} | σ
n
k
ω → ω and n
k
→ +∞} 6= ∅.
Lemma 14.4 Let h(X, Z
+
, π), (Ω, Z
+
, σ), hi be a non-autonomous dynamical
system and the following conditions be fulfilled:
1) (Ω, Z
+
, σ) is pseudo recurrent;
2) γ ∈ C(Ω, X) is an invariant section of the homomorphism h : X → Ω.
Then the autonomous dynamical system (γ(Ω), Z
+
, π) is pseudo recurrent too.
Proof. It is evident that the space γ(Ω) is compact, because Ω is compact and
γ ∈ C(Ω, X). We note that on the space γ(Ω) we have the dynamical system
(γ(Ω), Z
+
, bπ) defined by the homomorphism γ : Ω → γ(Ω). Namely, bπ
n
γ(ω) :=
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 14
478 Global Attractors of Non-autonomous Dissipative Dynamical Systems
γ(σ
n
ω) for all n ∈ Z
+
and ω ∈ Ω. Hence, bπ
n
γ(ω) = π
n
γ(ω) for all n ∈ Z
+
and
ω ∈ Ω. Now we will show that γ(Ω) =
{π
n
γ(ω
0
) | n ∈ Z
+
}. Really, let x ∈ γ(Ω).
Then there exists a unique point ω ∈ Ω such that x = γ(ω). Let {n
k
} ⊂ Z
+
be a
sequence such that σ
n
k
ω
0
→ ω. Then x = γ(ω) = lim
k→+∞
γ(σ
n
k
ω
0
) = lim
k→+∞
π
n
k
γ(ω)
and, consequently, γ(Ω) ⊂
{π
n
γ(ω
0
) | n ∈ Z
+
}. The inverse inclusion is trivial.
Thus, γ(Ω) =
{π
n
γ(ω
0
) | n ∈ Z
+
}. To finish the proof of the lemma it is sufficient
to note that N
ω
⊆ N
γ(ω)
for every point ω ∈ Ω and, consequently, every point γ(ω)
is stable in the sense of Poisson. The lemma is proved.
Lemma 14.4 implies that under the conditions of Theorem 14.2 (respectively of
Th. 14.3, 14.5 or Th. 14.9) equation (14.32) or equation (14.15)) admits a pseudo
recurrent integral manifold.
Therefore, we have the following result.
Theorem 14.15 Assume the driving dynamical system (Ω, Z
+
, σ) is pseudo re-
current, and assume the conditions of Theorem 14.2 (respectively of Th. 14.3, 14.5
or Th. 14.9) are satisfied. Then equation (14.32) (respectively, equation (14.34) or
equation (14.15)) admits a pseudo recurrent integral manifold.
14.7 Chaos in triangular maps
Let (X, ρ) be a metric space and (X, Z
+
, π) be a discrete dynamical system gener-
ated by positive powers of the map f : X → X, i.e. π(n, x) = f
n
x for all x ∈ X
and n ∈ Z
+
, where f
n
:= f
n−1
◦f.
Definition 14.15 A subset M ⊆ X is called transitive, if there exists a point
x
0
∈ X such that H(x
0
) :=
{π(n, x
0
) | n ∈ Z
+
} = M.
Definition 14.16 {p, q} ⊆ X is called a Li-Yorke pair, if simultaneously
lim inf
n→+∞
ρ(π(n, p), π(n, q)) = 0 and lim sup
n→+∞
ρ(π(n, p), π(n, q)) > 0.
Definition 14.17 A set M ⊆ X is called scrambled, if any pair of distinct points
{p, q} ⊆ M is a Li-Yorke pair.
Definition 14.18 A dynamical system (X, Z
+
, π) is said to be chaotic, if X
contains an uncountable subset M satisfying the following conditions:
(1) the set M is transitive;
(2) M is scrambled;
(3)
P (M) = M, where P (M) := {x ∈ M | N
x
6= ∅} (i.e. x ∈ P (M ), if x is stable
in the sense of Poisson) and by bar we denote the closure in X.
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Bibliography 479
Theorem 14.16 Let (X, Z
+
, π) and (Ω, Z
+
, σ) be two dynamical systems and ν :
X → Ω be a homeomorphism of (Ω, Z
+
, σ) onto (X, Z
+
, π). Assume that (Ω, Z
+
, σ)
is chaotic. Then the dynamical system (X, Z
+
, π) is chaotic too.
Proof. Let (Ω, Z
+
, σ) be chaotic and ν : X → Ω be a homeomorphism of (Ω, Z
+
, σ)
onto (X, Z
+
, π). Then there exists a subset A ⊆ Ω satisfying all the conditions of
Definition 14.17. Denote M := ν(A). Then the set M is transitive and
P (M) = M
(see the proof of Lemma 14.4). Note that the set M := ν(A) is uncountable,
because the homeomorphism ν : Ω → X transforms different points to different
points. Finally, to finish the proof of Theorem 14.16 it is sufficient to note that the
homeomorphism ν : Ω → X maps a Li-Yorke pair in (Ω, Z
+
, σ) to a Li-Yorke pair
in (X, Z
+
, π). The theorem is proved.
Using Theorem 14.16 and the results from sections 3 and 4 we will obtain some
criterions of the existence of chaotic sets for triangular maps. For example the
following statements are held.
Theorem 14.17 Let (Ω, Z
+
, π) be a chaotic dynamical system. Then under the
conditions of Theorem 14.3 equation (14.15) (respectively triangular map) admits a
compact invariant chaotic set.
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480 Global Attractors of Non-autonomous Dissipative Dynamical Systems
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