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

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Method of Lyapunov functions 207

where v = ˙x + F (x) and F (x) :=

h(t)dt. As usual, along with the system (5.72)

we consider its H-class

(

˙x = −F (x) + v

˙v = −x + g(t) (g ∈ H(p)).

(5.73)

Theorem 5.30 If h(x) > 0 (excepting maybe the discrete set of points), p(t) is

recurrent and F (+∞) = +∞ (or F (−∞) = −∞), then the equation (5.65) has a

unique bounded on R uniformly compatible and uniformly globally asymptotically

stable solution, i.e. (5.65) is convergent.

Proof. Reasoning in the same way that in

[

272

]

show that under the conditions of

the theorem there exists a closed contour that intersects all the solutions of every

system (5.73). According to Theorem 5.6 the system (5.72) is dissipative.

Let us show that the system (5.72) in fact is convergent. Let {x

(t), v

(t)}

(i = 1, 2) be two diﬀerent solutions of the system (5.73). Construct the function

ϕ(t) = |x

(t) − x

(t)|

+ |v

(t) − v

(t)|

and calculate its derivative. We will obtain:

˙ϕ(t) = −(x

(t) − x

(t))[F (x

(t)) −F (x

(t))]. (5.74)

Note that if ˙ϕ(t) ≡ 0 then x

(t) ≡ x

(t) and, consequently, v

(t) ≡ v

(t). To ﬁnish

the proof of the theorem is suﬃcient to refer to Theorem 5.23. 

Theorem 5.31 If ψ

(x) = h(x) − 2D (0 < D < 1) and ψ

(x) = g(x) −x satisfy

the conditions:

(1) |ψ

(x)| ≤ c

for all x ∈ R;

(2) |ψ

(

x) − ψ

(x)| ≤ c

|x) − x| for all ¯x, x ∈ R, where c

+ c

= c < mD =

√

1 − DD,

then the system (5.72) has a unique bounded on R uniformly compatible, uniformly

globally asymptotically stable solution.

Proof. Let

{ϕ(t, x

, y

, g)} = {(ϕ

(t, x

, y

, g), ϕ

(t, x

, y

, g))

(i = 1, 2, g ∈ H(f)) are two solutions of the system (5.73) and

(

u(t) := ϕ

(t, x

, y

, g) − ϕ

(t, x

, y

, g)

v(t) := ϕ

(t, x

, y

, g) −ϕ

(t, x

, y

, g).

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 5

208 Global Attractors of Non-autonomous Dissipative Dynamical Systems

The functions u and v satisfy the following system of equations











˙u = v

˙v = −u − 2Dv − [ψ

(ϕ

(t, x

, y

, g)) − ψ

(ϕ

(t, x

, y

, g))]−

[ψ

(ϕ

(t, x

, y

, g))ϕ

(t, x

, y

, g)−

(ϕ

(t, x

, y

, g))ϕ

(t, x

, y

, g)].

In the same way that in

[

272

]

we show that











|u(t)| < (1 +

)α[exp(−(D −

)t) + exp(−Dt)]

|v(t)| < (1 + c

)(1 +

)α[exp(−(D −

)t)

+ exp(−Dt)]

, (5.75)

where

α =

|u(0)|

+ 2D|u(0)||v(0)| + |v(0)|

≤

√

1 + D

|u(0)|

+ |v(0)|

From (5.75) follows that

|u(t)|

+ |v(t)|

≤ N exp(−(D −

)t)

|u(0)|

+ |v(0)|

where N > 0 is some constant depending only on c

, c

and D. Now to complete

the proof of the theorem is suﬃcient to refer to Theorem 2.15 and Corollary 2.6.

Remark 5.9 All the statements given in this chapter in the periodical case coin-

cide with previously established results (see

[

160

]

[

270

]

[

272

]

5.6 Construction of Lyapunov function for homogeneous systems

Let (X, h, Y ) be a locally trivial Banach ﬁbering, | · | : X → R

is a norm on

(X, h, Y ) coordinated with the metric ρ on X.

