On integral separation of nonautomous differential equations

Thesis Assurance i1.1 Definition and main properties of characteristic exponents 3 1.2 The Lyapunov spectrum of a linear system.. • Continuity of the Lyapunov exponent The thesis consits

HANOI PEDAGOGICAL UNIVERSITY 2

DEPARTMENT OF MATHEMATICS

Nguyen Thi Thu Huyen

ON INTEGRAL SEPARATION OF NONAUTONOMOUS DIFFERENTIAL EQUATIONS

GRADUATION THESIS

Hanoi – 2019

HANOI PEDAGOGICAL UNIVERSITY 2

DEPARTMENT OF MATHEMATICS

Nguyen Thi Thu Huyen

ON INTEGRAL SEPARATION OF NONAUTONOMOUS DIFFERENTIAL EQUATIONS

GRADUATION THESIS

SUPERVISOR: Assoc.Prof- Dr.Sc Doan Thai Son

Hanoi – 2019

Thesis Acknowledgement

I would like to express my gratitudes to the teachers of the

Depart-ment of Mathematics, Hanoi Pedagogical University 2, the teachers in

the Analytics group as well as the teachers involved The lecturers

have imparted valuable knowledge and facilitated for me to complete

the course and the thesis

In particular, I would like to express my deep respect and gratitude

to Assoc.Prof- Dr.Sc Doan Thai Son, who has direct guidance, help me

complete this thesis

Due to time, capacity and conditions are limited, so the thesis can

not avoid errors Then, I look forward to receiving valuable comments

from teachers and friends

Thesis Assurance

I assure that the data and the results of this thesis are true and not

identical to other topics I also assure that all the help for this thesis

has been acknowledged and that the results presented in the thesis has

been identified clearly

Ha Noi, May 5, 2019

Nguyen Thi Thu Huyen

Thesis Assurance i

1.1 Definition and main properties of characteristic exponents 3

1.2 The Lyapunov spectrum of a linear system 11

2.1 Characterization of continuity of Lyapunov spectrum 23

2.1.1 Simple Lyapunov spectrum 23

2.1.2 One point spectrum 25

2.2 Explicit examples 27

Theory exponent Lyapunov has a long history and is known to be an

important tool in studying the stability of differential equations

Specifi-cally, Lyapunov’s exponent measures the separation velocity of the

near-derivative solutions of the differential equation and when the exponent

is negative, the solutions converge when the time is infinite In fact,

there are many equations that we do not know exactly the vector field

and then the question is how the Lyapunov exponent will change if the

vector field of the system is small One of the profound results

associ-ated with this problem is integral separation which is a necessary and

sufficient condition for the continuity of the Lyapunov exponent when

these exponents separate Within the framework of the thesis, student

wished to present systematically the problem of continuity of the

expo-nent Lyapunov and its relation to integral separation To do this, the

thesis will focus on:

• Introduction sketchy to the Lyapunov exponent and basis results

• Continuity of the Lyapunov exponent

The thesis consits of two chapter :

Chapter 1: Lyapunov exponent and basis results

Chapter 2: Continuity of the Lyapunov exponent

I sincerely thanks Assoc.Prof- Dr.Sc Doan Thai Son devotedly

in-structed the author to read the materials and practice the research I

would also like to thank Assoc.Prof- Dr.Sc Doan Thai Son gave detailed

comments on how to present some results in the thesis

I sincerely thank the teachers of the Mathematics Department at

Hanoi Pedagogical University 2, especially the Analysis Team, for

cre-ating favorable conditions for me in the process of studying at the

Uni-versity and implementing the thesis

Ha Noi, 06/05/2019

Author of the thesis

Nguyen Thi Thu Huyen

Lyapunov exponent and basic

results

The aim of this chapter is to introduce the notion of Lyapunov exponent

of an arbitrary function Next, we study the Lyapunov spectrum of a

linear nonautonomous differential equation We refer the reader to [2, 3]

for more details

exponents

We start this section by giving a definition of the Lyapunov

character-istic exponent of a non-zero function Roughly speaking this exponent

measures the growth of this function in comparison with the exponential

function

Definition 1.1.1 (Lyapunov Characteristic exponent) For any

non-zero function f : R → R, its Lyapunov characteristic exponent (for short

characteristic exponent) is defined as

4 For a non-zero constant function c, we have χ[c] = 0

Now, we study the property of the Lyapunov characteristic exponent

of a function given as a sum or a product of a finite number of functions

For this purpose, we need the following technical lemma

Lemma 1.1.2 Let f : R → R be a non-zero function Then, χ[f ] =

α ∈ (−∞, ∞) if and only if for any ε > 0 the following two conditions

fix ε > 0 By (1.1), there exists T > 0 such that

(Sufficiency) Suppose that two statements (1) and (2) hold We will

show that χ(f ) = α By condition (1), for any ε > 0 there exists T such

that

|f (t)| < e(α+ε)t for all t ≥ T

of limit, for sufficient large k we have

Theorem 1.1.3 (Characteristic exponent of a sum of finite number

of functions) Let f1, , fn : R → R be non-zero functions Define

Then the following statements hold:

