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

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Analytic dissipative systems 127

Corollary 3.3 Suppose that for certain y

∈ Y H

) = Y is compact and ω

is minimal. If the conditions 2. and 3. of Theorem 3.5 are fulﬁlled, then the system

h(X, T

, π), (Y, T

, σ), hi is convergent.

Proof. This assertion follows from Theorem 3.4. In fact, under the conditions of

the Corollary 3.3 the system (Y, T

, σ) is compact dissipative and J

= ω

. 

Corollary 3.4 Let y

be asymptotic recurrent (i.e. there is a recurrent point

∈ Y such that lim

t→+∞

ρ(y

t, q

t) = 0). If Y := H

) and the conditions of

Theorem 3.5 are fulﬁlled, then the system h(X, T

, π), (Y, T

, σ), hi is convergent

and each point x ∈ X is asymptotic recurrent.

Proof. The ﬁrst statement of Corollary follows from Theorem 3.5 and Corollary

3.3. Let now x ∈ X

and y ∈ Y = H

). Since the point y

is asymptotic

recurrent, then the point y is so asymptotic recurrent, i.e. there is a recurrent

point q ∈ ω

such that ρ(yt, qt) → 0 as t → +∞. Taking into account that

h(X, T

, π), (Y, T

, σ), hi is convergent, then J

∩X

consists from the single point

p. Since in our conditions ρ(xt, pt) → 0 as t → +∞, then the point x is asymptotic

recurrent. 

Consider the equation

˙z = f(t, z) + r(t, z) (z ∈ C). (3.13)

It follows directly from Theorem 3.5 and Corollary 3.4 the following theorem.

Theorem 3.6 Let f, r ∈ C(R ×C

, C

) and the following conditions are fulﬁlled:

(1) equation (3.13) is dissipative;

(2) lim

t→+∞

|r(t, z)| = 0 uniformly in z on compacts from C

;

(3) f is holomorphic in z ∈ C

and recurrent (almost periodic, periodic) in t ∈ R

uniformly in z on the compacts from C

Then the equation

˙z = f (t, z)

has a unique bounded on R solution p ∈ C(R, C

), which is recurrent (almost peri-

odic, periodic) in t ∈ R and lim

t→+∞

|ϕ(t) − p(t)| = 0 for all solution ϕ of (3.13).

In the conclusion of this section we will give the following

Example 3.5 Consider the system of diﬀerential equations

(

˙z

= −2z

+ z

˙z

= −z

, z

∈ C). (3.14)

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128 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Dynamical system deﬁned by this system is C-analytic and dissipative. In ad-

dition, from

[

7, p.167-174

]

follows that the system (3.14) can not linearized by

biholomorphic transformation.

Thus, there are the nonlinear C-analytic dissipative systems, which can not

linearized by biholomorphic transformation.

Remark 3.2 For real analytic system the assertion analogous to example 3.5 does

not hold, as the example below shows.

Example 3.6 Consider the system of the diﬀerential equations

(

˙x = (1 −x

−y

´y = (1 − x

−y

(3.15)

It is possible to check by directly integration that the system (3.15) is dissipative

and its Levinson’s centre J = {(x, y) : x

+ y

≤ 1} contains a continuum points.

Thus , the analogous of Theorem 3.3 for real analytic system is not true.

3.3 Converse of Lyapunov’s theorem for C-analytic systems

Let R (C) be the set of all real (complex) numbers and C

be n-dimensional complex

space with norm kzk := max{|z

| : i = 1, ..., n}. Denote by A(R × W, C

) the

space of all continuous functions f : R × W → C

which are holomorphic in the

second variable, where W is a domain in C

containing 0 ∈ C

We consider the diﬀerential equation

= f(t, z), (3.16)

where f ∈ A(R × C

, C

) and f(t, 0) = 0. along with (3.16) we consider the

variational equation for the zero solution

= A(t)u, (3.17)

in which A(t) :=

∂f

∂t

(t, 0). We denote by z(t; t

, v) the solution of (3.16) passing

through v ∈ C

for t = t

Deﬁnition 3.4 We recall (see, for example,

[

273

]

) that the solution z = 0 of

(3.16) is said to be stable if for any ε > 0 and t

∈ R there exists a δ = δ(ε, t

) > 0

such that for all z ∈ B(0, δ) = {w| kwk < δ} and t ≥ t

one has kz(t; t

, v)k ≤ ε.