Deﬁnition 5.9 The autonomous dynamical system (X, R

, π) is said to be ho-

mogeneous of order m ∈ R

, if for any x ∈ X, t ∈ R

and λ > 0 the equality

π(t, λx) = λπ(λ

m−1

t, x) takes place.

Deﬁnition 5.10 h(X, T, π), ((Y, T, σ), hi is called homogeneous of order m = 1 if

the autonomous dynamical system (X, T, π) is homogeneous of order m = 1.

Theorem 5.32 For an autonomous homogeneous (of order m > 1 ) dynamical

system (X, R

, π) the following assertions are equivalent:

1. there exist positive numbers a

and b

(i = 1, 2) such that

|x|

1−m

+ b

1−m

≤ |π(t, x)| ≤ (a

|x|

1−m

+ b

1−m

(5.76)

for all t ≥ 0 and x ∈ X;

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Method of Lyapunov functions 209

2. for all k > m − 1 there exists a continuous function V : X → R

with the

following properties:

2.1. V (λx) = λ

k−m+1

V (x) for all λ ≥ 0 and x ∈ X;

2.2. α|x|

k−m+1

≤ V (x) ≤ β|x|

k−m+1

for all x ∈ X, where α and β are certain

positive numbers;

2.3. V

(x) = −|x|

for all x ∈ X, where V

(x) =

V (π(t, x)) |

t=0

for T = R

and V

(x) = V (π(1, x)) − V (x) for T = Z

Proof. We will show that from 1. follows 2. Let a and b are positive numbers,

such that the inequality (5.76) takes place, then for each k > m − 1 we deﬁne the

function V : X → R

by equality

V (x) =

+∞

|π(t, x)|

dt. (5.77)

First of all we note that by equality (5.77) it is deﬁned correctly the function

V : X → R

because the integral, which ﬁgures in the second number of equation

(5.77) is convergent, moreover it is uniformly convergent w.r.t. x on every bounded

set from X. Indeed, since

|x|

1−m

+ b

1−m

≤ |π(t, x)|

≤ (a

|x|

1−m

+ b

1−m

, (5.78)

+∞

|(a|x|

1−m

+ bt)

1−m

dt =

+∞

a|x|

1−m

dτ (5.79)

and

1−m

< −1, then the integral (5.79) is convergent; moreover, the convergence

is uniform on every bounded set from X.

We will show that the function V , deﬁned by equality (5.77), is our unknown

function. The continuity of V results from the continuity of mapping π : T×X → X

and uniform convergence of integral (5.79) w.r.t. x on every bounded set from X.

We now note that

V (λx) =

+∞

|π(t, λx)|

dt =

+∞

|π(λ

m−1

t, x)|

= λ

k−m+1

+∞

|π(τ, x)|

dt = λ

k−m+1

V (x)

for all λ > 0 and x ∈ X. It is not diﬃcult to show that the function V is positive

deﬁnite. In virtue of (5.78) and (5.79) we have

α|x|

k−m+1

≤ V (x) ≤ β|x|

k−m+1

(5.80)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 5

210 Global Attractors of Non-autonomous Dissipative Dynamical Systems

for every x ∈ X, where

α =

(m − 1)a

k−m+1

1−m

(k − m + 1)

and β =

(m − 1)a

k−m+1

1−m

(k − m + 1)

From (5.79)-(5.80) results that the function V satisﬁes and the condition 2.2.

Finally, we note that

V (π(t, x)) = −|π(t, x)|

(5.81)

and, consequently, V

(x) = −|x|

for all x ∈ X.

We will prove now that from condition 2. follows 1. In fact, we denote by

ψ(t) = V (π(t, x)), then in virtue of condition 2.3 we will have

(t) = −|π(t, x)|

(5.82)

for all t ≥ 0.

From the condition 2.2 we have |π(t, x)|

k−m+1

≥

ψ(t) and, consequently,

(t) ≤ −

k−m+1

ψ(t)

k−m+1

for all t ≥ 0. If x 6= 0, then ψ(t) = V (π(t, x)) > 0 for all t ≥ 0, therefore

V (π(t, x)) ≤ (V

−

m−1

k−m+1

(x) +

m − 1

k − m + 1

k−m+1

1−m

(5.83)

for all x ∈ X and t ≥ 0.

