Cheban D.N. Global Attractors Of Non-autonomous Dissipative Dynamical Systems

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xx Global Attractors of Non-autonomous Dissipative Dynamical Systems

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 0

Notations

∀ for all;

∃ there exists;

:= is equal (coincide) by the deﬁnition;

0 the number 0 and also the null element of all additive group

(semigroup);

N the set of all natural numbers;

Z the set of all integers;

Q the set of all rational numbers;

R the set of all real numbers;

C the set of all complex numbers;

S one of the sets R or Z;

−

) the set of all nonnegative (non-positive) numbers from the

set S;

X ×Y the product space of two sets;

the product space of n copies of the set M;

the real or complex n-dimensional Euclidean space;

} a sequence;

x ∈ X x is an element of the set X;

∂X the boundary of the set X;

X ⊆ Y the set X is a part of or coincides with the set Y ;

Y the union of the two sets X and Y ;

X \Y the complement of the set Y in the set X;

Y the intersection of the sets X and Y ;

∅ the empty-set;

(X, ρ) the complete metric space with the distance ρ;

xxi

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xxii Global Attractors of Non-autonomous Dissipative Dynamical Systems

the space of all closed bounded subsets of the metric

space X;

M the closure of the set M;

−1

the inverse mapping of the mapping f;

f(M ) the image of the set M ⊆ X by the mapping f : X → Y ,

i.e., {y ∈ Y : y = f (x), x ∈ M };

f ◦ g the composition of the mappings f and g, i.e., (f ◦g)(x) =

f(g(x));

the restriction of the mapping f on the set M;

f(·, x) the partial mapping, deﬁned by the function f if the second

variable takes a value x;

the identity mapping from the X into itself;

Im(f) range of values of the function f ;

D(f ) the domain of the deﬁnition of the function f;

|x| or ||x|| the norm of the element x;

(x, y) the ordered pair;

C(X, Y ) the set of all continuous mappings from X into Y with

compact-open topology;

(U, M) the set of all k-time continuously diﬀerentiable functions

f : U → M;

f : X → Y the mapping from X into Y ;

B(M, ε) the open ε-neighborhood of the set M in the metric

space X;

B[M, ε] the closed ε-neighborhood of the set M in the metric

space X;

{x, y, ..., z} the set consisting from the elements x, y, ..., z;

1, n the set consisting from the elements 1, 2, ..., n;

{x ∈ X|R(x)} the set of all elements of the set X, possessing the

property R;

−1

(M) the pro-image of the set M ⊆ Y by the mapping f : X → Y ,

i.e. {x ∈ X : f(x) ∈ M};

F (t, ·) := f

the partial mapping, deﬁned by the function f if the ﬁrst

variable takes the value t;

ρ(ξ, η) the distance in the metric space X;

lim

n→+∞

the limit of the sequence;

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Notations xxiii

↓ 0 the monotone decreasing to the 0 sequence;

lim

x→a

f(x) the limit of the mapping f as x → a;

: λ ∈ Λ} the union of the family of sets {M

}

∈ Λ;

: λ ∈ Λ} the intersection of the family of sets {M

}

∈ Λ;

(H, h·, ·i) Hilbert space with the scalar product h·, ·i;

˙ϕ :=

dϕ

the derivative of the function ϕ;

cardM the cardinality of the set M;

C(X) the family of all compacts from X;

B(X) the family of all bounded subsets from X;

λ(A) the measure of non-compactness of the set A ∈ B(X);

(X, T, π) a dynamical system;

(X, P ) cascade, generated by the mapping P ;

(α

) ω(α)-limit set of the point x;

the whole trajectories of (X, T, π) with condition ϕ

(0) = x.

the set of all whole trajectories ϕ

of (X, T, π).

