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

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Non-autonomous Dissipative Dynamical Systems 117

Note, that the equality (2.135) implies

lim

t→+∞

sup

|x|≤γ

|xt| = 0,

if we remark that xt = µ

− µ

+ µ

− µ

, where z = h(x), |x| ≤ γ and

x = w

− w

. From the equality (2.136) and Theorem 2.38 it follows the local

dissipativity of the dynamical system h(X, S

, π), (Y, S, σ), hi.

Conversely. Let h(X, S

, π), (Y, S, σ), hi be locally dissipative, then according to

Theorem 2.38 there exit positive numbers N and ν such that the equality (2.130)

holds. Denote by Γ(Z, W ) the space of all continuous sections of the homomorphism

% : Z → W endowed with the distance

d(ϕ, ψ) = max

z∈Z

ρ(ϕ(z), ψ(z)) = max

z∈Z

|ϕ(z) − ψ(z)|,

which converts it in the complete metric space. Deﬁne the mapping S

: Γ(Z, W ) →

Γ(Z, W ) by the equality (S

ϕ)(z) := µ

ϕ(λ

−t

z) (z ∈ Z, ϕ ∈ Γ(Z, W )) for all t ∈ S.

It is easy to check that the family of mappings {S

: t ∈ S

} forms a commutative

group with respect to composition. Note, that

d(S

, S

) = max

z∈Z

|µ

(λ

−t

z) − µ

(λ

−t

z)|

= max

z∈Z

|µ

(z) − µ

(z)| = max

z∈Z

|π

(ϕ

(z) − ϕ

(z))| ≤

−νt

max

z∈Z

|ϕ

(z) − ϕ

(z)|. (2.137)

From the equality (2.137) it follows that

d(S

, S

) ≤ Ne

−νt

max

z∈Z

|ϕ

(z) − ϕ

(z)|

for all ϕ

, ϕ

∈ Γ(Z, W ) and t ≥ 0 and, consequently, the mapping S

is contraction

for the suﬃcient large t. From this it follows that the commutative semi-group {S

t ∈ S

} of mappings has a unique ﬁxed point ϕ ∈ Γ(Z, W ), i.e. (S

ϕ)(z) = ϕ(z)

for all z ∈ Z and t ∈ S

and, consequently, µ

ϕ(z) = ϕ(λ

z) (z ∈ Z, t ∈ S). Let

w ∈ W and z = ρ(w), then

|µ

w − µ

ϕ(z)| = |π

(w − ϕ(z))| ≤ Ne

−νt

|w − ϕ(z)| . (2.138)

From (2.138) it follows that the compactly invariant set J = ϕ(Z) ⊂ W is globally

uniformly asymptotically stable, and in addition, J

= J ∩ W

= {ϕ(z)} for all

z ∈ Z. According to Theorem 2.14, the dynamical system h(W, S, µ), (Z, S, λ), %i

is convergent and, consequently, it is compactly dissipative and J = ϕ(Z) is the

Levinson center of (W, S, µ). To prove that (W, S, µ) is locally dissipative it is

suﬃcient to show that the set J = ϕ(Z) is uniformly contracting. We will show

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 2

118 Global Attractors of Non-autonomous Dissipative Dynamical Systems

that the equality

lim

t→+∞

β(µ

B(J, γ), J) = 0.

holds for all γ > 0. If we suppose that it is not so, then there exit ε

, γ

, w

∈

B(J, γ

) and t

→ +∞ such that

ρ(µ

, J) ≥ ε

. (2.139)

From the inequality (2.138) and the compactness of Z it follows that the sequence

{µ

} can be considered convergent. Put

w = lim

t→+∞

, then w ∈ J. Passing

to limit in (2.139) as n → +∞, we have

w 6∈ J. The obtained contradiction proves

the theorem. 

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 3

Chapter 3

Analytic dissipative systems

3.1 Skew-product dynamical systems and cocycles

Let Y be a compact metric space and W = E

. Consider the cocycle

, ϕ, (Y, T

, σ)i over the dynamical system (Y, T

, σ) with the ﬁber E

and the

non-autonomous dynamical system h(X, T

, π), (Y, T

, σ), hi associated by it. Then

(X, h, Y ), where X := E

× Y , is a ﬁnite-dimensional vectorial ﬁbering (a trivial

ﬁbering with the ﬁber E

). The function V : X ×X → R

deﬁned by the equality

V ((u

, y), (u

, y)) := |u

−u

|, where |·| is a norm on E

deﬁnes on X a Riemannian

metric.