From the condition 2.2 and the inequality (5.83) results that |π(t, x)| ≤

|x|

1−m

for all x ∈ X and t ≥ 0, where

= (αβ)

m−1

k−m+1

and b

= (α)

m−1

k−m+1

(β)

k−m+1

m − 1

k − m + 1

Analogously can be proved the second inequality. The theorem is completely

proved. 

Corollary 5.10 Let (X, R

, π) be an autonomous homogeneous (of order m >

1 ) dynamical system (X, R

, π) and the trivial motion of the dynamical system

(X, R

, π) is uniform asymptotic stable, then for every number k > m − 1 there

exists a continuous function V : X → R

which possesses properties 2.1-2.3 from

the theorem 5.32.

Proof. This assertion directly follows from Theorems 5.32 and 2.35. 

Theorem 5.33 Let a non-autonomous system h(X, T, π), ((Y, T, σ), hi be homo-

geneous of order m = 1. Then the conditions 1. and 2. are equivalent:

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Method of Lyapunov functions 211

1. there exist positive numbers N

and ν

(i = 1, 2) such that N

−ν

|x|≤

|π(t, x)| ≤N

−ν

|x| for all x ∈ X and t ≥ 0;

2. for each k > 0 there exists a continuous function V : X → R

satisfying the

following conditions :

2.1. V (λx) = λ

V (x) for all x ∈ X and λ > 0;

2.2. there exist positive numbers α ≥ 1 and β so that

α|x|

≤ V (x) ≤ β|x|

(5.84)

for every x ∈ X;

2.3. V

(x) = −|x|

for all x ∈ X.

Proof. Let us show that under the conditions of theorem from assumption 1.

follows 2. First of all let us show that the function V , deﬁned by equality (5.77) is

the unknown, in the case when T = R

. We note that by equality (5.77) is correctly

deﬁned the function V : X → R

, such that

−ν

|x|

≤ |π(t, x)|

≤ N

−ν

|x|

(5.85)

for all t ≥ 0, x ∈ X and, consequently, the integral in second member of equality

(5.77) is convergent uniformly w.r.t. x on every bounded set from X. In particularly,

the function V is continuous w.r.t. x ∈ X.

From (5.77) and (5.85) follows that

α|x|

≤ V (x) ≤ β|x|

(5.86)

for every x ∈ X, where α = N

(ν

−1

and β = N

(ν

−1

. From equality (5.79)

we have that V (λx) = λ

V (x) for every x ∈ X and λ > 0, and from (5.81) follows

the equality V

(x) = −|x|

Let us show that the inverse implication holds too . We now suppose that

the conditions 2.1-2.3 of Theorem hold. One denote by ψ(t) = V (π(t, x)), then the

equality (5.82) holds for all t ≥ 0. From condition 2.2 follows that |π(t, x)|

≥

ψ(t)

and, consequently,

(t) ≤ −

ψ(t) (5.87)

for all t ≥ 0. From (5.87) we have

V (π(t, x)) ≤ V (x)e

−

(5.88)

for any t ≥ 0 and x ∈ X. According to (5.86) and (5.88) we have

|π(t, x)| ≤ N

−ν

|x|

for any t ≥ and x ∈ X, where N

= (

)

and ν

βk

. Analogously can be

established the second inequality.

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212 Global Attractors of Non-autonomous Dissipative Dynamical Systems

If T = Z

, then we will deﬁne the function V : X → R

by equality

V (x) =

+∞

n=1

|π(n, x)|

. (5.89)

The series in second member of equality (5.89) is convergent uniformly w.r.t. x on

every bounded subset from X, because under the conditions of Theorem we have

|π(n, x)|

≤ N

−νkn

|x|

(5.90)

for all n ∈ Z

and x ∈ X and, consequently, the series (5.89) is majored by

geometric series with the denominator q = e

−νk

< 1. Thus the function V deﬁned

by equality (5.89) is continuous.

