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

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Method of Lyapunov functions 217

where ωt := σ(t, ω). Let ϕ(t, u, ω) be a solution of equation (5.106) satisfying

the initial condition ϕ(0, u, ω) = u, then it is not diﬃcult to see that the triple

hE, ϕ, (Ω, R, σ)i is a cocycle over (Ω, R, σ) with the ﬁber E. It is easy to check that

if a function f is homogeneous of order m = 1 w.r.t. u ∈ E, then the equality

ϕ(t, λu, ω) = λϕ(t, u, ω), holds for all t ≥ 0, λ > 0, u ∈ E and ω ∈ Ω.

Theorem 5.35 Let the function f ∈ C

(E ×F, F ) and Φ ∈ C

(F, F ) be regular,

Ω is compact and the function f is homogeneous (of order m = 1) w.r.t. variable

u ∈ E, then the following conditions are equivalent:

1. the trivial solution of equation (5.106) is uniform exponential stable, i.e. there

exist positive numbers N and ν such that

|ϕ(t, u, ω)| ≤ Ne

−νt

|u| (5.107)

for all u ∈ E, t ≥ 0 and ω ∈ Ω;

2. for all number k > 1 there exists a continuously diﬀerentiable function V :

E × Ω → R

satisfying the following conditions:

2.1. V (λu, ω) = λ

V (u, ω) for all u ∈ Ω and λ > 0;

2.2. there exist positive numbers α and β such that α|u|

≤ V (u, ω) ≤ β|u|

for all u ∈ E and ω ∈ Ω;

2.3.

(u, ω) :=

V (ϕ(t, u, ω), ωt)



t=0

= D

V (u, ω)f(u, ω)+

V (u, ω)Φ(ω) = −|u|

for all u ∈ E and ω ∈ Ω, where D

V (respectivelyD

V ) is a partial

Frechet’s derivative of function V w.r.t. variable u ∈ E (respectively ω ∈

Ω).

Proof. Let h(X, R

, π), (Ω, R, σ), hi be a non-autonomous dynamical system gen-

erated by diﬀerential system (5.104), where X = E × Ω, h = pr

and π = (ϕ, σ).

The triple (X, h, Ω) is the trivial ﬁber bundle with the ﬁber E, moreover the norm

of element x = (u, ω) ∈ E ×Ω is deﬁned by equality |x| = |u|, where |u| is a norm of

element u in the space E. Then under the conditions of Theorem the zero section of

non-autonomous dynamical system, constructed above, will satisfy the conditions

of Theorem 5.35. Thus to ﬁnish the proof of Theorem 5.35 it is suﬃcient to show

that the function V : E × Ω → R

from the theorem 5.33 under the conditions of

Theorem 5.35 will be continuously diﬀerentiable and satisﬁes the condition

(u, ω) = D

V (u, ω)f(u) + D

V (u, ω)Φ(ω)

for all (u, ω) ∈ E × Ω.

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218 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Let us show that a function V : E ×Ω → R

, deﬁned by equality (5.77), i.e.

V (u, ω) =

+∞

|ϕ(t, u, ω)|

dt (5.108)

for all (u, ω) ∈ E ×Ω (such that |π(t, x)| = |(ϕ(t, u, ω), ωt)| = |ϕ(t, u, ω)|) is contin-

uously diﬀerentiable. To this aim we will formally diﬀerentiate the equality (5.108)

w.r.t. variable u ∈ E, thus we obtain

V (u, ω)v = (5.109)

+∞

k|ϕ(t, u, ω)|

k−2

RehD

ϕ(t, u, ω)v, ϕ(t, u, ω)idt.