α-limit set of the whole trajectory ϕ

∈ Φ

;

(M) the stable manifold of the set M;

M st.L

the set M is stable in sense of Lagrange

in the positive direction;

) positive extension (positive limit extension) of the point x;

xt = π

x = π(t, x) the state of point x at the moment t;

: X

×X

→ X

the projection of X

×X

on the i-th (i = 1, 2) component

;

(M) the positive extension of the set M;

(M) the positive limit extension of the set M ;

the positive semi-trajectory of the point x;

(M) the positive semi-trajectory of the set M;

(x) the closure of positive semi-trajectory of the point x;

the trajectory of the point x;

H(x) the closure of trajectory of the point x;

Ω the closure of the union of all ω-limit points of (X, T, π);

β(A, B) the semi-deviation of the set A from the set B (A, B ∈ 2

);

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Chapter 1

Autonomous dynamical systems

1.1 Some notions, notations and facts from theory of dynamical

systems

1. Below we give some notions, denotations and facts from theory of dynamical

systems

[

32, 33, 258, 261, 300, 302, 304

]

which we will use in this book.

Let X be a topological space, R (Z) be a group of real (integer) numbers, R

) be a semi-group of the nonnegative real (integer) numbers, S be one of the

two sets R or Z and T ⊆ S (S

⊆ T) be a sub-semigroup of additive group S.

Deﬁnition 1.1 Triplet (X, T, π), where π : T ×X → X is a continuous mapping

satisfying the following conditions:

π(0, x) = x; (1.1)

π(s, π(t, x)) = π(s + t, x); (1.2)

is called a dynamical system. If T = R (R

) or Z (Z

), then the dynamical system

(X, T, π) is called a group (semi-group). In the case, when T = R

or R the

dynamical system (X, T, π) is called a ﬂow, but if T ⊆ Z, then (X, T, π) is called a

cascade (discrete ﬂow).

Sometimes, brieﬂy, we will write xt instead of π(t, x).

Below X will be a complete metric space with metric ρ.

Deﬁnition 1.2 The function π(·, x) : T → X is called a motion passing through

the point x at the moment t = 0 and the set Σ

:= π(T, x) is called a trajectory of

this motion.

Deﬁnition 1.3 A nonempty set M ⊆ X is called positively invariant (nega-

tively invariant, invariant) with respect to dynamical system (X, T, π) or, sim-

ple, positively invariant (negatively invariant, invariant), if π(t, M) ⊆ M (M ⊇

π(t, M), π(t, M) = M) for every t ∈ T.

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2 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Deﬁnition 1.4 A closed positively invariant set, which does not contain own

closed positively invariant subset, is called minimal.

It easy to see that every positively invariant minimal set is invariant.

Deﬁnition 1.5 A closed positively invariant (invariant) set is called indecompos-

able, if it can not be represented in the form of union of two nonempty disjoint

positively invariant (invariant) subsets.

Deﬁnition 1.6 Let M ⊆ X. The set

ω(M) :=

t≥0

[

τ≥t

π(τ, M)

is called ω-limit for M. If T = S, then the set

α(M) :=

t≤0

[

τ≤t

π(τ, M)

is called α-limit for M.

Denote by Σ(M ) := π(S, M) (Σ

(M) := π(S

, M)) and H(M) :=

π(S, M)

( H

(M) :=

π(S

, M) ). If M = {x}, then we put ω

:= ω({x}), α

α({x}), Σ

:= Σ

({x}), H

(x) := H

({x}) and H(x) := H({x}).

Let (X, T, π) be a dynamical system.

Deﬁnition 1.7 The point x ∈ X is called a τ-periodic (τ > 0, τ ∈ T) point, if

xt = x (xτ = x respectively) for all t ∈ T, where xt := π(t, x).

Deﬁnition 1.8 The number τ ∈ T is called ε > 0 shift (almost period) if

ρ(xτ, x) < ε (respectively ρ(x(τ + t), xt) < ε for all t ∈ T).