(i) The Lyapunov characteristic exponent f does not exceed the greatest

of the Lyapunov characteristic exponents of f1, , fn

(ii) If only one functions has the greatest exponent of f1, , fn then

Since ε is arbitrary, we get χ[f ] 6 α

(ii) By definition of α and the assumption only one functions has the

greatest exponent of f1, , fn, there exists ` ∈ {1, , n} such that

lim sup

m→∞

|f`(t)|

e(α−ε)t m = ∞

We have

The second term on the right-hand side of the last inequality tends to

zero and the first tends to infinity as m → ∞ By properties of limit,

for sufficient large m we obtain

!

> α − ε,Which together with (i) implies that

The proof is complete

Example 1.1.4 Using Theorem 1.1.3, we obtain that

χ[e2t+ e3t + e−t] = 3

Theorem 1.1.5 The characteristic exponent of the product of a finite

number of function does not exceed the sum of the characteristic

expo-nents of the cofactors, i.e., for any non-zero functions f1, , fn : R → R

Proof We assume that +∞ and −∞ are not simultaneously present

among characteristic exponents A direct computation yields that

= lim sup

t→∞

1t

The proof is complete

Example 1.1.6 1)χ[et.e−3t] = χ[e−2t] = −2 and χ[et.e−3t] = χ[et] +χ[e−3t] = 1 − 3 = −2 In this case a strict equality is realized in (2.2)2)χ[et sin t.e−t sin t] = χ[1] = 0

At the same time χ[et sin t.e−t sin t] < χ[et sin t] + χ[e−t sin t] = 2

In this case a strict inequality is realized in (2.2)

Next, we define the characteristic exponent of a matrix-valued

func-tions Throughout the content of this thesis, we restrict our attention to

only matrix-valued functions taking the value in Rn×n

Definition 1.1.7 (Lyapunov Characteristic Exponent of a

Matrix-Val-ued Function) For any matrix F (t) = {fij(t)}i,j=1, ,n, its Lyapunov acteristic exponent (for short characteristic exponent) is defined as:

char-χ[F ] = max

i,j χ[fij]

Now, we show that the characteristic exponent of a matrix-valued

function coincides with the characteristic exponent of its norm To do

this, we recall that all norms on a finite-dimensional linear space are

equivalent Then, in the following we endow Rn with the max norm, i.e.,

kAk = max

1≤i,j≤n|aij|

Theorem 1.1.8 The charactertistic exponent of a finite dimensional

matrix F (t) coincides with the characteristic exponent of its norm, i.e.,

χ[F ] = max

i,j χ[fij] 6 χ[kF (t)k] (1.3)

On the other hand,

kF (t)k 6 X

i,j

|fij(t)| Hence,

χ[F ] = χ[kF k]

The proof is complete

Consider a linear nonautonomous differential equation

Our aim in this subsection is to examine the characteristic exponents

of the solution of (1.4) To do this, we first prove the existence of the

global solution of (1.4)

Theorem 1.2.1 (Global unique and existence of solution for linear

nonautonomous differential equations) Let x0 ∈ Rn be arbitrary Then,system (1.4) with the initial value condition x(0) = x0 has a uniquesolution.

To prove the above theorem, we need to show the existence of a

unique continuous function x : [0, ∞) → Rn satisfying the followingintegral equation

x(t) = x0 +

Z t 0

The idea is to find a suitable contraction operator on a suitable space

such that its unique fixed point of this operator gives rise to the solution

of (1.5) Now, choose and fix an arbitrary γ > 0 Let

Lemma 1.2.2 The space (Cγ, k.kγ) is a Banach Space

Proof Firstly, we verify that (Cγ, k.kγ) is norm space Indeed, we checkthree properties of a norm space:

kf kγ = 0 ⇔ sup e−γtkf (t)k = 0 ⇔ f (t) = 0 ∀t ∈ [0; +∞)

kf + gkγ = sup e−γtkf + gk 6 sup e−γt(kf k + kgk)