Let us show that integral in (5.109) is uniformly convergent w.r.t. ω ∈ Ω and

u, v on every bounded set from E and since by formula (5.109) is really deﬁned a

derivative of Frechet of function V w.r.t. variable u ∈ E. First of all we note that

operator-function U(t, u, ω) := D

ϕ(t, u, ω) satisﬁes the operational equation

= B(t, u, ω)X, (5.110)

where B(t, u, ω) := D

f(ϕ(t, u, ω), ωt), and initial condition U(0, u, ω) = I (I

is the identity operator in E). According to Lemma 3.1 function B(t, u, ω) =

f(ϕ(t, u, ω), ωt) is homogeneous w.r.t. ϕ(t, u, ω) of order m = 1 and there exists

a number M > 0 such that

kB(t, u, ω)k ≤ M (5.111)

for all (u, ω) ∈ E × Ω and t ≥ 0. From the inequality (5.111) follows that

kU(t, u, ω)k ≤ e

kB(τ,u,ω)kdτ

≤ e

for all t ≥ 0 and (u, ω) ∈ E ×Ω and, consequently,

|ϕ(t, u, ω)|

k−2

|RehD

ϕ(t, u, ω)v, ϕ(t, u, ω)i| (5.112)

|ϕ(t, u, ω)|

k−1

ϕ(t, u, ω)k|v| ≤ M|v|N

k−1

−ν(k−1)t

|u|

for any t ≥ 0, u, v ∈ E and ω ∈ Ω. From the inequality (5.112) follows that for k > 1

the integral in the second hand of (5.108) is convergent, moreover uniformly w.r.t.

ω ∈ Ω and u, v on the every bounded subset from E. Thus, the function V (u, ω),

deﬁned by formula (5.108) is continuously diﬀerentiable w.r.t. variable u ∈ E and

by equality (5.109) it is deﬁned its derivative of Fr´echet w.r.t. variable u ∈ E.

Having formally diﬀerentiated the equality (5.108) w.r.t. variable ω ∈ F we will

obtain

V (u, ω)w = (5.113)

+∞

k|ϕ(t, u, ω)|

k−2

RehD

ϕ(t, u, ω)w, ϕ(t, u, ω)idt

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Method of Lyapunov functions 219

for all (u, ω) ∈ E × Ω. Let us show that integral in the second member of (5.113)

is uniformly convergent w.r.t. ω ∈ Ω and (u, w) ∈ E × F on every bounded subset

from E × F. First of all we note that from (5.107) we have

ϕ(t, u, ω) = u +

f(ϕ(τ, u, ω), ωτ)dτ

for all t ≥ 0 and (u, ω) ∈ E ×F and, consequently,

ϕ(t, u, ω) =

f(ϕ(τ, u, ω), ωτ)D

ϕ(τ, u, ω)

f(ϕ(τ, u, ω), ωτ)DΦ(ωτ)dτ. (5.114)

We denote by V(t, u, ω) the operator-function D

ϕ(t, u, ω), then from (5.114) follows

that V(t, u, ω) satisﬁes the operational equation

= B(t, u, ω)Y + F(t, u, ω), (5.115)

where F(t, u, ω) = D

f(ϕ(t, u, ω), ωt)DΦ(ωt), and initial condition

Y (0) = O (5.116)

(O is a null operator, acting from F into F ). From the equality (5.115) and the

condition (5.116) follows that

V(t, u, ω) =

U(t, τ, u, ω)F(τ, u, ω)dτ (5.117)

for all t ≥ 0 and (u, ω) ∈ E × Ω, where U(t, τ, u, ω) is a solution of operational

equation (5.110), satisfying the initial condition U(τ, τ, u, ω) = U(τ, u, ω), therefore

the estimate

kU(t, τ, u, ω)k ≤ e

kB(τ,u,ω)kdτ

≤ e

M(t−τ )

(5.118)

holds for any t ≥ τ(t, τ ∈ R

) and (u, ω) ∈ E × Ω.