From the homogeneity of (X, T, π) of order m = 1 and equality (5.89) follows

that V (λx) = λ

V (x) for all λ > 0 and x ∈ X. According to (5.89) and (5.90) we

have

α|x|

≤ V (x) ≤ β|x|

(5.91)

for any x ∈ X, where α = 1 and β = N

(1 −e

−νk

)

−1

. Finally, from equality (5.89)

follows that V

(x) = V (π(1, x)) −V (x) = −|x|

for all x ∈ X.

We now will show that from the conditions 2.1–2.3 of Theorem (in the case, when

T = Z

) follows the estimation (5.84). In fact, we denote by ψ(n) = V (π(n, x)),

then from the conditions 2.2–2.3 we have

∆ψ(n) = ψ(n + 1) − ψ(n) = −|π(n, x)|

≤ −

ψ(n). (5.92)

We may assume without loss of generality that β > 1, then from (5.92) we have

V (π(n, x)) ≤ γ

V (x) (5.93)

for all x ∈ X and n ∈ Z

, where γ = 1 − β

−1

. From (5.91) and (5.93) follows

the inequality (5.84), if we suppose that N = (

)

and ν = −

ln(1 − β

−1

). The

theorem is proved. 

Deﬁnition 5.11 The trivial section of ﬁber bundle (X, h, Y ) is called globally

uniformly asymptotically stable, if

(1) for arbitrary ε > 0 there is δ(ε) > 0 such that |π(t, x)| < ε for all t ≥ 0 and

|x| < δ;

(2) lim

t→+∞

|π(t, x)| = 0 for all x ∈ X, moreover this equality holds uniformly in x on

the bounded subsets from X.

Corollary 5.11 Let h(X, T, π), ((Y, T, σ), hi be homogeneous of order m = 1.

Then the following conditions are equivalent:

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Method of Lyapunov functions 213

(1) the zero section of ﬁbering (X, h, Y ) is uniformly asymptotically stable, i.e.

(a) for all ε > 0 there exists δ(ε) > 0 such that |π(t, x)| < ε for all t ≥ 0 and

|x| < δ;

(b) lim

t→+∞

|π(t, x)| = 0 for all x ∈ X, moreover this equality holds uniformly

w.r.t. x on every bounded set from X;

(2) there exist positive numbers N and ν such that |π(t, x)| ≤ Ne

−νt

|x| for all t ≥ 0

and x ∈ X;

(3) for every k > 0 there exists a continuous function V : X → R

satisfying the

conditions 2.1-2.3 of Theorem 5.33.

Proof. This assertion follows from the Theorems 5.33 and 2.34. 

Remark 5.10 In the case, when the space X is ﬁnite dimensional Theorem 5.32

and Corollary 5.10 (for autonomous system with T = R

) generalize and reﬁne

some results of V.I.Zubov (see, for example

[

337

]

, Theorems 36 and 37).

5.7 Diﬀerentiable homogeneous systems

Let H be a Hilbert space with scalar product h·, ·i and the norm |·| =

h·, ·i. We de-

note by C(E, B)(C

(E, B)) the space of all continuous ( diﬀerentiable continuous )

functions deﬁned on the space E wit values in some Banach space B.

The function f : E × Ω → F is called homogeneous of order k w.r.t. variable

u ∈ E if the equality f(λx, ω) = λ

f(x, ω) holds for all λ > 0, x ∈ E and ω ∈ Ω.

Lemma 5.6 The following assertions hold.

(1) Let Ω be a compact, f ∈ C(E × Ω, F ), f(0, ω) = 0 for all ω ∈ Ω, f (λu, ω) =

f(u, ω) (for every λ > 0and (u, ω) ∈ E ×Ω)). Then there exists M > 0 such

that |f(u, ω)| ≤ M|u|

for all (u, ω) ∈ E × Ω.