Deﬁnition 1.9 The point x ∈ X is called almost recurrent (almost periodic) if

for any ε > 0 there exists a positive number l such that on any segment of length

l, will be found an ε shift (almost period) of point x ∈ X.

Deﬁnition 1.10 If a point x ∈ X is almost recurrent and the set H(x) =

{xt | t ∈ T} is compact, then x is called recurrent.

Deﬁnition 1.11 The point x ∈ X is called stable in the sense of Poisson in the

positive direction if x ∈ ω

. If the dynamical system (X, T, π) is a two-sided one

and x ∈ α

, then the point x is called stable in the sense of Poisson in the negative

direction. If the point x is stable in the sense of Poisson in both directions, it is

called Poisson stable.

Deﬁnition 1.12 The continuous mapping ϕ : S → X with the properties:

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Autonomous dynamical systems 3

(1) ϕ(0) = x;

(2) π

ϕ(s) = ϕ(t + s) for all t ∈ T and s ∈ S

is called a whole trajectory of one-sided dynamical system (X, T

, π).

Denote by Φ

the family of all whole trajectory of dynamical system (X, T

, π)

passing through the point x ∈ X at the moment t = 0.

Deﬁnition 1.13 The motion π(·, x) : T

→ X of one-sided dynamical system

(X, T

, π) is called extendable on S, if there exists a whole trajectory ϕ passing

through point x.

We denote by α

:= {y : ∃t

→ −∞, t

∈ S and ϕ(t

) → y}, where ϕ is an

extension on S the motion π(·, x).

Deﬁnition 1.14 The set W

(Λ) (W

(Λ)), deﬁned by equality

(Λ) := {x ∈ X| lim

t→+∞

ρ(xt, Λ) = 0}

(Λ) := {x ∈ X| lim

t→−∞

ρ(xt, Λ) = 0}),

is called a stable (unstable) manifold of set Λ ⊆ X.

Deﬁnition 1.15 The set M is called:

− orbital stable, if for every ε > 0 there exists δ = δ(ε) > 0 such that ρ(x, M) < δ

implies ρ(xt, M) < ε for all t ≥ 0;

− attracting, if there exists γ > 0 such that B(M, γ) ⊂ W

(M);

− asymptotic stable, if it is orbital stable and attracting;

− global asymptotic stable, if it is asymptotic stable and W

(M) = X;

− uniform attracting, if there exists γ > 0 such that

lim

t→+∞

sup

x∈B(M,γ)

ρ(xt, M) = 0.

Theorem 1.1

[

305

]

Let M ⊆ X be a nonempty compact invariant set, then every

motion in M is extendable on S.

Theorem 1.2

[

305

]

The compact set M ⊆ X is negative invariant, if and only if

for every point x ∈ M there exists a whole trajectory ϕ passing through point x ∈ M

such that ϕ(t) ∈ M for all t ≤ 0.

Theorem 1.3

[

305

]

Union and closure of positively invariant (negatively invari-

ant, invariant) sets is positively invariant (negatively invariant, invariant).

Theorem 1.4

[

305

]

The following statements are true:

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4 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(1) ∀ x ∈ X the set ω

is positively invariant;

(2) ∀ x ∈ X and ϕ ∈ Φ

the set α

is positively invariant.

2. Below we indicate one general method of construction of dynamical systems

on the space of continuous functions. In this way we will get many well known

dynamical systems on the functional spaces (see, for example,

[

32, 300

]

Let (X, T, π) be a dynamical system on X, Y be a complete pseudo metric

space and P be a family of pseudo metric on Y . Denote by C(X, Y ) the family

of all continuous functions f : X → Y equipped with the compact-open topology.

This topology is given by the following family of pseudo metric {d

} (p ∈ P, K ∈

C(X)), where

(f, g) := sup

x∈K

p(f(x), g(x))

and C(X) a family of all compact subsets of X. We deﬁne for all τ ∈ T the mapping

: C(X, Y ) → C(X, Y ) by the following equality: (σ

f)(x) := f(π(τ, x)) (x ∈ X).