Thus, a function f (u, ω) is homogeneous w.r.t. u ( of order m = 1), then

f(u, ω) will be the same and, in virtue of Lemma 5.6 there exists a number

M > 0 such that

f(u, ω)k ≤ M

|u|

for all u ∈ E and ω ∈ Ω. Thus,

kF(t, u, ω)k = kD

f(ϕ(t, u, ω), ωt)DΦ(ωt)| ≤ L|ϕ(t, u, ω)| (5.119)

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220 Global Attractors of Non-autonomous Dissipative Dynamical Systems

for any t ≥ 0 and (u, ω) ∈ E × Ω, where L = M

max{kDΦ(ω)k : ω ∈ Ω}. From

(5.106),(5.117), (5.118) and (5.119) we have

|ϕ(t, u, ω)|

k−2

|RehD

ϕ(t, u, ω)wi| ≤ |ϕ(t, u, ω)|

k−1

ϕ(t, u, ω)k|w|

≤ |ϕ(t, u, ω)|

k−1

|w|

kU(t, τ, u, ω)kkF(τ, u, ω)kdτ

≤ |ϕ(t, u, ω)|

k−1

|w|

M(t−τ )

L|ϕ(τ, u, ω)|dτ ≤ N

k−1

−ν(k−1)t

|u|

k−1

|w|

M(t−τ )

LNe

−ντ

|u|dτ = N

−ν(k−1)t

−e

−νt

)

M + ν

|u|

|w|

for every t ≥ 0, u ∈ E and ω ∈ Ω. From this follows, that for suﬃciently large

k the integral (5.113) is uniformly convergent w.r.t. ω ∈ Ω and (u, w) ∈ E × F

on the every bounded subset from E × F . Therefore the function V (u, ω) deﬁned

by formula (5.108) is continuously diﬀerentiable w.r.t. variable ω ∈ Ω and by

formula (5.113) it is deﬁned its Frechet derivative w.r.t. variable ω ∈ Ω. So, the

function V : E × Ω → R

is continuously diﬀerentiable w.r.t. variable u ∈ E and

variable ω ∈ Ω both and, consequently, it is continuously diﬀerentiable in the sense

of Frechet.

Let us note that

V ((ϕ(t, u, ω), ωt) = D

V (ϕ(t, u, ω), ωt)D

ϕ(t, u, ω) +

V (ϕ(t, u, ω), ωt)

ωt = D

V (ϕ(t, u, ω), ωt)f(ϕ(t, u, ω), ωt) +

V (ϕ(t, u, ω), ωt)Φ(ωt)

for any t ≥ and (u, ω) ∈ E ×Ω and, consequently,

(u, ω) =

V (ϕ(t, u, ω), ωt)



t=0

= D

V (u, ω)f(u, ω) + D

V (u, ω)Φ(u).

Thus, under the conditions of Theorem from 1. follows 2.

For ﬁnishing the proof of the theorem it is suﬃcient to remark that according

to Theorem 5.33 from the conditions 2.1–2.3 of Theorem follows 1. The theorem is

completely proved. 

Remark 5.11 For ﬁnite dimensional systems the theorem 5.34 generalizes and

reﬁnes the theorem 38 from

[

337

]

5.8 Global attractors of quasi-homogeneous systems

Let E and F be ﬁnite-dimensional Banach spaces. Consider the diﬀerential equation

= f (x), (5.120)

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Method of Lyapunov functions 221

where f ∈ C(E, E) is regular and homogeneous of order m ≥ 1. Along with

equation (5.120) consider also perturbed equation

= f (x) + F (t, x) (5.121)

with the regular perturbation. Let us remember (see, for example

[

337

]

[

224

]

and

[

227

]

), that equation (5.121) is called quasi-homogeneous if

lim sup

|x|→+∞

|F (t, x)|

|x|

= c, (5.122)

where c is a certain nonnegative suﬃciently small constant. Moreover, the limiting

relation (5.122) holds uniformly w.r.t. t ∈ R.

Along with equation (5.121) consider the family of equations

= f(x) + G(t, x), (5.123)

where G ∈ H(f) :=

: τ ∈ R} and the bar denotes closure in C(R×E, E), C(R×

E, E) is equipped by topology of convergence on every compact from R×E. Let us

denote by (Y, R, σ) a dynamical system of translations on Y := H(F ), by ϕ(t, u, f +

G) the solution of equation (5.123), satisfying the condition ϕ(0, u, f + G) = u

and h(X, R

, π), (Y, R, σ), hi (i.e. h := pr

and π := (ϕ, σ)) a non-autonomous

dynamical system generated by equation (5.123).