6 sup e−γtkf k + sup e−γtkgk

= kf kγ + kgkγ ∀t ∈ [0; +∞) ,kαf kγ = sup e−γtkαf k = α sup e−γtkf k = αkf kγ, ∀t ∈ [0; +∞)

To complete the proof, we verify that the norm space (Cγ, k.kγ) iscomplete Now let Let {fn} be a Cauchy sequence in (Cγ, k.kγ) Then, weneed to construct f∗ ∈ Cγ such that lim fn = f∗ We do this throughoutthe following steps:

Step 1: For a fixed t ∈ [0, ∞) We will show that {fn(t)} is a cauchysequence in R Indeed, since {fn} is Cauchy in (Cγ, k.kγ), ∀ε > 0, ∃N ∈ Nsuch that ∀n, m > N , we have:

that f∗ is continuous at t For this purpose, let {tm}∞m=1 be a sequencesuch that lim

m→∞tm = t We need to show that lim

m→∞f∗(tm) = f∗(t).Indeed, let ε > 0 be arbitrary but fixed Since (fn) is a Cauchy sequence

in Cγ, there exists K ∈ N such that

which together with continuity of the function fK proves the continuity

of the function f∗ Similarly, we aslo have kf∗kγ < ∞ Therefore,

f∗ ∈ Cγ

Step 3: To complete the proof, we verify that

lim

n→∞kfn− f∗kγ = 0Since {fn} is Cauchy sequence, so ∀ε > 0, ∃N ∈ N,such that ∀n, m > N,

kfn− f∗kγ < ε for all n ≥ N.

Consequently,

lim

n→∞kfn− f∗kγ = 0The proof is complete

We are now in the position to prove the global uniqueness existence

Tf ∈ Cγ Since A(s)f (s) is continuous function, then by the mental Theorem of Calculus, Tf(t) is a also continuous Morover, wehave:

eγskf kγ ds < ∞,

which implies that kTf(t)kγ < +∞

Step 2 We will show that for a suitable choice of γ, the linear operator

T is contractive Indeed, using the same estimate as in Step 1 we have

kTf − Tgkγ ≤ M sup

t≥0

e−γt

Z t 0

eγskf − gkγ ds

γ kf − gkγ,which implies that if γ > M the operator T is contractive

Step 3 So far, T is contractive therefore by Banach Fixed Point

The-orem, T has a unique fixed point To conclude the proof, we will show

that ξ(t) is solution of (1.4) satisfying x(0) = x0 if and only if ξ(t) is afixed point of T

(Necessity) Suppose ξ(t) is solution of (1.4) with ξ(0) = x0, i.e., ˙ξ(t) =A(t)ξ(t)

Thus, ξ(t) is a fixed point of T

(Sufficiency) Suppose that ξ(t) satisfies:

Therefore, ξ(t) is solution of (1.4) with ξ(0) = x0.

This complete the proof

So far, we have proved that for any initial valued problem x(0) = x0equation (1.4) has a unique solution In the remainning of this chapter,

we examine the characteristic exponents of solutions of (1.4) Firstly,

we show that the characteristic exponent of a non-zero slution of (1.4)

is finite

Theorem 1.2.3 Any non-trivial solution x(t) of the linear system (1.4)

has a finite characteristic exponent, i.e −M ≤ χ[x] ≤ M

Proof Let Rn be endowed with the Euclidean norm A driect tion yields that

t

Z

t0

2M dτ

−M 6 lim sup

t→∞

1

t ln kx(t)k 6 M

That means −M 6 χ[x] 6 M The proof is complete

Next, we show that the set of Lyapunov characteristic exponents of all

non-zero solutions of (1.4) has no more than n elements Before proving

this fact, we need the following technical lemmas

Lemma 1.2.4 Vector functions x1(t), , xm(t) defined on [0; ∞) andhaving different finite characteristic exponents are linearly independent

Proof Let us consider the linear combination of these vectors with a

nontrivial set of coefficient is

m

P

i=1

cixi(t) According to Theorem 1.1.3,the characteristic exponent of this sum is equal to max

i χ[χi(t)] That is,χ[

m

X

k=1

ckxk(t)] = χ[0] = −∞

This is contradiction Therefore, {xi}m

i=1 is linearly independent

Lemma 1.2.5 System (1.4) has n solutions x1(t), , xn(t) which arelinearly independent

Proof Suppose the contrary, i.e., there exists n+1 solutions x1(t), , xn+1(t)

of (1.4) which are linearly dependently Then, there exists c1, , cn+1such that

Hence,

ξ(t) ≡ x1(t),

c1x1(t) + c2x2(t) + + cn+1xn+1(t) = 0.