If one can choose δ(ε, t

) independent of t

∈ R, then the solution z = 0 is called

uniformly stable.

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Analytic dissipative systems 129

Deﬁnition 3.5 The solution z = 0 of (3.16) is said to be attracting if for any

∈ R one can ﬁnd an η = η(t

) > 0 and for any ε > 0 and kvk < η one can ﬁnd a

σ = σ(t

, ε, v) > 0 such that kz(t; t

, v)k < ε for all t ≥ t

+ σ. If one can choose η

independent of t

and σ, but dependent only on ε, then the solution z = 0 is called

uniformly attracting.

Deﬁnition 3.6 The solution z = 0 is said to be asymptotically stable (uniformly

asymptotically stable) if it is stable (uniformly stable) and is attracting (uniformly

attracting).

Theorem 3.7 The following assertions hold:

(1) if the zero solution of (3.16) is stable (uniformly stable), then the zero solution

of (3.17) is also stable (uniformly stable);

(2) if the zero solution of (3.16) is attracting (uniformly attracting), then so is the

zero solution of (3.17).

Proof. Let the solution z = 0 of (3.16) be stable (uniformly stable), ε > 0, t

∈ R

and δ = δ(ε, t

) > 0 (δ = δ(ε) > 0) be a positive number from the stability (uniform

stability) condition. By the theorem on analytic dependency on the initial data

[

122

]

the function z(t; t

, v) is holomorphic in v for ﬁxed t and t

in the domain of the

deﬁnition of z(t; t

, v). We compute the linear part of the Taylor expansion at the

point v = 0 of the function z(t; t

, v). According to the variation of parameters

formula we have

z(t; t

, v) = U(t, t

)v +

U(t, τ )F (τ, z(τ ; t

, v))dτ,

where U (t, τ) := U(t, A)U

−1

(τ, A), U(t, A) being the Cauchy operator of (3.17) and

F (t, z) := f(t, z) −A(t)z.

We note that z(t; t

, v) = v +

f(τ, z(τ; t

, v))dτ and hence

∂z

∂v

(t; t

, v)



v=0

= E +

∂f

∂z

(τ, z(τ; t

, v))



v=0

∂z

∂v

(τ; t

, v)



v=0

dτ,

where E is the identity matrix. Thus, V (t, t

) = U(t, t

) satisﬁes the matrix

diﬀerential equation

∂V

∂t

= A(t)V and the initial condition V (t

, t

) = E so

V (t, t

) = U(t, t

). Since

∂

∂v





U(t, τ )F (τ, z(τ ; t

, v))dτ





v=0

U(t, τ )

∂F

∂z

(τ, 0) · U(τ, t

)dτ = 0,

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130 Global Attractors of Non-autonomous Dissipative Dynamical Systems

the linear part of the Taylor expansion in v of z(t; t

, v) for any t

∈ R and t ≥ t

is U(t; t

)v. On the other hand, this linear part can be computed from the Cauchy

integral formula

[

207

]

. Since in C

the l

∞

norm is chosen, the domain B(0, δ) is an

open polydisk and one can write

U(t, t

)w =

(2πi)

·. . . ·

z(t; t

, v)

· v

·. . . ·v

k=1

·. . . ·dv

. (3.18)

The stability (uniform stability) of the zero solution of (3.17) follows directly from

(3.18) if the zero solution of (3.16) is stable (uniformly stable).

If the zero solution of (3.16) is attracting (uniformly attracting), then passing to

the limit in (3.18) and considering Lebesgue’s theorem on passage to the limit under

the integral sign as t → +∞ we get the analogous property for the zero solution of

(3.17). The theorem is proved. 

Corollary 3.5 If the zero solution of (3.16) is asymptotically stable (uniform

asymptotically stable), then so is the zero solution of (3.17).

Remark 3.3 1. If (3.16) is autonomous, Theorem 3.7 is a result of Lyubich

[

241

]

2. For a real analytic system the assertion analogous to Theorem 3.7, does not

hold, as the example ˙y = −y

shows.