We note that the family of mappings{σ

: τ ∈ T} possesses the following properties:

a. σ

= id

C(X,Y )

;

b. ∀τ

, τ

∈ T σ

◦σ

= σ

+τ

;

c. ∀τ ∈ T σ

is continuous.

Lemma 1.1 The mapping σ : T × C(X, Y ) → C(X, Y ), deﬁned by the equality

σ(τ, f) := σ

f (f ∈ C(X, Y ), τ ∈ T), is continuous.

Proof. Let f ∈ C(X, Y ), τ ∈ T and {f

}, {τ

} are arbitrary directions which

converge to f and τ respectively. Then

(σ(τ

, f

), σ(τ, f)) = sup

x∈K

p(σ(τ

, f

)(x), σ(τ, f)(x)) =

sup

x∈K

p((f

(π(τ

, x)), f(π(τ, x))) : x ∈ K} ≤

sup

x∈K

p(f

(π(τ

, x)), f(π(τ

, x))) + sup

x∈K

p(f(π(τ

, x)), f(π(τ, x))) ≤

sup

x∈K, s∈Q

p(f

(π(s, x)), f(π(s, x))) + sup

x∈K

p(f(π(τ

, x)), f(π(τ, x)))

≤ sup

m∈π(Q,K)

p(f

(m), f(m)) + sup

x∈K

p(f(π(τ

, x)), f(π(τ, x))) =

π(Q,K)

, f) + sup

x∈K

p(f(π(τ

, x)), f(π(τ, x))), (1.3)

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Autonomous dynamical systems 5

where Q =

{τ

}. Passing to limit in the inequality (1.3) we obtain the necessary

statement. 

Corollary 1.1 The triple (C(X, Y ), T, σ) is a dynamical system on C(X, Y ).

Remark 1.1 Lemma 1.1 is true, if X is an arbitrary Hausdorﬀ topological space.

Consider now same examples of dynamical systems of form (C(X, Y ), T, σ) useful

in the applications.

Example 1.1 Let X = T and we denote by (X, T, π) a dynamical system

on T, where π(t, x) := x + t. Dynamical system (C(T, Y ), T, σ) is called Bebu-

tov’s dynamical system

[

300

]

(dynamical system of translations or shifts dynamical

system).

Deﬁnition 1.16 We will say that the function ϕ ∈ C(T, Y ) possesses the prop-

erty A, if with this property possesses the motion σ(·, ϕ) : T → C(T, Y ) in the

dynamical system of Bebutov (C(T, Y ), T, σ), generated by the function ϕ. In qual-

ity of property A may be periodicity, almost periodicity, recurrence, asymptotic

almost periodicity etc.

Example 1.2 Let X := T × W , where W some metric space and by (X, T, π)

we denote a dynamical system on X deﬁned in the following way: π(t, (s, w)) :=

(s + t, w). Using the general method proposed above we can deﬁne on C(T ×W, Y )

a dynamical system of translations (C(T × W, Y ), T, σ).

The function f ∈ C(T × W, Y ) is called almost periodic (recurrent, asymptotic

almost periodic etc) with respect to t ∈ T uniform on w on every compacts from

W , if the motion σ(·, f) is almost periodic (recurrent, asymptotic almost periodic,

etc.) in the dynamical system (C(T × W, Y ), T, σ).

Example 1.3 Let W := C

, Y := C

and A(T × C

, C

) be a family of all

functions f ∈ C(T × C

, C

) which are holomorphic with respect to second vari-

able. It easy to verify that the set A(T ×C

, C

) is closed and invariant subset of

dynamical system (C(T ×C

, C

), T, σ) and, consequently, on A(T ×C

, C

) it is

induced a dynamical system of translations (A(T ×C

, C

), T, σ).