Therefore,{xi}n+1i=1 is linearly independent The proof is complete

Theorem 1.2.6 The number of elements of the spectrum doew not

Thus, spectrum of (1.4) consists of finite element α1 6 α2 6 6

αm, (m 6 n), where αi = χ[xi(t)] So, the number of elements of thespectrum does not exceed n

Continuity of the Lyapunov

where Q ∈ C([0; +∞)), with sup kQ(t)k 6 σ The Lyapunov spectrum

of (2.3) is denoted by with the spectrum:

−∞ < λ10 6 λ2 0

Under the influence of perturbation Q(t) , the characteristic exponents

of system (2.3) vary, generally speaking, discontinuously

Definition 2.0.1 The characteristic exponents of system (2.1) are said

to be stable if for any ε > 0 there exists a σ > 0 such that the inequality

Our aim in this chapter is to give a characterization of the stability of

Lyapunov spectrum of a linear planar system We distinguish the cases

simple spectrum, i.e., λ1 < λ2 and one point spectrum, i.e λ1 = λ2

(Section 2.2)

spec-trum

2.1.1 Simple Lyapunov spectrum

Firstly, we have the following result for a sufficient condition of continuity

of simple Lyapunov spectrum

Theorem 2.1.1 If system (2.1) has different characteristic exponent

λ1 < λ2, then their stability follows from the existence of a Lyapunovtransformation x = L(t)z of this system to a diagonal system

˙z = diag[p1(t), p2(t)]z = P (t)z (2.6)

Where the functions p1(t) and p2(t) are integrally separared, i.e.,

It turns out the integral separatedness is also a necessary condition

for stability of simple Lyapunov spectrum

Theorem 2.1.2 If the exponents λ1 < λ2 of a two dimensional system:

λ1 = lim

t→∞

1t

t

Z

0

p2(τ )dτ

Proof The idea of the proof is as follows Assume that the diagond is

not integrally separated Fixing β ∈ (λ1 λ2), we show that under thisassumption for any σ > 0 the perturbation:

kQ(t)k < σ

Can be chosen such that there exists a solution y(t) of the perturbed

sytem with x[y] = β, and this contradicts the stability of the

expo-nents The method for constructing these perturbations was developed

by Millionshchikov and is called the medthod of rotations or the method

of perturbations by rotations

2.1.2 One point spectrum

We pass to the case of a two-dimensional diagonal system with equal

characteristic exponent To formulate the characterization of continuity

of the spectrum in this case, we first need to introduction the notion of

where t > s > 0 and the quantities dr,ε, DR,ε do not depend on growth

of the solutions from below and from above, respectively

Definition 2.1.4 The number Ω defined as:

Ω = inf

R { lim

t→∞

1t

t

Z

0

is called the upper central exponent of system (2.1) Here the infimum

is taken over the set of all the upper functions R of system

Definition 2.1.5 The number ω defined as

t

Z

0

Here r(t) is the set of all the lower functions of system (2.1)

Theorem 2.1.6 If a linear two-dimensional has equal characteristic

exponents λ1 = λ2 = λ, then they are stable if and only if

ω = λ = Ω

Proof Necessity Let the exponent of the system be stable but:

max{Ω − λ, λ − ω} = γ > 0

For the sake of definiteness, we assume that Ω − λ = γ > 0 According

to Millionshchikov’s Theorem (see [1]), there exists a perturbation Q(t)

of the linear system such that

kQ(t)k < σ, t > 0

Where σ is arbitrary small, and the perturbed system has a

character-istic exponent λ0 > Ω This contradicts the stabililty of the exponents.Similar arguments can be carried out for the case λ − ω = γ > 0 The

fact that ω is attainable was established by Millionshchikov

Hence

Ω − λ = λ − ω = 0

Or

ω = λ = Ω

Consider a linear nonautonomous differential equation

Our aim in this subsection is to examine the characteristic exponents

of the solution of (1.4) To this, we... the remainning of this chapter,

we examine the characteristic exponents of solutions of (1.4) Firstly,

we show that the characteristic exponent of a non-zero slution of (1.4)

is... is called the medthod of rotations or the method

of perturbations by rotations

2.1.2 One point spectrum

We pass to the case of a two-dimensional diagonal system with equal