HANOI PEDAGOGICAL UNIVERSITY 2
DEPARTMENT OF MATHEMATICS
————oOo————

NGUYEN THI KIM OANH

LINEARIZATION AND HYPERBOLICITY OF
THE AUTONOMOUS NONLINEAR
EQUATIONS

GRADUATION THESIS

HANOI, 01/2019


HANOI PEDAGOGICAL UNIVERSITY 2
DEPARTMENT OF MATHEMATICS
————oOo————

GRADUATION THESIS

LINEARIZATION AND HYPERBOLICITY OF
THE AUTONOMOUS NONLINEAR
EQUATIONS

Supervisor : Dr. TRAN VAN BANG
Student

:

NGUYEN THI KIM OANH

Class

: K41CLC

HANOI, 01/2019


1

Thesis Assurance
I assure for this is my research thesis which is completed under the
guidance of Dr. Tran Van Bang. The results presented in the thesis are
honest, and have never been published in any other thesis.
Student

Nguyen Thi Kim Oanh


1

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

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 so
that I could complete this thesis.
Due to limited time, capacity and conditions, the discourse cannot
avoid errors. Therefore, I look forward to receiving valuable comments

from teachers and friends.

Student

Nguyen Thi Kim Oanh


Linearization and hyperbolicity

NGUYEN THI KIM OANH

Preface
Differential equations are an important discipline of mathematics and
have many applications in the fields of science and technology, which are
considered as the bridge between theory and application. Thus, the differential equation is a subject that is widely taught in universities at
home and abroad
Stability theory is one of the important qualitative properties in the
study of differential equations. For linear systems of equations, we have
an explicit criterion for studying the stability of trivial solution of the
system of equations by examining the sign of the real part of matrix [A]
is eigen values. However, many differential equations are expressed in
non-linear forms, for instance in quasi linear one- the defferential equation system which has the form x = A(t)x + f (t), where [A]n×n is a
square matrix.
A fairly useful method for studying the stability of nonlinear systems
is the linearization method, we perform a transformation convert nonlinear defferential equations to linear form. Then, stability of nonlinear
systems solutions is evaluated through the stability of linearized systems solutions. Through this thesis, I focus on the stability of nonlinear
systems based on the linearization method, while studying the stability
characteristics for solution of differential equations based on the stable
manifold of that system.


1


Contents

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

References

32

2


Chapter 1
PRELIMINARIES
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 → Rn
where the dot denoted by the differentiation with respect to time t. A
particularly simple example of differential equation is the linear differential equation
x˙ = Ax,

(1.1)

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

∞
k=0

k

(tA)
k!

= I + tA +

(tA)
2!

2

k


+ ... +

(tA)
k!

+ ....

Theorem 1.1.1. (Local existence and uniqueness )
Suppose x˙ = f (x, t) and f : Rn × R −→ Rn is continuously differentiable. Then there exists maximal t1 > 0, t2 > 0 such that a solution x(t)
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Linearization and hyperbolicity

NGUYEN THI KIM OANH

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 C r (r times continuously differentiable) and r ≥ 1,
in some neighbourhood of (x0 , t0 ). Then there exists > 0 and δ > 0 such
that if |x −x0 | < there is a unique solution x(t) defined on [t0 −δ, t0 +δ]
with x(t0 ) = x . Solutions depend continuously on x and on t.

1.2

Flows

In this section, we see that solutions to differential equations can
be represented as curves in some appropriate space. Consider the Autononoous equation
x˙ = f (x), x ∈ Rn .