Theorem 5.36 Let f ∈ C

(E, E) be regular, homogeneous of order m ≥ 1 and

the zero solution of equation (5.120) be uniformly asymptotically stable. If f + F ∈

C(R ×E, E) is regular, equation (5.121) is quasi-homogeneous, then the following

assertions take place:

(1) a set I

= {u ∈ E|sup{|ϕ(t, u, f + G)| : t ∈ R} < +∞} is not empty, compact

and connected for each G ∈ H(F );

(2) ϕ(t, I

, f + G) = I

σ(t,f +G)

for all t ∈ R

and G ∈ H(F );

(3) a set I =

|G ∈ H(F )} is compact and connected;

(4) the equalities

lim

t→+∞

β(ϕ(t, M, f + G

−t

), I

) = 0

and

lim

t→+∞

β(ϕ(t, M, f + G), I) = 0

take place for all G ∈ H(F ) and bounded subset M ⊆ E.

Proof. Let h(X, R

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

erated by equation (5.121). Thus, f ∈ C

(E, E) and the zero solution of equation

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222 Global Attractors of Non-autonomous Dissipative Dynamical Systems

(5.120) is uniformly asymptotically stable, then according to Theorem 5.34 by equal-

ity (5.77) it is deﬁned a continuously diﬀerentiable function V : E → R

, satisfying

the conditions 2.1–2.3 of Theorem 5.34. Let us deﬁne a function V : X → R

the following way: V(x) = V (u) for all x = (u, G) ∈ E × H(F ). We note that

(x) =

V (ϕ(t, u, f + G))



t=0

(5.124)

= DV (ϕ(t, u, f + G))

ϕ(t, u, f + G)



t=0

= DV (u)(f(u) + G(0, u))

= DV (u)f (u) + DV (u)G(0, u) = −|u|

+ DV (u)G(0, u).

In virtue of condition 2.1 of Theorem 5.34 the function V is homogeneous of order

k − m + 1, then according to Lemma 5.6 its Frechet derivative DV (u) will be

homogeneous of order k − m and, consequently, there exists a number L > 0 such

that

kDV (u)k ≤ L|u|

k−m

(5.125)

for all u ∈ E. From equality (5.122) follows that for all ε > 0 there exists r = r(ε) >

0 such that

|G(t, u)| ≤ (c + ε)|u|

(5.126)

for all |u| ≥ r, t ∈ R and G ∈ H(F ). From (5.125)–(5.126) we have

|DV (u)G(0, u)| ≤ L(c + ε)|u|

(5.127)

and, consequently, from (5.124),(5.127) we will obtain

(x) ≤ |u|

(−1 + L(c + ε)) = −γ|x|

for each x ∈ X

, where γ = 1 − l(c + ε) > 0 since c and ε are a suﬃciently small

positive numbers. To ﬁnish the proof of Theorem it is suﬃcient to refer to Theorems

2.24- 2.25. 

Theorem 5.37 Let f ∈ C

(E, E), Φ ∈ C

(F, F ) and f + F ∈ C

(E × F, E) be

regular, Ω ⊆ F be a compact invariant set of dynamical system (5.105), the function

f be homogeneous (of order m > 1) and the zero solution of equation (5.105) be

uniformly asymptotically stable. If

|F (u, ω)| ≤ c|u|

for all |u| ≥ r and ω ∈ Ω, where rand c are certain positive numbers, and moreover

c is suﬃciently small, then the following assertions take place:

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Method of Lyapunov functions 223

(1) a set I

= {u ∈ E|sup{|ϕ(t, u, ω)| : t ∈ R} < +∞} is not empty, compact and

connected for each ω ∈ Ω, where ϕ(t, u, ω) is a solution of equation

= f(u) + F (u, ωt)

satisfying the initial condition ϕ(0, u, ω) = u;

(2) ϕ(t, I

, ω) = I

σ(t,ω)

for all t ∈ R

and ω ∈ Ω;

(3) a set I =

| ω ∈ Ω} is compact and connected;

(4) the equalities

lim

t→+∞

β(ϕ(t, M, ω

−t

), I

) = 0 (5.128)

and

lim

t→+∞

β(ϕ(t, M, ω), I) = 0 (5.129)

take place for all ω ∈ Ω and bounded subset M ⊆ E.