(2) If the function f ∈ C

(E × Ω, F ) is homogeneous (of order m ≥ 1), then

the function D

f(·, ω) : E → L(E, F ) (ω ∈ Ω) will be homogeneous (of order

m−1), where L(E, F ) is the space of all linear continuous operators A : E → F ;

(3) The function f ∈ C

(E, F ) is homogeneous (of order m) if and only if f satisﬁes

the equation Df(x)x = mf(x) for all x ∈ E, where Df(x) is the derivative of

Frechet of function f ∈ C

(E) in the point x;

Proof. First of all we note that in conditions of Lemma 5.6 there exists δ

> 0

such that |f(u, ω)| ≤ 1 for all |u| ≤ δ and ω ∈ Ω. If we suppose that it is not so,

then there exist δ

→ 0, |u

| < δ

and ω

∈ Ω such that |f(u

, ω

)| > 1. Since Ω is

compact we may assume that the sequence {ω

} is convergent. Let ω

→ ω

, then

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 5

214 Global Attractors of Non-autonomous Dissipative Dynamical Systems

according to continuity of f we have |f(0, ω

)| ≥ 1. On the other hand f(0, ω

) = 0.

The obtained contradiction proves the required assertion.

Thus there exists δ > 0 such that |f (u, ω)| ≤ 1 for all |u| ≤ δ and ω ∈ Ω and,

consequently, we have

|f(u, ω)| ≤ |f(

|u|

uδ

|u|

, ω)| =

|u|

|f(

δu

|u|

, ω)| ≤

|u|

Let f ∈ C

(E, F ) and f be homogeneous (of order m), then

f(x + h) − f (x) = Df (x)h + r(x, h) (5.94)

and |r(x, h)| → 0 as |h| → 0. We now will replace x by λx in equality (5.94), then

we obtain

f(λx + h) − f (λx) = Df(λx) + r(λx, h)

and, consequently,

f(x + u) −f(x) = λ

−m+1

Df (λx)h + λ

−m

r(λx, λu) (5.95)

for all λ > 0, where u = λ

−1

h. From (5.95) follows, that λ

−m+1

Df (λx) = Df (x),

i.e. Df(λx) = λ

m−1

Df (x) for any λ > 0.

Finally, let us prove the third assertion of Lemma. Let f ∈ C

(E, F ) be a

homogeneous function (of order m). Having diﬀerentiated the identity f(λx) =

f(x) w.r.t. λ and taking λ = 1 one obtain the identity Df (x)x = mx. Let now

the identity Df(x)x = mx take place. We denote by ϕ(λ) = λ

−m

f(λx)(λ > 0).

Thus Df (λx)λx = mf (λx), then

(λ) = −mλ

−m+1

f(λx) + λ

−m

Df (λx)x =

−mλ

−m−1

f(λx) + λ

−m−1

mf(λx) = 0

for all λ > 0 and, consequently, ϕ(λ) = const for all λ > 0. Thus ϕ(1) = f(x), when

ϕ(λ) = f (x) for any λ > 0, i.e. f(λx) = λ

f(x) for every λ > 0 and for all x ∈ E.

The lemma is proved. 

The function f ∈ C(E) is said to be regular, if for the diﬀerential equation

= f(x) (5.96)

the conditions of existence, uniqueness on R

and continuous dependence on initial

data are fulﬁlled, i.e. by equation (5.96) it is generated a semigroup dynamical

system (E, R

, π), where π(t, x) is the solution of equation (5.96) with initial con-

dition π(0, x) = x.

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Method of Lyapunov functions 215

Theorem 5.34 Let f ∈ C

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

then the following conditions are equivalent:

1. the zero solution of equation (5.96) is uniform asymptotic stable;

2. for all suﬃciently large k there exists a continuously diﬀerentiable function

V : E → R

satisfying the following conditions:

2.1. V (λx) = λ

k−m+1

V (x) for all λ ≥ 0 and x ∈ E;

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

k−m+1

≤ V (x) ≤

β|x|

k−m+1

for all x ∈ E;

2.3. V

(x) = DV (x)f (x) = −|x|

for any x ∈ E, where DV (x) is a derivative

of Frechet of function V in the point x.