(1.2)

Definition 1.2.1. The curve (x1 (t), ..., xn (t)) in Rn is an integral curve
of equation (1.2) iff
(x˙ 1 (t), ..., x˙ n (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 differential
equation 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
ϕ(x, t) = f (ϕ(x, t))
dt
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Linearization and hyperbolicity

NGUYEN THI KIM OANH

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 = x0 etA . Then the flow ϕ(x0 , t) = x0 etA .
Hence the flow ϕ(x, t) = xetA .
We will go to check properties of the flow in this case,we have:
i, ϕ(x, 0) = xe0 = x.
ii, We have ϕ(x, t) = etA x.
Then ϕ(x, t + s) = xe(t+s)A ;
ϕ(ϕ(x, t), s) = ϕ(x, t)esA = xetA esA = xe(t+s)A ;
ϕ(ϕ(x, s), t) = ϕ(x, s)etA = xesA etA = xe(t+s)A
Therefore ϕ(x, t + s) = ϕ(ϕ(x, t), s) = ϕ(ϕ(x, s), t).
Definition 1.2.3. A point x is stationary point of the flow iff ϕ(x, t) = x,
for all t. Thus, at a stationary point f (x) = 0.

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Linearization and hyperbolicity

NGUYEN THI KIM OANH

Example 1.2.2. Consider the equation


x˙ = −x

x(0) = x
0

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) ∀t

ϕ(x, t + s) = 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 of
the differential equation and is a closed curve in phase space.
Example 1.2.3. Consider the differential equations


x˙1 = x2

(1.3)


x˙ = −x .
2
1




0 1
 and the initial condition is
For this equation, the matrix A = 
−1 0

 
a
x(0) =   .
b
We can transform (1.3) into n second order differential equation for x1
x¨1 + x1 = 0
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Linearization and hyperbolicity

NGUYEN THI KIM OANH

The characteristics equation is λ2 +1 = 0 ⇒ λ = ±i. Hence, the solution
of this ordinary differential equation is x1 = c1 cost + c2 sint. This implies
that x2 = c2 cost-c1 sint. Using the unitial condition, we get

 x1 = x1 cos t + x − 2 sin t
 x = x − 1 sin t − x − 2 cos t
2



and the flow ϕ(x, t) = 

x1 cos t + x2 sin t





x1 sin t − x2 cos t
It is easy to see that every point x is periodic of period 2π of the flow
ϕ, because


 

x1 cos(t + 2π x2 sin(t + 2π)
x cos t x2 sin t
= 1
 = ϕ(x,
ϕ(x, t + 2π) = 
x1 sin(t + 2π −x2 cos(t + 2π)
x1 sin t −x2 cos t.

and ϕ(x, t + s) = ϕ(x, t), ∀0 < s < 2π.

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) =

ϕ(x, t)
t∈R

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


γ + (x) =
t≥0

ϕ(x, t)
t≤0

Definition 1.3.2. The w−limit set of x, Λ(x), and the α−limit set of
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 →
∞}.
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Linearization and hyperbolicity

NGUYEN THI KIM OANH

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 of
points 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 set
are the same, ∂B(0, x ). Indeed,
suppose ϕ(x, t0 ) = y ∈ ∂B(0, x ). By choosing
tn = t0 + n2π → +∞ as n → ∞, we have
ϕ(x, tn ) = ϕ(x, t0 + n2π) = ϕ(x, t0 ) = y, for all n.
So,ϕ(x, tn ) → y as → ∞.
Similarly, with tn = t0 − n2π → −∞ as n → ∞, we also get

ϕ(x, tn ) = ϕ(x, t0 − n2π) = ϕ(x, t0 ) = y, ∀n, and
ϕ(x, tn ) → y as → ∞.
Hence, Λ(x) = ∂B(0, x ) = A(x).

1.4

Stability

Consider the nonlinear differential equation
x˙ = f (x, t), x ∈ Rn .
Definition 1.4.1. A point x is Liapounov stable(start near stay near)
iff for all

> 0, ∃δ > 0 so that if |x − y| < δ then |ϕ(x, t) − ϕ(y, t)| <

, ∀t ≥ 0.
Definition 1.4.2. A point x is quasi-asymptotically stable (tends to
eventually) iff there exsist δ > 0 such that if |x − y| < δ then
|ϕ(x, t) − ϕ(y, t)| −→ 0, as t −→ ∞.

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Linearization and hyperbolicity

NGUYEN THI KIM OANH

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 must

exists a neighbourhood of the point such that all points in this neighbourhood 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 of
the equation x˙ = f (x), so for all > 0, there exists δ > 0 such that
|y − x| < δ ⇒ |ϕ(y, t) − x| < , ∀t ≥ 0
and ∃δ > 0 such that |y − x| < δ ⇒ |ϕ(y, t) − x| → 0 as t → ∞,
then
Dx = {y ∈ Rn | lim |ϕ(y, t) − x| = 0}
t→∞

is called the domain of asymptotic stability of x. If Dx = Rn then x is
globally asymptotically stable.
Definition 1.4.5. (Normal forms )
Let P be an 2x2 matrix with a repeated real eigenvalue λ. Then the
characteristic polynomial of P is (s − λ)2 = 0. Since P satisfies its own
characteristic equation, this implies that
(P − λI)2 x = 0
for all x ∈ R. If λ is a double eigenvalue of P then there is a change of
coordinates which brings P into one of the two cases


 
λ 0
λ 1
.

 or 
0 λ
0 λ

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Linearization and hyperbolicity

NGUYEN THI KIM OANH

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


ρj −ωj

B=
ωj ρ j

10


Chapter 2
LINEARIZATION AND
HYPERBOLICITY
2.1

Poincare’s Linearization theorem

Consider the equation x˙ = f (x), f (0) = 0 and f is analytic on Rn . We

want 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 problem
depends 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
k

mk ≥ 2 such that
n

(m, k) =

mk λk = λs
k=1

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

n
1 mk

is called the order

of the resonance.

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

NGUYEN THI KIM OANH

coordinates.
Assum that this linear coordinate change has already been made, so if
Df (0) has distinct eigenvalues (which may be complex) then the coordinate system (x1 , ...xn ) is such that

λi xi + higher order terms

then group together the higher order terms by order
x˙ = Df (0) + Vr (x) + Vr+1 (x)

(2.1)

We have r ≥ 2 and Vj contant terms only of order j. Then we let
Mr = m ∈ Z n |mi ≥ 0,

mk ≥ r, then we rewrite the following

am x m

Vr (x) =

(2.2)

m∈Mr

mn
1

wherw m = (m1 , ..., mn ), xm = xm
1 ...xn and am is an dimensional vec-

tor
For example: if n=3 and r=3 then Mr = {(0, 0, 3), (1, 1, 1), (3, 0, 0)}, in
coordinate x = (x1 , x2 , x3 )
v3 (x) = a(3,0,0) x31 + a(1,1,1) x1 x2 x3 + a(0,0,3) x33

A near identity change of coordinates y = x + ... should be made such
that
y˙ = Df (0)y + Vr+1 (y) + ...
12

(2.3)


Linearization and hyperbolicity

NGUYEN THI KIM OANH

When we repeat the argument for the terms of order r+1, having killed
terms of order r, r + 1, ... successively, y˙ = Df (0)y. Thus, we have
ami xm + vr+1,i (x) + ...

x˙i = λi xi +

(2.4)

m∈Mr


and the coodinate change
bmi xm

(2.5)

bmi y m + O(|y|)r+1

(2.6)

yi = x i +
m∈Mr

with inverse
xi = y i −
m∈Mr

and we can choose the coeffiicients bmi such that
y˙i = λi yi + vr+1,i (x) + ...

(2.7)

we need to differentiate yi and differentiate xm with respect to time.
Now,
d m
x =
dt

n

k=1


x˙k m
x =
xk

n

mk λk xm + O(|x|)r+1 .

(2.8)

k=1

So, let y in terms of x we can differentiating the expression and writing
(m, λ) =

n
1 mk λk ,

bmi (m, λ)xm + O(|x|r+1 ).

y˙ i = x˙ i +
m∈Mr

13

(2.9)


Linearization and hyperbolicity


NGUYEN THI KIM OANH

Substituting for x˙ i from (2.4) gives
bmi (m, λ)xm + O(|x|r+1 ).

ami xm +

y˙ i = λi xi +
m∈Mr

m∈Mr

Substituting x in terms of y using (2.6)
bmi y m ) +

y˙ i = λi (yi −
m∈Mr

bmi (m, λ)y m + O(|y|r+1 ).

ami y m +
m∈Mr

m∈Mr

(2.10)
Consider a single sum, put the first and third sums, we find that
ami y m + O(|y|r+1 ).


bmi y m (−λi + (m, λ))y m +

y˙ i = λi yi +
m∈Mr

m∈Mr

(2.11)
If we choose the coefficients bmi such that
bmi =

ami
λi − (m, λ)

(2.12)

and we have
y˙ = λi yi + O(|y|r+1 )

(2.13)

y˙ = λi yi + Vr+1,i (y) + ...

(2.14)

or

are all terms of order r disappear
We can only choose this value of bmi provided
λi − (m, λ) = 0


(2.15)

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

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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 is
resonant of order 6 since λ1 =2 λ1 +4 λ2 (or λ2 =λ1 +5λ2 ).Similarly, If
i) 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 no
integers m1 , m2 ≥ 0 such that m1 + m2 ≥ 2 and either
7 = m1 7 + m2 5 ⇔ (m1 − 1)7 + m2 5 = 0
or
5 = m1 7 + m2 5 ⇔ m1 7 + (m2 − 1)5 = 0
The 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 converges if the eigenvalues (λi ) are non-resonant and either Re λi > 0 for
i = 1, ..., n or Reλi < 0 for i = 1, ..., n.
If some eigenvalues have negative real parts others positive real parts

then the statement is more complicated.
Definition 2.1.2. The n-tuple λ = (λ1 , ..., λn ) satisfiles the Siegel condition if there exists C > 0 and v such that for all i = 1, ..., n
|λi − (m, λ)| ≥

C
|m|v

for all m = (m1 , ..., mn ), where (mi ) are non-negative interger with
|m| =

n
1 mi

≥ 2.
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Linearization and hyperbolicity

NGUYEN THI KIM OANH

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

Theorem 2.1.1. Suppose that x˙ = f (x), f(0)=0 and Df(0) is not resonant. Then if Df(0) is diagonal there exists a formal near identity change
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 stationary
point are non-resonant and either Re λi > 0 for i = 1, ..., n or Re λi < 0
for i=1,...,n, or λi satisfies a Siegel condition, then the power series of
Theorem 2.1.1 converges on some neighbourhood of the stationary point.

Example: Consider
x˙ = x, y˙ = 2y + x2
With linearization at the origin given by
x˙ = x, y˙ = 2y
. The eigenvalues of the linearization are (λ1 , λ2 )=(1,2) and this is resonant of order two since 2λ1 = λ2 . Solution curves of the linearized
equation lie on solution of
dy
2y
=
dx
x
which we can easily solve to obtain a family of parabolae,
y = kx2 .

(2.16)

Similary, the full problem has solution which lie on curves defined by
dy
2y
=
+x
dx
x
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Linearization and hyperbolicity

NGUYEN THI KIM OANH


multiplying through by the integrating factor x−2
x−2

dy
d −2
2y
2y
=
(x y) + 3 = 3 + x−1 .
dx dx
x
x

Hence the solutiom curves are of the form
y = x2 (log |x| + k).

2.2

(2.17)

Hyperbolic stationary points and the manifoid
theorem

Definition 2.2.1. A stationary x is hyperbolic iff Df (0) has no zero or
purely imaginary eigenvalues.

There are two improtant theorems for hyperbolic stationary points,
the stable manifold theorem and Hartman’s theorem.
Let U be some neighbourhood of a stationary point, x. So, by analogy
with the definition of the invariant manifolds for linear systems we can

s
(x), by
define the local stable manifold of x, Wloc
s
Wloc
(x) = {y ∈ U |ϕ(y, t) → x as t → ∞, ϕ(y, t) ∈ U for all t ≥ 0}

and

u
Wloc
(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 of
the same dimension as the stable and unstable manifold of the linearized
equationy˙ = Df (x)y if x is hyperbolic, and that they are tangential to
the linearized manifolds at x.