Proof. The proof of formulated assertion is carried out using the same scheme as

in Theorem 5.36, therefore its proof may be omitted. 

Theorem 5.38 Let f ∈ C

(E × F, E), Φ ∈ C

(F, F ) and f + F ∈ C

(E ×F, E)

be regular, Ω ⊆ F be a compact invariant set of dynamical system (5.105), function

f be homogeneous (of order m = 1) w.r.t. variable u ∈ E and the zero solution of

equation (5.106) be uniformly asymptotically stable.

|F (u, ω)| ≤ c|u| (5.130)

for all |u| ≥ r and ω ∈ Ω, where r and c are certain positive numbers, and moreover

c is suﬃciently small. Then the following assertions take place:

(1) the set I

= {u ∈ E|sup{|ϕ(t, u, ω)| : t ∈ R} < +∞} is not empty, compact

and connected for each ω ∈ Ω, where ϕ(t, u, ω) is a solution of equation

= f(u, ωt) + F (u, ωt) (5.131)

satisfying the initial condition ϕ(0, u, ω) = u;

(2) ϕ(t, I

, ω) = I

σ(t,ω)

for all t ∈ R

and ω ∈ Ω;

(3) the set I =

|ω ∈ Ω} is compact and connected;

(4) the equalities (5.128) and (5.129) take place for every bounded subset M ⊆ E

and ω ∈ Ω.

Proof. Let h(X, R

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

erated by equation (5.131). Since f ∈ C

(E × F, E) and the zero solution of equa-

tion (5.106) is uniformly asymptotically stable, then according to Theorem 5.35 by

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224 Global Attractors of Non-autonomous Dissipative Dynamical Systems

equality

V (u, ω) =

+∞

|ϕ(t, u, ω)|

it is deﬁned a continuously diﬀerentiable function V : X = E ×Ω → R

, satisfying

the conditions 2.1-2.3 of Theorem 5.33. Let us remark that

(u, ω) =

V ((ϕ(t, u, ω), ωt)



t=0

= {D

V (ϕ(t, u, ω), ωt)

ϕ(t, u, ω)

V (ϕ(t, u, ω), ωt)

ωt}



t=0

= D

V (u, ω)[f(u, ω) + F (u, ω)]

V (u, ω)Φ(ω) = −|u|

+ D

V (u, ω)F (u, ω). (5.132)

Since the function V (u, ω) is homogeneous of order k > 1 w.r.t. variable u ∈ E,

then in virtue of Lemma 5.6 there exists a number L > 0 such that

V (u, ω)k ≤ L|u|

k−1

(5.133)

for all u ∈ E and ω ∈ Ω. From (5.130) and (5.133) follows, that

V (u, ω)F (u, ω)| ≤ cL|u|

(5.134)

for each |u| ≥ r and ω ∈ Ω and, consequently, according to (5.132), (5.134) we have

(u, ω) ≤ −|u|

+ cL|u|

= −γ|u|

for all ω ∈ Ω and |u| ≥ r, where γ = 1 − cL > 0 since a number c is suﬃciently

small. For ﬁnishing the proof of the theorem it is suﬃciently to refer to Theorem

2.24, 2.25 and 5.3. The theorem is proved. 

Remark 5.12 The dissipativity of equation (5.121) is established in

[

337

]

(theorem 39 and corollary 1).

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Chapter 6

Dissipativity of some classes of equations

6.1 Diﬀerence equations

All the results about diﬀerential equations, which are presented in previous chap-

ters, for diﬀerence equations hold too, because they were formulated and proved

for general non-autonomous dynamical systems, both for dynamical systems with

continuous time and those with discrete time. Below we will give some of them that

we need to study periodical systems with impulse.