Example 1.4 Let W and X be complete pseudo metric spaces

[

213

]

. Denote by

P(T × W, X) a set of all continuous functions f : T × W → X which are bounded

on every bounded subset from T × W and continuous w.r.t. t ∈ T uniformly on

w on bounded subsets W equipped with the topology of uniform convergence on

bounded subsets from T×W . This topology can by generate by the following family

of pseudo metrics {d

} (p ∈ P, B ∈ B(T ×W )), where

(f, g) = sup

(t,w)∈B

p(f(t, w), g(t, w)), (1.4)

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6 Global Attractors of Non-autonomous Dissipative Dynamical Systems

B(T ×W ) is a family of all bounded subsets of T × W and P is a family of pseudo

metric on X. We note that the set P(T × W, X) endowed with the family of

pseudo metric {d

|p ∈ P, B ∈ B(T × W )} become a complete pseudo metric

space, invariant with respect to translations on t ∈ T. For each τ ∈ T we denote

by f

the translation of function f ∈ P(T × W, X) on τ w.r.t. variable t ∈ T,

i.e. f

(t, w) = f(t + τ, w) ( (t, w) ∈ T × W ). Now we deﬁne a mapping σ :

T×P(T×W, X) → P(T×W, X) as following: σ(τ, f ) := f

for all f ∈ P(T×W, X)

and τ ∈ T. It is easy to see that σ(0, f) = f and σ(τ

, σ(τ

, f)) = σ(τ

+ τ

, f) for

all f ∈ P(T × W, X) and τ

, τ

∈ T. Using the same reasoning as in the lemma 1.1

it is possible to verify the mapping σ is continuous and, consequently, the triplet

(P(T × W, X), T, σ) is a dynamical system on P(T × W, X).

Remark 1.2 If the function f ∈ C(T × W, X) is uniform continuous on every

bounded subset from T×W , then f ∈ P(T×W, X). Denote by U (T×W, X) the set of

all functions f ∈ C(T ×W, X) which are bounded and uniform continuous on every

bounded subset from T×W. Then the set U(T×W, X) is invariant w.r.t. translations

on t ∈ T and it is closed in P(T × W, X) and, consequently, on U(T × W, X) it is

induced a dynamical system of translations (U(T ×W, X), T, σ) (see, for example,

[

32, 300

]

and

[

302

]

Deﬁnition 1.17 Let (X, T

, π) and (Y, T

, σ) (S

⊆ T

⊆ S) be

two dynamical systems. The mapping h : X → Y is called a homomorphism

(respectively isomorphism) of dynamical system (X, T

, π) on (Y, T

, σ), if the

mapping h is continuous (respectively homeomorphic) and h(π(x, t)) = σ(h(x), t)

( t ∈ T

, x ∈ X). In this case a dynamical system (X, T

, π) is an extension of

dynamical system (Y, T

, σ) by homomorphism h, but a dynamical system (Y, T

, σ)

is called a factor of dynamical system (X, T

, π) by homomorphism h. Dynamical

system (Y, T

, σ) is called also a base of extension (X, T

, π).

Deﬁnition 1.18 The triplet h(X, T

, π), (Y, T

, σ), hi, where h is a homo-

morphism from (X, T

, π) on (Y, T

, σ) and (X, h, Y ) is a local-trivial ﬁbering

[

190

]

is called a non-autonomous dynamical system.

Remark 1.3 In the latter years in the works of I.U.Bronsteyn and his col-

laborators (see, for example,

[

32, 33, 34, 158

]

) an extension is called a triplet

h(X, T, π), (Y, T, h), hi, i.e. the object which we name here a non-autonomous

dynamical system.

Deﬁnition 1.19 The triplet hW, ϕ, (Y, T

, σ)i (or shortly ϕ), where (Y, T

, σ) is

a dynamical system on Y , W is a complete metric space and ϕ is a continuous

mapping from T

×W ×Y in W , possessing the following conditions:

a. ϕ(0, u, y) = u (u ∈ W, y ∈ Y );