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Linearization and hyperbolicity

NGUYEN THI KIM OANH

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 negative real part and a source if all the eigenvalues of Df (0) have strictly

positive real parts.Otherwise x0 is a saddle.

In Sections 2.1.2 and 2.1 that for small perturbations of the defining
equations, a source remains a source, a sink remains a sink and a saddle
remains a saddle. Furthermore, as one would expect, if x0 is a sink then
it is asymptotic stable
If we choose a coordinate system for which the linear part of the differential equation at the origin is in normal form we can always arrange

For the differential equation to be of the form
x˙ = Ax + g1 (x, y), y˙ = −By + g2 (x, y)

(2.18)

where x ∈ Rns , y ∈ Rns and both the square matrices A and B have
eigenvalues with positive real parts. The function gi (x, y), i = 1, 2,
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Linearization and hyperbolicity

NGUYEN THI KIM OANH

contain the nonlinear parts of the equation, together with their first
derivaties at the origin, (x, y) = (0, 0). Hence
E s (0, 0) = {(x, y)|x = 0} and E u (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)


∂Si
(0) = 0, 1 ≤ i ≤ nu , 1 ≤ j ≤ ns .
∂yi

(2.20)

where

Similary we can write the unstable manifold as
yi = Uj (x),

∂Si
(0) = 0, 1 ≤ i ≤ nu , 1 ≤ j ≤ ns
∂yi

(2.21)

We begin by expanding Uj as a power series in x, so
umj xm

Uj (x) =

(2.22)

r≥2 m∈Mr

If B is diagonal then with eigenvalues (λi ), i = 1, 2, ..., ns then
y˙ j = −λj yj + g2j (x, y)


(2.23)

and unstable manifol y = U (x) so
y˙ j = −λj Uj (x) + g2j (x, U (x)).

(2.24)

On the other hand
d
y˙ j = Uj (x) =
dt

nu

x˙ k
k=1

∂
Uj (x).
∂xk

(2.25)

Consider (2.24) and (2.25), we find that
nu

−λi Uj (x) = g2j (x, U (x)) =

x˙ k
k=1


19

∂
Uj (x)
∂xk

(2.26)


Linearization and hyperbolicity

NGUYEN THI KIM OANH

An example may make this clearer
Example: Consider the equations
x˙ = x, y˙ = −y + x2
This has a unique stationary point at (x, y) = (0, 0) and the equation in
form near the stationary point, the linearized equation is
x˙ = x, y˙ = −y.
Giving a saddle at the origin
E s (0, 0) = {(x, y)|x = 0} and E u (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
(0) = 0
∂x


and
uk xk

U (x) =
k≥2

Now
y˙ = −y + x2 = −

kuk xk
k≥2

on the unstable manifold and also
y˙ = x˙

∂U
(x) =
∂x

kuk xk
k≥2

Equating terms of order x2 , x3 and so on gives
−u2 + 1 = 2u2 , and − uk = kuk , k ≥ 3
Hence u2 = 31 , uk = 0 for k ≥ 3 and so
1
u
Wloc
(0, 0) = {(x, y)|y = x3 }
3

20


Linearization and hyperbolicity

NGUYEN THI KIM OANH

Theorem 2.2.1. (Stable manifold theorem)
Suppose that the origin is a hyperbolic stationary point for x˙ = f (x) and
E s and E u are the stable and unstable manifolds of the linear system x˙ =
s
Df (0)x. Then there exists local stable and unstable manifolds Wloc
(0)
u
and Wloc
(0) of the same dimension as E s and E u respectively. These

manifolds are (respectively) tangential to E s and E u at the origin and
as smooth as the original function f .
Suppose that xo is a hyperbolic stationary point, then there are three
s
u
possibilities. Either Wloc
(x0 ) = ∅, or Wloc
(x0 ) = ∅, or both manifolds

are non-empty. These three possibilities are given names: x0 is called a
source, sink or saddle respectively.

Theorem 2.2.2. (Hartman’s theorem) If x=0 is a hyperbolic stationary

point 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 that
the 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 and
the phase portrait obtainned when the Jacobian at the stationary point
has real eigenvalues.

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