From Theorem 2.23 follows the following statement.

Theorem 3.1 Let Y be a compact and hE

, ϕ, (Y, T

, σ)i be a cocycle over the

dynamical system (Y, T

, σ) with the ﬁber E

. Then the following statements are

equivalent:

1. There exists a positive number R such that

lim sup

t→+∞

|ϕ(t, u, y)| < R

for all u ∈ E

” y ∈ Y .

2. There is a positive number r

such that for all u ∈ E

and y ∈ Y there exists

τ = τ (u, y) > 0 for which |ϕ(τ, u, y)| < r

3. There is a positive number r

such that

lim inf

t→+∞

|ϕ(t, u, y)| < r

for all u ∈ E

and y ∈ Y .

4. There exists a positive number R

and for all R > 0 there is l(R) > 0 such that

|ϕ(t, u, y)| ≤ R

for all t ≥ l(R), u ∈ E

, |u| ≤ R and y ∈ Y .

Taking into consideration Lemma 2.14 we obtain that every condition 1.-4. that

ﬁgures in Theorem 3.1 is equivalent to the dissipativity of the non-autonomous

119

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120 Global Attractors of Non-autonomous Dissipative Dynamical Systems

dynamical system h(X, T

, π), (Y, T

, σ), hi associated by the cocycle hE

, ϕ,

(Y, T

, σ)i over (Y, T

, σ) with the ﬁber E

We will give below an example of a skew-product dynamical system which plays

an important role in the study of non-autonomous diﬀerential equations.

Example 3.1 Let E

be an n-dimensional real or complex Euclidean space. Con-

sider the diﬀerential equation

= f(t, u), (3.1)

where f ∈ C(R × E

, E

). Along with the equation (3.1) we consider its H-class

[

]

[

137

]

[

238

]

[

300

]

[

302

]

, i.e. the family of the equations

= g(t, v), (3.2)

where g ∈ H(f) =

: τ ∈ R} and f

(t, u) = f(t + τ, u), where the bar indicating

closure in the compact-open topology. We will suppose that the function f is

regular, that is for every equation (3.2) the conditions of existence, uniqueness and

extendability on R

are fulﬁlled. Denote by ϕ(·, v, g) the solution of (3.2) passing

through the point v ∈ E

for t = 0. Then the mapping ϕ : R

×E

×H(f) → E

satisﬁes the following conditions (see, for example,

[

]

[

290

]

[

291

]

1) ϕ(0, v, g) = v for all v ∈ E

and g ∈ H(f );

2) ϕ(t, ϕ(τ, v, g), g

) = ϕ(t + τ, v, g) for each v ∈ E

, g ∈ H(f ) and t, τ ∈ R

;

3) ϕ : R

×E

×H(f) → E

is continuous.

Denote by Y := H(f) and (Y, R

, σ) a dynamical system of translations on

Y , induced by the dynamical system of translations (C(R × E

, E

), R, σ). The

triple hE

, ϕ, (Y, R

, σ)i is a cocycle over (Y, R

, σ) with the ﬁber E

. Hence,

the equation (3.1) generates a cocycle hE

, ϕ, (Y, R

, σ)i and the non-autonomous

dynamical system h(X, R

, π), (Y, R

, σ), hi, where X := E

× Y , π := (ϕ, σ) and

h := pr

: X → Y .

Deﬁnition 3.1 Recall that the equation (3.1) is called dissipative

[

137

]

[

271

]

[

325

]

[

326

]

, if for all t

∈ R and x

∈ E

there exists a unique solution

x(t; x

, t

) of the equation (3.1) passing through the point (x

, t

) and if there exists

a number R > 0 such that lim

t→+∞

sup |x(t; x

, t

)| < R for all x

∈ E

and t

∈ R.

In other words, for every solution x(t; x

, t

) there is an instant t

= t

+ l(t

, x

such that |x(t; x

, t

)| < R for all t ≥ t

. If the number l(t

, x

) can be chosen

independent on t

, then the equation (3.1) is called uniformly dissipative

[

137

]

Below we will establish the relation between the dissipativity of the equation

(3.1) and the dissipativity of the non-autonomous dynamical system generated by

the equation (3.1).

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Analytic dissipative systems 121

Lemma 3.1 Let f ∈ C(R × E

, E

) be regular and H(f) be compact. Then the

following aﬃrmations are equivalent:

(1) Equation (3.1) is uniformly dissipative; that is, there are numbers R > 0 and

l(x

) > 0 such that

|x(t; x

, t

)| < R (t ≥ t

+ l(x

), x

∈ E

). (3.3)

(2) There is a positive number r such that

lim sup

t→+∞

|ϕ(t, x

, g)| < r (x

∈ E

, g ∈ H(f)). (3.4)

Proof. Let x

∈ E

and g ∈ H(f ). If g = f

for some τ ∈ R. Then

ϕ(t, x

, f

) = x(t + τ, x

, τ) (3.5)

and the inequality (3.4) follows from (3.3). For this aim it is suﬃcient to take r > R.

Consider the case when g = lim

k→+∞

and τ

→ +∞ (or −∞). By Theorem 3.2

from

[

186

]

follows that the sequence {ϕ(t, x

, f

)} converges to a certain solution

of (3.2) and this solution passes through the point (x

, 0) ∈ E

×R. Note, that the

solution passing through the point (x

, 0) is unique because the function f is regular.

Thus, ϕ(t, x

, g) = lim

k→+∞

ϕ(t, x

, f

) uniformly with respect to t on every compact

from R

. From (3.3) and (3.5) follows that |ϕ(t, x

, g)| = lim

k→+∞

|ϕ(t, x

, f

)| ≤ R

(t ≥ l(x

)) and, consequently, the inequality (3.4) holds.

Conversely. Suppose that the inequality (3.4) holds. Then the non-autonomous

dynamical system generated by the equation (3.1) is dissipative and by Theorem

3.1 (the condition 4.) there is R

> 0 such that for all x

∈ E

there exists

l(x

) = l(|x

|) > 0 for which

|ϕ(t, x

, g) ≤ R

(3.6)

for all t ≥ l(x

) and g ∈ H(f). Note that

x(t; x

, t

) = ϕ(t − t

, x

, f

), (3.7)

and taking into account (3.6) we obtain (3.3). For this aim it is suﬃcient to take

R > R

. The lemma is proved. 

Thus, for the equation (3.1) (f is regular and H(f) is compact) we established

that it is uniformly dissipative if and only if the non-autonomous dynamical system

generated by this equation is dissipative.

We note that the notions of the dissipativity and uniform dissipativity for the

equation (3.1) with periodic right hand side are equivalent, but for the equations

with almost periodic right hand side they are diﬀerent. This assertion can be

illustrated by the following example.

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122 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Example 3.2 Consider the linear scalar equation

= a(t)x, (3.8)

where the function a ∈ C(R, R) is almost periodic such that all the solutions of

the equation (3.8) tend to 0 as t → +∞, but the trivial solution of (3.8) is not

uniformly stable. The equations (3.8) with the indicated above properties exist

(see, for example

[

123

]

and also the example 13.2). It is clear that along with the

equation (3.8) all translations of the equation (3.8), that is

= a

(t)x,

where a

(t) = a(t + τ) (t, τ ∈ R), possess the same property. Therefore, from

the relation (3.7) follows that lim

t→+∞

|x(t; x

, t

)| = 0 (x(t; x

, t

) is the solution of

the equation (3.8) passing through the point (x

, t

) ), i.e. the equation (3.8) is

dissipative.

We will prove that the equation (3.8) is not uniformly dissipative. In fact, if we

suppose that is not true, then by Lemma 3.1 the non-autonomous dynamical system

generated by the equation (3.8) will be dissipative. In virtue of Theorem 2.38 the

trivial solution of (3.8) is uniformly asymptotically stable. But this contradicts to

the choice of the function a. The desired example is constructed.

Taking into account the fact established above, we will call the equation (3.1)

dissipative if the non-autonomous dynamical system generated by this equation is

dissipative.

An important class of dissipative equations are equations with convergence

[

137

]

[

270

]

The equation (3.1) possesses the property of convergence if it has a unique

bounded on R solution which is globally uniformly asymptotically stable. This

unique bounded solution is called a limit regime for the equation (3.1).

From the results of

[

]

[

290

]

[

291

]

follows that, if the equation (3.1) is convergent,

then every equation from the family (3.2) possesses the same property. Thus, the

non-autonomous dynamical system generated by the equation (3.1) is convergent.

On the other hand, it is easy to construct examples of the equation which is not

convergent, but the non-autonomous dynamical system associated by this equation

is convergent. Note, that it is possible that the equation (3.1) has the limit regime

which is not a solution of this equation, but if the right hand side of the equation

(3.1) is stable in the sense of Poisson in positive direction, then the phenomenon

indicated above is not true. That is in this case the equation (3.1) is convergent if

and only if the corresponding non-autonomous dynamical system generated by this

equation is convergent.

Below we will call the equation (3.1) convergent if the non-autonomous

dynamical system generated by this equation is convergent.

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Analytic dissipative systems 123

Example 3.3 Consider the system of diﬀerential equations

(

= F (y, u)

= G(y),

(3.9)

where Y ⊆ E

, G ∈ C(Y, E

) and F ∈ C(Y × E

, E

). Suppose that for the

system (3.9) the conditions of the existence, uniqueness and extendability on R

are fulﬁlled. Denote by (Y, R

, σ) a dynamical system on Y generated by the second

equation of the system (3.9) and by ϕ(t, u, y) – the solution of the equation

= F (σ(t, y), u) (3.10)

passing through the point u ∈ E

for t = 0. Then the mapping ϕ : R

×E

×Y →

satisﬁes the conditions 1) and 2) from Example 3.1 and, consequently, the system

(3.9) generates a non-autonomous dynamical system h(X, R

, π), (Y, R

, σ), hi

(where X := E

×Y , π := (ϕ, σ) and h := pr

: X → Y ).

We will give some generalization of the system (3.9). Namely, let (Y, R

, σ) be

a dynamical system on the metric space Y . Consider the system

(

= F (σ(y, t), u)

y ∈ Y,

(3.11)

where F ∈ C(Y × E

, E

). Suppose that for the equation (3.10) the conditions

of the existence, uniqueness and extendability on R

are fulﬁlled. The system

h(X, R

, π), (Y, R

, σ), hi, where X := E

×Y , π := (ϕ, σ), ϕ(·, x, y) is the solution

of (3.10) and h := pr

: X → Y is a non-autonomous dynamical system generated

by the equation (3.11).

3.2 C-analytic systems

Deﬁnition 3.2 We will call that the non-autonomous dynamical system

h(X, T

, π), (Y, T

, σ), hi is C-analytic, if the following conditions are fulﬁlled:

a. X := C

× Y and π((z, y), t) := (ϕ(t, z, y), σ(t, y)) for all t ∈ T and (z, y) ∈

×Y i.e. the non-autonomous dynamical system h(X, T

, π), (Y, T

, σ), hi is

generated by the cocycle hC

, ϕ, (Y, T

, σ)i over (Y, T

, σ) with the ﬁber C

;

b. for each y ∈ Y and t ∈ T

the mapping U(t, y) : C

→ C

(U(t, y) := ϕ(t, ·, y))

is holomorphic.

It follows from the results above that, when Y is compact, all three types of dis-

sipation properties for C-analytic systems are equivalent. In addition a C-analytic

system is dissipative if and only if there is a positive number r (independent of both

z and y) such that lim sup

t→+∞

|ϕ(t, z, y)| ≤ r, where | ·| denotes the norm in C

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124 Global Attractors of Non-autonomous Dissipative Dynamical Systems

An important property of C-analytic dissipative systems is described in

Theorem 3.2 If Y is compact, then the C-analytic dissipative dynamical system

h(X, T

, π), (Y, T

, σ), hi is uniformly stable in the positive direction on compacts of

, i.e., for arbitrary ε > 0 and r > 0, there is a δ = δ(ε, r) > 0 such that the

inequality |z

−z

| < δ (|z

|, |z

| ≤ r) implies that |U(t, y)(z

) −U(t, y)(z

)| ≤ ε for

all y ∈ Y and t ∈ T

Proof. We ﬁrst prove that, for arbitrary a > 0, there is a compact K

⊂ C

such that U(t, y)B[0, a] ⊆ K

for all t ≥ 0 and y ∈ Y . In fact, Theorem 3.1

implies that there is b > 0 such that, for arbitrary a > 0, there is l

≥ 0 for which

U(t, y)B[0, a] ⊆ B[0, b] for t ≥ l

and y ∈ Y . Let K

:= ϕ([0, l

], B[0, a], Y )∪B[0, b];

then K

is the required set.

Now let r > 0; then the foregoing implies that the family of holomorphic map-

pings {U(t, y)} of the space C

into itself is uniformly bounded (with respect to

t ≥ 0 and y ∈ Y ) on the ball B[0, r], and the generalized Vitali theorem

[

167

]

implies that this family is equicontinuous in B[0, r], i.e. for arbitrary ε > 0 there is

δ(ε, r) > 0 such that |U(t, y)z

−U(t, y)z

| < ε for all (t, y) ∈ T

×Y if |z

−z

| < δ

(|z

|, |z

| < r). This proves the theorem. 

Theorem 3.3 Let h(X, T

, π), (Y, T

, σ), hi be a C-analytic non-autonomous dis-

sipative dynamical system and let Y be a compact minimal set; then

(1) for each y ∈ Y the set J

:= J ∩ X

:= h

−1

(y)) contains only the single

point x

, where J is the centre of Levinson of the system (X, T

, π);

(2) for all ε > 0 there is δ(ε) such that the inequality ρ(x, x

) < δ (x ∈ X

) implies

that ρ(xt, x

) < ε for t ≥ 0.

(3) lim

t→+∞

ρ(xt, x

h(x)t

) = 0 for x ∈ X, uniformly with respect to x on compacts from

Proof. Since Y is compact, it follows from the preceding remark that the non-

autonomous system h(X, T

, π), (Y, T

, σ), hi is compact dissipative, and Theorem

3.2 implies that it is uniformly positively stable on compacts from X. Hence, if

Y is a compact minimal set, then all conditions of Theorem 2.4 are satisﬁed, and

applying this theorem in the case under consideration we obtain assertions 2. and

3. To complete the proof of the theorem it remains to establish 1.

Thus let y ∈ Y . According to Theorem 2.4, the set J

is connected because

:= C

×{y} (y ∈ Y ) is connected. In proving Theorem 2.4 we established that

the semigroup P

contains an idempotent element u such that J

= u(X

), and

that there are t

→ +∞ for which u = lim

k→+∞

. Let x ∈ X

, that is, x := (z, y)

(z ∈ C

); then u(x) = u(z, y) = (ξ

(z), y), where ξ

(z) := lim

k→+∞

U(t

, y)z and this

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 3

Analytic dissipative systems 125

relation holds uniformly on compacts in C

. Since the mappings U (t

, y) : C

→ C

(k = 1, 2, . . .) are holomorphisms, the same is true of their limit ξ

. In proving

Theorem 2.4 we showed that the idempotent element u acts on J

in the same

fashion as the identity mapping, i.e. u(x) = x (x ∈ J

); hence ξ

(z) = z for all z

such that (z, y) ∈ J

. Thus, if (z, y) ∈ J

, then z ∈ C

is a ﬁxed point of ξ

. Since

is a holomorphism, it has at most a ﬁnite number of ﬁxed points in the interior

of each bounded set. It follows that J

contains a ﬁnite number of points, and its

connectivity implies that it contains only one point x

. This proves the theorem.

Corollary 3.1 If Y is compact and minimal, then C-analytic non-autonomous

dynamical system h(X, T

, π), (Y, T

, σ), hi is convergent.

Corollary 3.2 Under the conditions of Theorem 3.3 the dynamical system

(X, T

, π) induces on the centre of Levinson J of the system (X, T

, π) a dynamical

system (J, T

, bπ) which is isomorphic to (Y, T

, σ). In particular, the point x

recurrent (almost periodic, quasi-periodic, periodic) if y ∈ Y is recurrent (almost

periodic, quasi-periodic, periodic).

Proof. This follows from Theorem 3.3 (see also Corollary 3.1). It is suﬃcient to

note that the mapping p : Y → J, deﬁned by the relation p(y) := x

= {x

}),

is a homomorphism of (Y, T

, σ) onto (J, T

, bπ). 

The general results obtained in this section can be used in the study of the

dissipative diﬀerential equations with the recurrent (with respect to time t) right

side if it is holomorph with respect to spatial variable z. Before formulate the

respectively results we will give the following example

Example 3.4 Denote by A(R ×C

, C

) the set of all continuous with respect to

t ∈ R and holomorphic with respect to z ∈ C

functions f : R×C

→ C

, equipped

with the topology of convergence uniform on the compacts from R × C

. Consider

the equation (3.1) with right side f from A(R ×C

, C

). Along with equation (3.1)

we consider its H - class, i.e. the family of equations

˙z = g(t, z) (g ∈ H(f )), (3.12)

where H(f) =

: τ ∈ R}, f

(t, z) = f(t + τ, z) and the bar indicating closure in

the space A(R × C

, C

). Let ϕ(·, z, g) be the solution of (3.12) passing through

the point z for t = 0 and deﬁned on R

Note the following properties of the mapping ϕ : R

×C

×H(f) → C

a. ϕ(0, z, g) = z (z ∈ C

, g ∈ H(f));

b. ϕ(τ, ϕ(t, z, g), g

) = ϕ(t + τ, z, g) (τ ∈ R

, t ∈ R, z ∈ C

and g ∈ H(f ));

c. the mapping ϕ : R

×C

×H(f) → C

is continuous and, for ﬁxed t ∈ R

and

g ∈ H(f ) the mapping U(t, g) := ϕ(t, ·, g) : C

→ C

is holomorphic

[

122

]

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 3

126 Global Attractors of Non-autonomous Dissipative Dynamical Systems

We write (Y, R, σ) (Y := H(f )) for the dynamical system of translations on H(f)

and deﬁne on X := C

×Y the semi-group dynamical system (X, R

, π) as follows:

π(τ, (z, g)) := (ϕ(τ, z, g), g

). Put h := pr

: X → Y (h is a homomorphism of

(X, R

, π) onto (Y, R, σ)). Equation (3.12) therefore generates the non-autonomous

dynamical system h(X, R

, π), (Y, R, σ), hi. Property c. of ϕ implies that the non-

autonomous dynamical system constructed is C-analytic and application to it of

Theorem 3.3 yields

Theorem 3.4 If f : R × C

→ C

is analytic in z ∈ C

and continuous in t ∈ R

and f recurrent with respect to t ∈ R uniformly with respect to z on compacts from

(in particular, it is almost periodic or periodic), then the dissipativity of equation

(3.12) implies that it is convergent, i.e. (3.12) has a unique solution p : R → C

, bounded on R and uniformly asymptotically stable in the large; moreover, p is

recurrent (almost periodic or periodic) if f has this property.

Proof. This is a direct consequence of Theorem 3.3 Corollaries 3.1 and 3.2. It should

be kept in mind that the recurrence of f means that the set H(f ) is compact and

minimal with respect to the dynamical system of translations on H(f). 

Remark 3.1 The ”local” variant of Theorem 3.3 (and so of Theorem 3.4) holds

too, i.e., Theorem 3.4 remains in force when C

is replaced everywhere by a domain

W ⊆ C

(W can be both bounded and unbounded and can coincide with C

). Hence

Theorem 3.4 (see also Corollary 3.2) implies that the autonomous equation ˙z = f (z)

cannot have limit cycles in W ⊆ C if f is holomorphic on W .

Deﬁnition 3.3 h(X, T

, π), (Y, T

, σ), hi is said to be C-analytic on the set M ⊆

X, if M is invariant and the system h(M, T

, π), (

Y , T

, σ), hi is C-analytic, where

Y := h(M), but (M, T

, π) is a restriction (X, T

, π) on M (analogous is deﬁned

(

Y , T

, π)).

Theorem 3.5 Suppose that the following conditions are fulﬁlled:

1. (Y, T

, σ) is compact dissipative and its Levinson’s centre J

is a minimal set;

2. (X, T

, π) is compact dissipative and J

is its Levinson’s centre;

3. h(X, T

, π), (Y, T

, σ), hi is C-analytic on J

Then the set J

∩X

contains only one point for every y ∈ J

and, consequently,

the system h(X, T

, π), (Y, T

, σ), hi is convergent.

Proof. Since J

is nonempty, compact and J

is minimal, then h(J

) = J

. From

the C-analyticity of h(X, T

, π), (Y, T

, σ), hi on J

follows, according to Theorem

3.3, that J

∩X

contains only one point for any y ∈ J

. Thus a non-autonomous

dynamical system h(X, T

, π), (Y, T

, σ), hi is convergent. 