Proof. We suppose that the zero solution of equation (5.96) is uniformly asymp-

totically stable. Denote by (E, R

, π) a semigroup dynamical system, generated

by equation (5.96). Thus, f ∈ C

(E, F ) is homogeneous (of order m > 1), then

according to Theorem 3.4

[

337

]

the equality π(t, λx) = λπ(λ

m−1

t, x) holds for all

x ∈ E, λ ≥ 0 and t ∈ R

, i.e. a dynamical system (E, R

, π) is homogeneous ( of

order m > 1 ). According to Theorem 5.32 and Corollary 5.10 by equality (5.95) it

is deﬁned a continuous function V : E → R

, satisfying the condition 2.1–2.3. of

Theorem 5.34. Let us show that under conditions of Theorem 5.34 function V will

be continuously diﬀerentiable and that the equality V

(x) = DV (x)x holds. To this

aim we will formally diﬀerentiate the equality (5.77) w.r.t. x, then we will have

DV (x)u =

+∞

k|π(t, x)|

k−2

RehD

π(t, x)u, π(t, x)idt. (5.97)

Now we will show that the integral ﬁguring in the second hand of formula (5.97),

is uniformly convergent w.r.t. x and u on the every ball from E and, consequently,

the equality (5.97) really deﬁnes a derivative of function V . We note that operator-

function U(t, x) = D

π(t, x) satisﬁes the operational equation

= A(t, x)X,

where A(t, x) = D

f(π(t, x)), and initial condition U(0, x) = I (I−is identity

operator in E). According to Lemma 5.6 the function A(t, x) = D

f(π(t, x)) is

homogeneous w.r.t. π(t, x) of order m − 1 and there exists a number M > 0 such

that

kA(t, x)k ≤ M|π(t, x)|

m−1

(5.98)

for all x ∈ E and t ≥ 0. Since the zero solution of equation (5.96) is uniformly

asymptotically stable, in virtue of Corollary 5.10 there exist positive numbers a and

b such that

|π(t, x)| ≤ (a|x|

1−m

+ bt)

1−m

(5.99)

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216 Global Attractors of Non-autonomous Dissipative Dynamical Systems

for any x ∈ E and t ≥ 0 and, consequently, from (5.98)– (5.99) we have

kA(t, x)k ≤ M(a|x|

1−m

+ bt)

−1

. (5.100)

Since

kU(t, x)k ≤ e

kA(τ,x)k,dτ

, (5.101)

from (5.100) and (5.101) we have

kU(t, x)k ≤ (a|x|

1−m

+ bt)

−1

|x|

m−1

)

(5.102)

for every t ≥ 0 and x ∈ E. By virtue of Inequality (5.102) we obtain

|π(t, x)|

k−2

|RehD

π(t, x)u, π(t, x)i| ≤ |π(t, x)|

k−1

π(t, x)k|u| (5.103)

≤ (a|x|

1−m

+ bt)

−

k−1

m−1

−

|x|

(m−1)

|u|

for all t ≥ 0 and x, u ∈ E. From estimation (5.103) follows that the integral in

second member of equality (5.97) is uniformly convergent w.r.t. x and u on the each

ball from E for suﬃciently large k and, consequently, the function V , deﬁned by

equality (5.77), under the conditions of Theorem 5.34 is continuously diﬀerentiable.

We now note, that

(x) =

V (π(t, x))



t=0

= DV (π(t, x))f (π(t, x))



t=0

= DV (x)f (x)

for all x ∈ E. On the other hand, according to Theorem 5.32 V

(x) = −|x|

. For

ﬁnishing the proof of the theorem it is suﬃcient to notice that in conformity with

Theorem 5.32 the inverse statement holds, i.e. conditions 2.1–2.3 of Theorem 5.34

imply 1. The theorem is completely proved. 

Now let f ∈ C

(E × F, E) and Φ ∈ C

(F, F ). Consider the diﬀerential system

(

= f (u, ω)

= Φ(ω)

(ω ∈ Ω), (5.104)

where Ω is a certain diﬀerentiable compact sub-manifold in F, f and Φ are regular

functions, i.e. for the diﬀerential system (5.104) and equation

= Φ(ω) (5.105)

are fulﬁlled the conditions of existence, uniqueness and continuous dependence on

initial data on the R

for (5.104) and on the R for (5.105). We denote by (Ω, R, σ)

a group dynamical system, generated by equation (5.105), then the system (5.104)

may be written in the form of non-autonomous equation

= f(u, ωt), (ω ∈ Ω) (5.106)