3. Theorem 3.7 is true also for the diﬀerence equations.

Along with equation (3.16) we consider its H-class

= g(t, z) (g ∈ H(f)). (3.19)

Let Θ := {(0, g)|g ∈ H(f )} ⊆ C

× H(f ), W

(Θ) := {(v, g)|(v, g) ∈ C

H(f), lim

t→+∞

kϕ(t, v, g)k = 0} and h(X, R

, π), (Y, R, σ), hi be a non-autonomous

dynamical system generated by the equation (3.16) (see the example 3.1). If the

zero solution of (3.16) is uniform asymptotically stable and the set H(f) is compact,

then the compact invariant set Θ ⊆ X = C

× H(f) is asymptotically stable and

its domain of attraction W

(Θ) is open. The following theorem holds.

Theorem 3.8 Let f ∈ A(R × C

, C

), H(f) be compact and the zero solution of

(3.16) be uniform asymptotically stable, then the set W

(Θ) is unbounded.

Proof. Let as assume that W

(Θ) is bounded, then the set M =

(Θ) ⊆ X is

compact invariant subset (X, R

, π). In virtue of Theorem 3.7 the zero solution of

(3.17) is also uniform asymptotically stable and hence one can ﬁnd positive numbers

N and ν such that

kU(t, t

)k ≤ Ne

−ν(t−t

)

(3.20)

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Analytic dissipative systems 131

for all t ≥ t

(t, t

∈ R). It follows from (3.20) that

kU(t, A)k ≥ N

−1

−νt

(3.21)

for all t ≤ 0.

On the other hand, it follows from (3.18) and the boundedness of W

(Θ) that

sup

t∈R

kU(t, A)k < +∞,

which contradicts (3.21). The theorem is proved. 

Corollary 3.6 Let f ∈ A(R × C

, C

), H(f ) be compact and the zero solu-

tion of (3.16) be uniform asymptotically stable. Then the set W

(0) = {v | v ∈

, lim

t→+∞

kϕ(t, v, f)k = 0} is unbounded.

Let Ω be a metric space and (Ω, R, σ) be a dynamical system on Ω. Consider

the diﬀerential equation

˙x = f(ωt, x) (3.22)

where ωt := σ(t, ω), f ∈ A(Ω × C

, C

), x ∈ C

and f(ω, 0) = 0 for all ω ∈ Ω. The

results presented above are true also for the equation (3.22). We will formulated

some of them.

Theorem 3.9 If the zero solution of (3.22) is uniform asymptotically stable, then

the zero solution of

˙y = A(ωt)y, (3.23)

where A(ω) :=

∂f(ω, x)

∂x



x=0

, is also uniform asymptotically stable.

Theorem 3.10 Let the zero solution of (3.22) be uniform asymptotically stable

and Ω be compact. Then the set

(Θ) = {(x, ω) ∈ C

×Ω | lim

t→+∞

|ϕ(t, x, ω)| = 0}

is unbounded, where ϕ(t, x, ω) is the solution of (3.22) passing through the point

x ∈ C

for t = 0 and deﬁned on R

Proof. These statements can be proved as Theorem 3.7 and 3.8. 

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132 Global Attractors of Non-autonomous Dissipative Dynamical Systems

3.4 On the structure of compact attracting sets of C-analytic systems

The present section is devoted to the structure of compact invariant sets of the

system

= f (t, z) (3.24)

with right-hand side f holomorphic with respect to z and almost periodic with re-

spect to t. We show that an asymptotically stable compact invariant set of system

(3.24) consists of almost periodic solutions to this system. This problem is studied

and solved in the frame of general non-autonomous dynamical systems. We give

also the description of the structure of uniform asymptotically stable compact in-

variant set of arbitrary C–analytic non-autonomous dynamical system with compact

minimal base.

We recall that the set K ⊆ X is called orbitally stable with respect to non-

autonomous system (2.1), if for any ε > 0 there exist δ(ε) > 0 such that the

inequality ρ(x, K

) < δ (x ∈ X, y = h(x)) implies ρ(xt, K

) < ε for all t ≥ 0.

In addition, if there exist γ > 0 such that ρ(xt, K

) → 0 for t → +∞ and every

x ∈ K

such that ρ(x, K

) ≤ γ then it is said that K orbitally asymptotically stable.

Theorem 3.11 Assume that h(X, S

, π), (Y, S, σ), hi is a non-autonomous C-

analytic dynamical system and the following conditions are satisﬁed:

(1) Y is compact and locally connected and (Y, S, σ) is minimal, i.e., Y = H(y) =

{yt |t ∈ S} for all y ∈ Y ;

(2) M ⊂ X is nonempty, compact, orbitally asymptotically stable, and invariant

with respect to h(X, S

, π), (Y, S, σ), hi;

(3) M orbitally asymptotically stable.

Then h(M, S, π), (Y, S, σ), hi is a distal covering of ﬁnite multiplicity, i.e.,

(a) for any y ∈ Y the set M

= {m |m ∈ M, h(m) = y} consists of ﬁnitely many

points;

(b) inf

t∈S

ρ(x

t, x

t) > 0 for all x

, x

∈ M (x

6= x

, h(x

) = h(x

)).

Proof. Since the space Y is locally connected and the bundle (X, h, Y ) is locally

trivial, the space X is locally connected as well. By Proposition 1.18

[

]

, the set

M has ﬁnitely many connected components M

, M

, . . . , M

. We write

(M) := {x |x ∈ X

= h

−1

(y), lim

t→+∞

ρ(xt, M

) = 0}

and note that W

(M) is open in X

. By theorem 1.36, for any compact set K ⊂

(M) := ∪{W

(M) |y ∈ Y }, the set Σ

(K) := ∪{π

K |t ≥ 0} is relatively

compact.

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Analytic dissipative systems 133

We set K

= W

) ∩ K and note that K

(i ∈

1, p) are closed, K

∩ K

= ∅,

if i 6= j, and K = K

∪ K

∪ . . . K

. Let γ > 0 (γ < α = min

i6=j

ρ(K

, K

)) and

l > 0 such that we have B(M, γ), B(K, γ) ⊂ W

(M) and |xt| ≤ l for all t ≥ 0

and x ∈

B(K, γ). By the Cauchy integral formula

[

167

]

there exists a number

L = L(l, γ) > 0 such that

ρ(π

, π

) ≤ Lρ(x

, x

) (3.25)

for all t ≥ 0 and x

, x

∈ K

(i ∈ {1, 2, . . . , p}, h(x

) = h(x

)).

Let us now show that for any ε > 0 and any compact set K ⊂ W

(M), there

exists δ(ε, K) > 0 such that ρ(x

, x

) < δ (x

, x

∈ K, h(x

) = h(x

)) the relation

implies

ρ(x

t, x

t) < ε (3.26)

for all t ≥ 0. If we suppose the contrary, then there exist ε

, δ

↓ 0, K

⊂

(M), {x

} ⊂ K

(i = 1, 2; h(x

) = h(x

)) and t

→ +∞ such that

ρ(x

, x

) < δ

and ρ(x

, x

) ≥ ε

. (3.27)

Since K

is compact we can assume that the sequences {x

} (i = 1, 2) are conver-

gent. Let x

:= lim

n→+∞

= lim

n→+∞

and x

∈ K

:= K ∩ X

. Hence W

(M) =

i=1

), there exists i

∈ {1, 2, . . . , p} such that x

∈ W

). Since

) is open set in X

, there exists δ

> 0 such that B

, δ

) := {x |x ∈

, ρ(x, x

) < δ

} ⊂ W

). For suﬃciently large n we have ρ(x

, x

) <

Without loss of generality we can assume that {x

} ⊂ K

(i = 1, 2); then by (3.25)

we have the estimate

ρ(x

, x

) ≤ Lρ(x

, x

) <

which contradicts (3.27). The obtained contradiction completes the proof of our

statement.

Now we will prove the ﬁrst statement of the theorem. Note that it follows from

inequality (3.25) implies that an invariant set M is distal in the negative direction

with respect to h. Now, since Y is minimal, it follows from Lemma 2.8

[

]

(see

also Lemma 1

[

238, p.104

]

) that M is two-sided distal.