Consider a diﬀerence equation

u(k + 1) = Φ(k, u(k)) (Φ ∈ C(Z ×E

, E

)). (6.1)

Denote by ϕ(·, u, Φ) a solution of the equation (6.1) passing through the point

u for k = 0. Suppose that Φ is p–periodic in k ∈ Z, where p ∈ Z. We set H(Φ) :=

{Φ

= σ(Φ, m) : 0 ≤ m ≤ p −1}, and denote by (Y, Z, σ) a dynamical system of

translations on Y := H(Φ) induced by (C(Z×E

, E

), Z, σ). Let X := E

×Y and

deﬁne a mapping π : X × Z

→ X by the equality π((u,

Φ), k) := (ϕ(k, u,

Φ),

From general properties of solutions of the diﬀerence equation (6.1) follows that

(X, Z

, π) is a semigroup dynamical system and



(X, Z

, π), (Y, Z, σ), h



(h :=

: X → Y ) is a non-autonomous p–periodic dynamical system. Applying to the

above constructed non-autonomous dynamical system the results of section 4.6, we

get some assertions for the equation (6.1). Before formulating some statements of

this type we will deﬁne the notion of dissipativity for the equation (6.1).

Deﬁnition 6.1 By analogy with the case of diﬀerential equations, the diﬀerence

equation (6.1) is said to be dissipative, if there is a positive number r such that

lim sup

k→+∞

|ϕ(k, v,

Φ| < r

for all v ∈ E

and

Φ ∈ H(Φ).

We deﬁne a mapping P : E

→ E

in the following way: P(v) := ϕ(p, v, Φ) and

denote by (E

, P ) the cascade generated by positive powers of P.

225

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226 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Theorem 6.1 The equation (6.1) is dissipative if and only if the cascade (E

, P)

is dissipative.

Theorem 6.2 If the equation (6.1) is dissipative, then:

1. there exists h > 0 such that for all a > 0 there is k(a) ∈ Z

for which

B[0, a] ⊆ B[0, h] for all k ≥ k(a);

2. the mapping P has at least one ﬁxed point u

∈ E

through which there passes

a p–periodic solution of (6.1).

Theorem 6.3 If the equation (6.1) is dissipative, then:

1. there is a nonempty compact set I ∈ E

which possesses the following proper-

ties:

1.a. P(I) = I;

1.b. for all ε > 0 there is δ(ε) > 0 such that ρ(u, I) < δ implies ρ(P

u, I) < ε

for all k ∈ Z

;

2. the set J := ∪{J

:= ϕ(k, I, Φ) : k ∈ Z

} is nonempty, compact and possesses

the following properties:

2.a. the set J is invariant with respect to (6.1), i.e. v ∈ J

implies

ϕ(k, v, Φ

) ∈ J for all k ∈ Z

;

2.b. J

k+p

= J

for all k ∈ Z

;

2.c. for all ε > 0 there exists δ(ε) > 0 such that ρ(ϕ(m, u, Φ), J

) < δ implies

ρ(ϕ(k, u, Φ), J

) < ε for all k ≥ m.

Theorem 6.4 Let the equation (6.1) be dissipative and the function Φ : Z×C

→

be holomorphic in second variable, then the Levinson’s center of this equation

consists from a single p–periodic trajectory, i.e., the equation (6.1) is convergent.

Theorems 6.1 – 6.4 directly follow from the results of section 4.6. Concern-

ing Theorem 6.4 it is necessary to add that when the right hand side of (6.1) is

holomorphic, then the solution ϕ(·, z, Φ) (z ∈ C

) is holomorphic too.

Consider the quasi-linear equation

u(k + 1) = A(k)u(k) + F (k, u(k)), (6.2)

where A(k) is a non-stationary n × n–matrix and the function F ∈ C(Z ×E

, E

)

satisﬁes some condition of ”smallness”.

Denote by U(k, A) the Cauchy’s matrix for linear equation

u(k + 1) = A(k)u(k).

Theorem 6.5 Let A(k) and F (k, u) be p-periodic in k ∈ Z

. Suppose that the

following conditions hold: