Linearization and hyperbolicity of the autonomous nonlinear equations

DEPARTMENT OF MATHEMATICS————oOo———— NGUYEN THI KIM OANH LINEARIZATION AND HYPERBOLICITY OF THE AUTONOMOUS NONLINEAR EQUATIONS GRADUATION THESIS HANOI, 01/2019... DEPARTMENT OF MATHEMATI

DEPARTMENT OF MATHEMATICS

————oOo————

NGUYEN THI KIM OANH

LINEARIZATION AND HYPERBOLICITY OF

THE AUTONOMOUS NONLINEAR

EQUATIONS

GRADUATION THESIS

HANOI, 01/2019

DEPARTMENT OF MATHEMATICS

————oOo————

GRADUATION THESIS

LINEARIZATION AND HYPERBOLICITY OF

THE AUTONOMOUS NONLINEAR

EQUATIONS

Supervisor : Dr TRAN VAN BANG

HANOI, 01/2019

Thesis Assurance

I assure for this is my research thesis which is completed under theguidance of Dr Tran Van Bang The results presented in the thesis arehonest, and have never been published in any other thesis

Student

Nguyen Thi Kim Oanh

Before presenting the main content of the thesis, I would like to press my gratitude to the mathematics teachers, Hanoi Pedagogical Uni-versity 2, teachers in the Analysis group as well as the teachers involved.Teaching has dedicatedly conveyed valuable knowledge and created fa-vorable conditions for me to successfully complete the course and thesis

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

to Dr Tran Van Bang, who directly instructed, just told to help me sothat I could complete this thesis

Due to limited time, capacity and conditions, the discourse cannotavoid errors Therefore, I look forward to receiving valuable commentsfrom teachers and friends

Student

Nguyen Thi Kim Oanh

Differential equations are an important discipline of mathematics andhave many applications in the fields of science and technology, which areconsidered as the bridge between theory and application Thus, the dif-ferential equation is a subject that is widely taught in universities athome and abroad

Stability theory is one of the important qualitative properties in thestudy of differential equations For linear systems of equations, we have

an explicit criterion for studying the stability of trivial solution of thesystem of equations by examining the sign of the real part of matrix [A]

is eigen values However, many differential equations are expressed innon-linear forms, for instance in quasi linear one- the defferential equa-tion system which has the form x0 = A(t)x + f (t), where [A]n×n is asquare matrix

A fairly useful method for studying the stability of nonlinear systems

is the linearization method, we perform a transformation convert linear defferential equations to linear form Then, stability of nonlinearsystems solutions is evaluated through the stability of linearized sys-tems solutions Through this thesis, I focus on the stability of nonlinearsystems based on the linearization method, while studying the stabilitycharacteristics for solution of differential equations based on the stablemanifold of that system

non-1 PRELIMINARIES 3

1.1 Differential equation 3

1.2 Flows 4

1.3 Limit sets and trajectories 7

1.4 Stability 8

2 LINEARIZATION AND HYPERBOLICITY 11 2.1 Poincare’s Linearization theorem 11

2.2 Hyperbolic stationary points and the manifoid theorem 17 2.3 Persistence of hyperbolic stationary points 22

2.4 Structural stability 23

2.5 Nonlinear sink 24

2.6 The proof of the stable manifold theorem 29

In this chapter, we discuss about the differential equation including

of flows, trajectory and the stability

1.1 Differential equation

Consider the differential equation in the form

˙x = f (x, t), x ∈ Rn, f : Rn×R →Rnwhere the dot denoted by the differentiation with respect to time t Aparticularly simple example of differential equation is the linear differ-ential equation

where A is an n × n matrix with constant coefficients With the initialcondition at t = 0 is x0, the equation (1.1) has solutions x = etAx0,Where etA =

∞

P

k=0

(tA)kk! = I + tA + (tA)

2

2! + + (tA)

k

k! +

Theorem 1.1.1 (Local existence and uniqueness )

Suppose ˙x = f (x, t) and f : Rn ×R −→ Rn is continuously tiable Then there exists maximal t1 > 0, t2 > 0 such that a solution x(t)

differen-with x(t0) = x0 exists and is unique for all t ∈ (t0 − t1, t0 + t1).

Theorem 1.1.2 (Continuity of solutions )

Suppose that f is Cr (r times continuously differentiable) and r ≥ 1,

in some neighbourhood of (x0, t0) Then there exists  > 0 and δ > 0 suchthat if |x0−x0| <  there is a unique solution x(t) defined on [t0−δ, t0+δ]with x(t0) = x0 Solutions depend continuously on x0 and on t

In this section, we see that solutions to differential equations can

be represented as curves in some appropriate space Consider the tononoous equation

Definition 1.2.1 The curve (x1(t), , xn(t)) in Rn is an integral curve

of equation (1.2) iff

( ˙x1(t), , ˙xn(t)) = f (x1(t), , xn(t))for all t ∈ I On the other words, (x1(t), , xn(t)) is solution of (1.2)

on I Thus the tangent to the integral curve at (x1(t0), , xn(t0)) is

f (x1(t0), , xn(t0))

Definition 1.2.2 Consider ˙x = f (x) The solution of this differentialequation defines a flow, ϕ(x, t) satisfies ϕ(x, t) is solution of the equation(1.2) with the initial condition x(0) = x

Hence

d

dtϕ(x, t) = f (ϕ(x, t))

for all t and ϕ(x, 0) = x.

Then the solution x(t) with x(0) = x0 is ϕ(x0, t)

Lemma 1.2.1 (Properties of the Flow)

(i) ϕ(x, 0) = x;

(ii) ϕ(x, t + s) = ϕ(ϕ(x, t), s) = ϕ(ϕ(x, s), t) = ϕ(x, s + t)

Example 1.2.1 Consider the equation

˙x = Ax with x(0) = x0.The solution of equation is x = x0etA Then the flow ϕ(x0, t) = x0etA.Hence the flow ϕ(x, t) = xetA

We will go to check properties of the flow in this case,we have:

Example 1.2.2 Consider the equation







˙x = −xx(0) = x0

We have x is stationary point iff x = ϕ(x, t)

⇔x = xe−t∀t

⇔x(e−t − 1) = 0 ∀t

⇔x = 0

Hence, the flow has unique stationary point, that is x = 0

Definition 1.2.4 A point x is periodic of (minimal) period T iff







ϕ(x, t + T ) = ϕ(x, t) ∀tϕ(x, t + s) 6= ϕ(x, t) f or all 0 ≤ s < T

The curve Γ = {y|y = ϕ(x, t), 0 ≤ t < T } is called a periodic orbit ofthe differential equation and is a closed curve in phase space

Example 1.2.3 Consider the differential equations

The characteristics equation is λ2+1 = 0 ⇒ λ = ±i Hence, the solution

of this ordinary differential equation is x1 = c1cost + c2sint This implies

that x2 = c2 cost-c1 sint Using the unitial condition, we get

1.3 Limit sets and trajectories

Consider ˙x = f (x) with x(0) = x0, or equivalently the flow ϕ(x0, t)

Definition 1.3.1 The trajectory through x is the set γ(x) = S

t∈R

ϕ(x, t)and the positive semi-trajectory, γ+(x), and the negative semi-trajectory,γ−(x)

x, A(x), are the sets

Λ(x) = {y ∈ Rn|∃tn with tn → ∞ and ϕ(x, tn) → y as n → ∞}

and A(x) = {y ∈ Rn|∃sn with sn → −∞ and ϕ(x, sn) → y as n →

∞}

In other words the w−limit set, Λ(x) is the set of points which x tends

to (i.e the limit points of γ+(x)), and the α−limit set, A(x) is the set ofpoints that trajector, through x, tends to in backward time

Example 1.3.1 Consider the differential equation in the Example 1.2.3

We can verify that for all x ∈ R2, the limit set Λ(x) and the α−limit setare the same, ∂B(0, kxk) Indeed,

, ∀t ≥ 0

Definition 1.4.2 A point x is quasi-asymptotically stable (tends toeventually) iff there exsist δ > 0 such that if |x − y| < δ then

|ϕ(x, t) − ϕ(y, t)| −→ 0, as t −→ ∞

Definition 1.4.3 A point x is asymptotically stable (tends to directly)iff it is both Liapounov stable and quasi-asymptotically stable.

Note: If a stationary point is asympotically stable then there mustexists a neighbourhood of the point such that all points in this neigh-bourhood tend to the stationary point

The largest neighbourhood for which this is true is called the domain

of (asymptotic) stability of this point

Definition 1.4.4 Let x be an asymptotically stable stationary point ofthe equation ˙x = f (x), so for all  > 0, there exists δ > 0 such that

|y − x| < δ ⇒ |ϕ(y, t) − x| < , ∀t ≥ 0and ∃δ > 0 such that |y − x| < δ ⇒ |ϕ(y, t) − x| → 0 as t → ∞,

Definition 1.4.5 (Normal forms )

Let P be an 2x2 matrix with a repeated real eigenvalue λ Then thecharacteristic polynomial of P is (s − λ)2 = 0 Since P satisfies its owncharacteristic equation, this implies that

(P − λI)2x = 0for all x ∈ R If λ is a double eigenvalue of P then there is a change ofcoordinates which brings P into one of the two cases

In both cases, solving the differential equation ˙x = Ax in this choice

of coordinates system is easy

If P has distinct eigenvalues, the matrix Λ is

Λ = diag(λ1, , λk, B1, , Bm)where (λi) are the real eigenvalues and Bj are the matrices

LINEARIZATION AND

HYPERBOLICITY

2.1 Poincare’s Linearization theorem

Consider the equation ˙x = f (x), f (0) = 0 and f is analytic onRn Wewant to determine condition for there to exist a change of coordinates

in some neighbourhood of the origin such that the defferential equation

in these new coordinates is the linear system ˙y = Df (0)y This problemdepends upon the eigenvalues of the matrix Df (0)

Definition 2.1.1 Suppost that the eigenvalues of Df (0) are (λ1, λn).Then

Df (0) is resonant if there exist non-negative integers (m1, mn) with

for some s ∈ {1, 2, n} The quantity |m| = Pn

1 mk is called the order

of the resonance

The problem associated with resonance is one of the convergence ofthe power series expansion of the new coordinates in terms of the old

Assum that this linear coordinate change has already been made, so if

Df (0) has distinct eigenvalues (which may be complex) then the dinate system (x1, xn) is such that

coor-λixi + higher order terms

then group together the higher order terms by order

When we repeat the argument for the terms of order r+1, having killedterms of order r, r + 1, successively, ˙y = Df (0)y Thus, we have

˙

xi = λixi+ X

m∈Mr

amixm + vr+1,i(x) + (2.4)and the coodinate change

Substituting for ˙xi from (2.4) gives

˙

y = λiyi+ Vr+1,i(y) + (2.14)are all terms of order r disappear

We can only choose this value of bmi provided

i.e if λ is not resonant of order r So provided that λ is not resonant oforder r, we can use a near identity change of coordinates with terms oforder r such that all the terms of order r are killed We can now repeat

this argument with the terms of order r + 1 and so on.

Exercise: Consider the case of systems in R2 and suppose thatDf (0)has eigenvalues λ1 and λ2 Then if λ1= 8 and λ2=-2 the system isresonant of order 6 since λ1=2 λ1+4 λ2 (or λ2=λ1+5λ2 ).Similarly, Ifi) λ1=1, λ2=-1, the system is resonant of oeder 3 because λ1=2λ1+ λ2

or λ2= λ1+2λ2;

ii) λ1=7, λ2=5, the system is not resonant, because there are are nointegers m1, m2 ≥ 0 such that m1 + m2 ≥ 2 and either

7 = m17 + m25 ⇔ (m1 − 1)7 + m25 = 0or

5 = m17 + m25 ⇔ m17 + (m2 − 1)5 = 0The change of coordinate is given, implicitly, by a formal power series.This only gives a true change of coordinate if the power series converges

in some neighbourhood of the origin

Poincar´e was able to prove that the power series of Theorem 2.1 verges if the eigenvalues (λi) are non-resonant and either Re λi > 0 for

|m| = Pn

1 mi ≥ 2

Thus the eigenvalue satisifiles Siegel’s condition if |λi− (m, λ)| is ciently far from zero.

suffi-Theorem 2.1.1 Suppose that ˙x = f (x), f(0)=0 and Df(0) is not nant Then if Df(0) is diagonal there exists a formal near identity change

reso-of coosdinates y=x+ for which ˙x = Df (0)

Theorem 2.1.2 (Poincar´e’s Linearization Theorem) If the eigenvalues(λi),i=1, ,n, of the linear part of an analytic vector field at a stationarypoint are non-resonant and either Re λi > 0 for i = 1, , n or Re λi < 0for i=1, ,n, or λi satisfies a Siegel condition, then the power series ofTheorem 2.1.1 converges on some neighbourhood of the stationary point.Example: Consider

˙x = x, ˙y = 2y + x2With linearization at the origin given by

˙x = x, ˙y = 2y The eigenvalues of the linearization are (λ1, λ2)=(1,2) and this is res-onant of order two since 2λ1 = λ2 Solution curves of the linearizedequation lie on solution of

dy

dx =

2yxwhich we can easily solve to obtain a family of parabolae,

multiplying through by the integrating factor x−2

Wlocs (x) = {y ∈ U |ϕ(y, t) → x as t → ∞, ϕ(y, t) ∈ U for all t ≥ 0}and

Wlocu (x) = {y ∈ U |ϕ(y, t) → x as t → −∞, ϕ(y, t) ∈ U for all t ≤ 0}The stable manifold theorem states that these manifolds exists and are ofthe same dimension as the stable and unstable manifold of the linearizedequation ˙y = Df (x)y if x is hyperbolic, and that they are tangential tothe linearized manifolds at x

Figure 2.1:

Definition 2.2.2 Suppose that x0 is a hyperbolic stationary point of

˙x = f (x) and let Df (0) denote the Jacobian matrix of f evaluated at

x0 Then x0 is a sink if all the eigenvalues of Df (0) have strictly tive real part and a source if all the eigenvalues of Df (0) have strictlypositive real parts.Otherwise x0 is a saddle

nega-In Sections 2.1.2 and 2.1 that for small perturbations of the definingequations, a source remains a source, a sink remains a sink and a saddleremains a saddle Furthermore, as one would expect, if x0 is a sink then

contain the nonlinear parts of the equation, together with their firstderivaties at the origin, (x, y) = (0, 0) Hence

Es(0, 0) = {(x, y)|x = 0} and Eu(0, 0) = {(x, y)|y = 0}

Since the stable and unstable manifolds are smooth and are tangential

to these manifolds at the origin, so the stable manifold is given by

xi = Si(y), i = 1, , nu (2.19)where

∂Si

∂yi(0) = 0, 1 ≤ i ≤ nu, 1 ≤ j ≤ ns. (2.20)Similary we can write the unstable manifold as

An example may make this clearer

Example: Consider the equations

˙x = x, ˙y = −y + x2This has a unique stationary point at (x, y) = (0, 0) and the equation inform near the stationary point, the linearized equation is

˙x = x, ˙y = −y

Giving a saddle at the origin

Es(0, 0) = {(x, y)|x = 0} and Eu(0, 0) = {(x, y)|y = 0}

By the stable manifold theorem we know that the nonlinear system has

a local unstable manifold of the form

y = U (x), ∂U

∂x(0) = 0and

U (x) = X

k≥2

ukxkNow

−u2 + 1 = 2u2, and − uk = kuk, k ≥ 3Hence u2 = 13, uk = 0 for k ≥ 3 and so

Wlocu (0, 0) = {(x, y)|y = 1

3x

3}

Theorem 2.2.1 (Stable manifold theorem)

Suppose that the origin is a hyperbolic stationary point for ˙x = f (x) and

Es and Eu are the stable and unstable manifolds of the linear system ˙x =

Df (0)x Then there exists local stable and unstable manifolds Wlocs (0)and Wlocu (0) of the same dimension as Es and Eu respectively Thesemanifolds are (respectively) tangential to Es and Eu at the origin and

as smooth as the original function f

Suppose that xo is a hyperbolic stationary point, then there are threepossibilities Either Wlocs (x0) = ∅, or Wlocu (x0) = ∅, or both manifoldsare non-empty These three possibilities are given names: x0 is called asource, sink or saddle respectively

Theorem 2.2.2 (Hartman’s theorem) If x=0 is a hyperbolic stationarypoint of ˙x = f (x) then there is a continuous invertible map,h, defined

on some neighbourhood of x = 0 which takes orbits of the nonlinear flow

to those of the linear flow exp(tDf (0)) This map can be chosen so thatthe parametrization of orbits by time is preserved

Note: The map is only continuous (not nesessary differentiable) and

so it does not distinguish between, for example, a logarithmic spiral andthe phase portrait obtainned when the Jacobian at the stationary pointhas real eigenvalues