Let α > 0, y ∈ Y . We set E

{π

(M)

|t ≥ α}, where the bar denotes

the closure in the compact-open topology, and P

:= {ξ |ξ ∈ E

, ξ(W

(M)) ⊆

(M)}. As in Lemma 2.7, we can establish that P

is a nonempty compact

topological semigroup. By Lemma 4.11

[

]

, there exists an idempotent u ∈ P

i.e., u

= u. Since u ∈ P

⊆ E

and Y is minimal, there exists t

→ +∞

such that u = lim

n→+∞

(M)

and hence u(W

(M)) ⊆ M

. Indeed, we have

= u(W

(M)), because M is two-sided distal and the idempotent u actcs on M

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134 Global Attractors of Non-autonomous Dissipative Dynamical Systems

as the identity mapping. Now let x ∈ W

(M) and p = u(x) ∈ M

. Let us show

that

lim

t→+∞

ρ(xt, pt) = 0. (3.28)

Indeed, since u(p) = u(u(x)) = u(x), such that ρ(xt

, pt

) → 0 as n → +∞. Let

ε > 0, K = H

(x) =

{xt | t ≥ 0} and δ = δ(ε, K) > 0 be such that (3.26) holds for

ρ(x

, x

) < δ (x

, , x

∈ K and h(x

) = h(x

)). Choose n

such that ρ(xt

, pt

) < δ

for all n ≥ n

; then we have

ρ(x(t

+ t), p(t

+ t)) < ε

for all t ≥ 0 and thus relation (3.28) holds.

Consider the open covering {B(p, γ) | p ∈ M

} of the set M

. Since M

is com-

pact, this covering contains a ﬁnite sub-covering {B(p

, γ)| i =

1, m}. Note that

u(B(p

, γ)) ⊆ M

and p

∈ u(B(p

, γ)). Since M

⊆ ∪{B(p

, γ) | i ∈

1, m}, we have

= u(M

) ⊆ ∪{u(B(p

, γ))|i ∈

1, m} ⊆ M

, and therefore ∪{u(B(p

, γ))|i =

1, m} = M

. Since u is holomorphic and u(x) = x for all x ∈ M

, the set

is analytic. Since M

is compact, it consists of ﬁnitely many points

[

167

]

Let M

= {x

, x

, . . . , x

m(y)

}. Consider the non-autonomous dynamical system

h(M, S, π), (Y, S, σ), hi. As proved above, this system is two-sided distal. By Propo-

sition 4

[

238

]

, the mapping H : Y → 2

(H(y) := M

) is continuous in the Haus-

dorﬀ metric, and therefore cardM

= n(y) does not depend on y ∈ Y . Thus, the

theorem is completely proved. 

Corollary 3.7 Assume that the conditions of Theorem 3.11 are satisﬁed and that

Y consists of recurrent (Bohr almost periodic, quasi-periodic, periodic, stationary)

motions. Then the set M consists of recurrent (Bohr almost periodic, quasi-periodic,

periodic, stationary respectively) motions.

Proof. This assertion follows from Theorem 3.11, Theorem 3

[

238

]

and Lemma

[

238

]

. 

Theorem 3.12 Under the conditions of Theorem 3.11 the following assertions

hold:

(1) the set M is uniform stable in the sense of Lyapunov, i.e. ∀ε > 0 ∃ δ(ε) > 0

such that ρ(x, p

) < δ (x ∈ X

, p ∈ M

) implies ρ(xt, pt) < ε for all t ≥ 0;

(2) for each x ∈ W

(M) there is p ∈ M

(y := h(x)) such that

lim

t→+∞

ρ(xt, pt) = 0.

In particular, if Y consists from the recurrent (almost periodic, quasi-periodic, pe-

riodic, stationary) points, then the set W

(M) consists of asymptotically recurrent

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Analytic dissipative systems 135

(asymptotically almost periodic, asymptotically quasi-periodic, asymptotically peri-

odic, asymptotically stationary) points.

Proof. The ﬁrst statement of Theorem follows directly from (3.26). The second

assertion follows from Theorem 3.11 and Lemma 4

[

]

. 

Corollary 3.8 Let (C

, S, π) be an autonomous C-analytic dynamical system,

M ⊂ C be a nonempty compact invariant orbitally asymptotically stable set, then

M consists of ﬁnitely many stationary points {x

, x

, . . . , x

Let f ∈ A(R × C

, C

). Consider the diﬀerential equation

= f(t, z). (3.29)

Along with equation (3.29), we consider its H-class

= g(t, z) (g ∈ H(f )). (3.30)

Deﬁnition 3.7 The set M ⊆ C

is said to be invariant for equation (3.29), if for

any point v ∈ M there exists g ∈ H(f ), such that the solution ϕ(t, v, g) to equation

(3.30) is deﬁned on the entire R and ϕ(t, v, g) ∈ M for all t ∈ R.

Deﬁnition 3.8 A compact set M ⊂ C

invariant with respect to (3.29) is said

to be orbitally stable if for any ε > 0, there exists δ(ε) > 0 such that the relation

ρ(v, M

) < δ implies ρ(ϕ(t, v, g), M

) < ε for all t ≥ 0, where M

= {x | x ∈

M, ϕ(t, x, g) ∈ M for all t ∈ R} and g

- is a translation of g in ﬁrst variable on t.

We write W

(M) = {v | v ∈ C

, lim

t→+∞

ρ(ϕ(t, v, g), M

) = 0} and W

(M) =

∪{W

(M) |g ∈ H(f)}.

Deﬁnition 3.9 A compact set M, invariant with respect to equation (3.29), is

said to be asymptotically orbitally stable if it is orbitally stable and there exists

γ > 0 such that B(M, γ) = {x | ρ(x, M ) < γ} ⊆ W

(M).

Applying Theorem 3.11 and Corollary 3.7, 3.8 to the constructed non-

autonomous C-analytic system generated by equation (3.29), we obtain the following

statement.

Theorem 3.13 Let f ∈ A(R × C

, C

) be regular and recurrent (Bohr almost

periodic, quasi-periodic, periodic, stationary) with respect to t ∈ R uniformly with

respect to z on compact sets from C

, M ⊆ C

and let the following conditions be

satisﬁed:

(1) M is a compact set invariant with respect to equation (3.29);

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136 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(2) M is orbitally asymptotically stable,

Then for any g ∈ H(f), the set M

consists of ﬁnitely many recurrent (Bohr al-

most periodic, quasi-periodic, periodic, stationary respectively) solutions of equation

(3.30).

Remark 3.4 A statement similar to Theorem 3.13, is valid for diﬀerence equa-

tions as well.

3.5 Dynamical systems in spaces of sections

Let h(X, S

, π), (Y, S, σ), h)i be a non-autonomous dynamical system. We recall

that a mapping γ : Y → X is called a section (selector) of the homomorphism h, if

h(γ(y)) = y for all y ∈ Y . The section γ of the homomorphism h is called invariant

if γ(σ(t, y)) = π(t, γ(y)) for all y ∈ Y and t ∈ S.

Denote by Γ = Γ(Y, X) family of all continuous sections of h, i.e. Γ(Y, X) =

{γ ∈ C(Y, X) : h ◦γ = Id

}. We will suppose that Γ(Y, X) 6= ∅. This condition is

fulﬁlled in the many important case for the applications.

We endow the space Γ = Γ(Y, X) with the uniform-convergence on compacts in

Y .

Let K ⊆ Y be a nonempty compact from Y , then the relation

(γ

, γ

) := max

y∈K

ρ(γ

(y), γ

(y))

deﬁnes a pseudo-metric p

on Γ, and the family of pseudo-metrics {p

: K ∈

C(Y )}, where C(Y ) is the family of nonempty compact subsets of Y , deﬁnes a

uniform structure on Γ (a uniform-convergence topology on compacts from Y ).

We consider the special case in which Y =

∞

k=1

, where {Q

} is an expanding

sequence of nested compacts. In this situation it is suﬃcient to use a countable

number of pseudo-metrics P

= {p

: k = 1, 2 . . .}, where

(γ

, γ

) := max

y∈Q

ρ(γ

(y), γ

(y)).

It can be easily veriﬁed that this family of pseudo-metrics separates the points of

Γ, and p

(γ

, γ

) ≤ p

(γ

, γ

) ≤ . . . p

(γ

, γ

) ≤ . . .. Hence the topology deﬁned by

this family of pseudo-metrics is metrizable. For example the topolgy deﬁned by the

metric

d(γ

, γ

) :=

∞

(γ

, γ

)

1 + p

(γ

, γ

)

is consistent with the topology of Γ deﬁned by the family of pseudo-metrics P