LECTURE 31: STABILITY AND CLASSIFICATION OF ISOLATED CRITICAL POINTS
MINGFENG ZHAO
November 19, 2014
Linearization

 x0 = f (x, y)
Definition 1. Suppose (x0 , y0 ) is an equilibrium solution to the autonomous system
. Let u = x − x0
 y 0 = g(x, y)

 x0 = f (x, y)
and v = y − y0 , the linearization at (x0 , y0 ) of the autonomous system
is:
 y 0 = g(x, y)

u0

v
0


=
fx (x0 , y0 ) fy (x0 , y0 )

u

gx (x0 , y0 )
gy (x0 , y0 )

.
v
Example 1. Find all critical points and linearizations at these critical points for the system x0 = (1 − y)(2x − y),
y 0 = (2 + x)(x − 2y).
Let’s solve
(1 − y)(2x − y) = 0,
and
(2 + x)(x − 2y) = 0.
Then there are four critical points:
(−2, 1),
(2, 1),
(−2, −4)
and
(0, 0).
Let f (x, y) = (1 − y)(2x − y) = 2x − y − 2xy + y 2 and g(x, y) = (2 + x)(x − 2y) = 2x − 4y + x2 − 2xy, then the


f (x, y)
 is:
Jacobian matrix of 
g(x, y)


2 − 2y
−1 − 2x + 2y

.
2 + 2x − 2y
−4 − 2x
Therefore, we know that
1
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MINGFENG ZHAO
I. At (−2, 1) the linearization is:

u0

v0


=
0
5
−4
0

u


.
v
II. At (2, 1) the linearization is:

u0

v0


=
0
−3
4
−4

u


.
v
III. At (−2, −4) the linearization is:

u0

v0


=
10 −5

u

6
0
2
−1
2
−4

.
v
IV. At (0, 0) the linearization is:

u0

v0


=

u


.
v
Isolated critical points and almost linear systems
Definition 2. Let f~(~x) be a vector function and ~a be a critical point of f~, that is, f~(~a) = 0, then
I. We say that the critical point ~a is isolated if it is the only critical point in a small “neighborhood” of ~a.
II. We say that the system ~x0 = f~(~x) is almost linear at critical point ~a if the critical point ~a is isolated, and the
Jacobian at ~a is invertible.
Remark 1. If the Jacobian matrix at ~a is invertible, then ~a is isolated. Therefore, it suffices to verify that the Jacobian
at ~a is invertible.
NonExample 1. It’s easy to see (0, 0) is a critical point of the system x0 = x2 , y 0 = y 2 , but the system is


0 0
 which is not invertible.
not almost linear at (0, 0), because the Jacobian matrix at (0, 0) is 
0 0
Stability and classification of isolated critical points
Definition 3. Let ~a be a critical point of f~, that is, ~x(t) ≡ a is an equilibrium solution to the system ~x0 = f~(~x. Then
LECTURE 31: STABILITY AND CLASSIFICATION OF ISOLATED CRITICAL POINTS
3
I. We say that the critical point ~a is stable if for any given > 0, there exists some δ > 0 such that for any
solutions ~x(t) to the initial value problem ~x0 = f~(~x), ~x(0) = ~x0 with |~x0 − ~a| < δ, we have
|~x(t) − ~x0 | < ,
for all t ≥ 0.
II. We say that the critical point ~a is unstable if ~a is not stable.
III. We say that the critical point ~a is asymptotically stable if ~a is stable, and there exists some δ > 0 such that for
any solutions ~x(t) to the initial value problem ~x0 = f~(~x), ~x(0) = ~x0 with |~x0 − ~a| < δ, we have
lim ~x(t) = ~a.
t→∞
The following table shows the behavior of an almost linear system near an isolated critical point:
Eigenvalues of the Jacobian matrix
Behavior
Stability
I.
real and both positive
source(unstable node)
unstable
II.
real and both negative
sink (stable node)
asymptotically stable
saddle
unstable
III. real and opposite signs
V.
complex with positive real part
spiral source
unstable
VI.
complex with negative real part
spiral sink
asymptotically stable
Example 2. Find all critical points and classify the stabilities of these critical points for the system x0 = −y − x2 ,
y 0 = −x + y 2 .
Let’s solve −y − x2 = 0 and −x + y 2 = 0, then
(x, y) = (0, 0),
and
(x, y) = (1, −1).
That is, all critical points of the system x0 = sin(πy) + (x − 1)2 , y 0 = y 2 − y are:
(0, 0) and (1−, 1) .

Let f (x, y) = −y − x2 and g(x, y) = −x + y 2 , then Jacobian matrix of 
f (x, y)
g(x, y)
know that


 is 
−2x
−1
−1
2y

. Therefore, we
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MINGFENG ZHAO

I. At (0, 0) the linearization is 
u0
v
0


=
0
−1
−1

u

0


. Notice that 
v

system is almost linear at (0, 0). Since the eigenvalues of 
0
−1
−1
0
0
−1
−1
0

 is invertible, then the

 are λ1 = −1 and λ2 = 1, then (0, 0) is
an unstable equilibrium point of the system, solutions behaves like a saddle near (0, 0).

 




u0
2 −1
u
2 −1
=

. Notice that 
 is invertible, then the
II. at (1, −1) the linearization is 
0
v
−1 −2
v
−1 −2


2 −1
 are λ1 = −1 and λ2 = −3, then
system is almost linear at (1, −1) . Since the eigenvalues of 
−1 −2
(1, −1) is an asymptotically stable equilibrium point of the system, solutions behaves like a sink near (0, 0).
Figure 1. Phase portrait with some trajectories of x0 = −y − x2 , y 0 = −x + y 2
Example 3. Find all critical points and classify the stabilities of these critical points for x0 = y + y 2 ex , y 0 = x.
Let’s solve y + y 2 ex = 0 and x = 0, then
(x, y) = (0, 0),
and
(x, y) = (0, −1).
That is, all critical points of the system x0 = y + y 2 ex , y 0 = x are:
(0, 0) and (0, −1) .
LECTURE 31: STABILITY AND CLASSIFICATION OF ISOLATED CRITICAL POINTS

Let f (x, y) = y + y 2 ex and g(x, y) = x, then Jacobian matrix of 
f (x, y)


 is 
y 2 ex
1 + 2yex
1
0
g(x, y)
5

. Therefore, we
know that

I. At (0, 0) the linearization is 
u0
v0


=
0
1
1
0

u



. Notice that 
v

almost linear at (0, 0) . Since the eigenvalues of 
0
1
1
0
0
1
1
0

 is invertible, then the system is

 are λ1 = −1 and λ2 = 1, then (0, 0) is an unstable
equilibrium point of the system, solutions behaves like a saddle near (0, 0).






u0
1 −1
u
1
 = 

. Notice that 
II. at (0, −1) the linearization is 
v0
1 0
v
1


1 −1
 are λ1 =
system is almost linear at (0, −1) . Since the eigenvalues of 
1 0
−1

 is invertible, then the
0
√
√
1
3
1
3
+i
and λ2 = − i
,
2
2
2
2
then (0, −1) is an unstable equilibrium point of the system, solutions behaves like a spiral source near (0, 0).
Figure 2. Phase portrait with some trajectories of x0 = y + y 2 ex , y 0 = x
Exercise 1. Find all critical points and classify the stabilities of these critical points for x0 = 2 sin(x) cos(y), y 0 =
cos(x) sin(y).
Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C.
Canada V6T 1Z2
E-mail address: mingfeng@math.ubc.ca