LECTURE 33: APPLICATIONS OF NONLINEAR SYSTEM
MINGFENG ZHAO
December 02, 2015
Pendulum
Suppose we have a mass m (in kilograms) on a pendulum of length L (in meters). Let θ(t) be the angle between the
vertical line and the pendulum at time t (in seconds), then
θ00 +
g
sin(θ) = 0 ,
L
which is a conservative equation.
Figure 1. Various possibilities for the motion of the pendulum
Let ω = θ0 , then we can study the following two dimensional system:

θ

0

 =
ω
ω


− Lg
sin(θ)
For critical points: Let’s solve
ω = 0,
and
−
g
sin(θ) = 0.
L
Then all critical points are:
k ∈ Z.
(kπ, 0),
1
2
MINGFENG ZHAO

Notice that the Jacobian matrix of 

ω
− Lg
 is:
sin(θ)



0
Since det 
1
− Lg
1
cos(θ)
0

.

 = g cos(kπ) 6= 0, then the system is almost linear at (kπ, 0). Hence we know that at
L
0
− Lg cos(kπ)
(kπ, 0), linearization is:

u

0

 =
v

 1
Since cos(kπ) =
 −1
0
if k is even
0
1
− Lg cos(kπ)
0

u


.
v
, then we know that
if k is odd
I. At (2kπ, 0), the linearization is:

u

0

 =
v

0
1
− Lg
0

u


.
v


r
r
0
1
g
g
 has eigenvalues λ1 = i
 has eigenvalues
Since 
and λ2 = −i
. Since 
L
L
− Lg 0
− Lg cos(θ) 0
r
r
g
g
cos(θ) and λ2 = −i
cos(θ) when θ is close to (2kπ, 0). So we know that the system has a
λ1 = i
L
L
stable center at critical point (2kπ, 0).
0
1

II. At ((2k + 1)π, 0), the linearization is:

u

0
 =
v

Since 
0
g
L
1

r
 has eigenvalues λ1 =
0
at critical point ((2k + 1)π, 0).
For trajectories: Let’s solve

0
1
g
L
0

u

g
and λ2 = −
L

.
v
r
g
. Then the system has an unstable saddle point
L
dω
g sin(θ)
=− ·
, then the trajectories satisfy:
dθ
L
ω
1 2 g
ω − cos(θ) = C.
2
L
LECTURE 33: APPLICATIONS OF NONLINEAR SYSTEM
3
Then
r
w=±
2g
cos(θ) + C.
L
Figure 2. Phase plane diagram and some trajectories of the nonlinear pendulum equation
Now for any initial condition (θ(0), ω(0)) = (θ0 , 0), we have C = −
r
0
θ =ω=±
2g
cos(θ0 ), that is,
L
r
2g
2g
2g p
cos(θ) −
cos(θ0 ) = ±
· cos(θ) − cos(θ0 ).
L
L
L
Let’s compute the period T of θ(t). It’s easy to see that T equals 4 times of the time from θ0 to 0. So we only
consider θ0 = ω is positive case, that is,
dθ
= θ0 = ω =
dt
r
2g p
· cos(θ) − cos(θ0 ).
L
Then
dt
=
dθ
s
L
1
·p
.
2g
cos(θ) − cos(θ0 )
Integrate the above identity over [0, θ0 ], we have
Z θ0 s
T
L
1
=
·p
dθ.
4
2g
cos(θ) − cos(θ0 )
0
That is, the period is:
s
T =4
L
2g
Z
θ0
0
1
·p
dθ .
cos(θ) − cos(θ0 )
From the above formula, we know that
lim T = ∞ .
θ0 →π
4
MINGFENG ZHAO
In fact,
Z
0
π
Z
1
p
cos(θ) + 1
dθ
π
=
0
Z
1 − cos(u)
π
=
0
=
1
p
1
√
2
1
q
Z
0
2 sin2
π
u
2
1
sin
u
2
du Let u = π − θ
du
du
Z π
1
1
≥ √
u du Since sin(x) ≤ x for all x ≥ 0
2 0 2
√ Z π1
=
2
du
0 u
=
∞.
Recall the general solution to the linearized pendulum equation θ00 +
g
√
√
θ = 0 is: θ(t) = C1 cos ( gLt) + C2 sin ( gLt).
L
So the period Tlinear for the the linearized pendulum equation is:
Tlinear
2π
= p g = 2π
L
s
L
.
g
Let’s compare T and Tlinear :
Figure 3. Plot of T and Tlinear with
g
T − Tlinear
= 1 (left), and
(right)
L
T
Predator-prey or Lotka-Volterra systems
Let
x =
number of the prey(e.g., hares)
LECTURE 33: APPLICATIONS OF NONLINEAR SYSTEM
y
=
5
number of the predator(e.g., foxes)
Recall the population model: the rate of growth of the population is proportional to the size of the
population.
In the predator-prey model, we have
x0
= ax − (by)x = (a − by)x
y0
=
(cx)y − dy = (cx − d)y
for some positive constants a, b, c, d > 0.
Critical points: Let’s solve
(a − by)x = 0,
(cx − d)y = 0.
and
Then we have two critical points:
(x, y) = (0, 0),

The Jacobian matrix of 
(a − by)x
(cx − d)y
and
(x, y) =
d a
,
c b
.

 is:

a − by
−bx
cy
cx − d



Then
I. At (0, 0), the linearization is:

u

0

 =
v

Since det 
a
0
0
−d
a
0
0
−d

u


.
v


 = −ad 6= 0, then the system is almost linear at (0, 0). Since 
a
0
0
−d

 has eigen-
values λ1 = a and λ2 = −d, then the solutions behaves like a saddle point near (0, 0), which is unstable .
d a
II. At
,
, the linearization is:
c b

u

v
0

 =
0
− bd
c
ac
b
0

u

v

.
6
MINGFENG ZHAO


0
d
a
 = −ad 6= 0, then the system is almost linear at
Since det 
,
. Since 
c b
ac
ac
0
b
b
√
√
d a
eigenvalues λ1 = i ad and λ2 = −i ad, then the solutions behaves like a center near
,
.
c b
Let’s study the trajectories for x, y > 0, that is, we need to solve
0
− bd
c

dy
dx
− bd
c

 has
0
(cx − d)y
y
cx − d
=
·
.
(a − by)x
a − by
x
=
This a separable equation, then
a − by
cx − d
dy =
dx.
y
x
So we get
a ln y − by = cx − d ln x + C.
That is, we have
y a e−by = ecx x−d eC .
So we get get an implicit equation for the trajectories:
C=
y a xd
= y a xd e−cx−by .
ecx+by
This constant C is called the constant of motion. That is, the trajectories for the predator-prey system are
y a xd
the level curves of cx+by .
e
Figure 4. The phase portrait (left) and graphs of x and y for a sample trajectory (right) for x0 =
(0, 4 − 0.01y), y 0 = (0.003x − 0.3)y
LECTURE 33: APPLICATIONS OF NONLINEAR SYSTEM

 x0 = (a − by)x
Theorem 1. The solutions to the system
 y 0 = (cx − d)y
7
behaves like a center near
d a
,
, which is a
c b
stable center.
y a xd
Proof. We are going to show that the function g(x, y) = cx+by = y a xd e−cx−by will attain its maximum at
e
d a
,
for x, y > 0. Let f (x, y) = ln g(x, y) = a ln y + d ln x − cx − by, so it suffices to prove that f (x, y) will
c b
d a
,
for x, y > 0. In fact, we know that
attain its maximum at
c b
So only critical point for f (x, y) is
x→∞
lim
|(x,y)|→∞
=
fy
=
d
−c
x
a
− b.
y
d a
,
. By L’Hopital’s Rule, we know that
c b
lim
So we get
fx
xd
ya
= lim by = 0.
cx
y→∞ e
e
g(x, y) = 0, which implies that
lim
|(x,y)|→∞
f (x, y) = −∞.
So we know that f (x, y) must attain its maximum point in x, y > 0, which must be a critical point of f . So
d a
f (x, y) will attain its maximum at
,
for x, y > 0
c b
Problems you can do:
Lebl’s Book [2]: All Exercises on Page 305, Page 306, Page 312, Page 313, Page 321 and Page 322.
Braun’s Book [1]: All exercises on Page 393 and Page 398. Read all materials in Section 4.3 and Section 4.4.
References
[1] Martin Braun. Differential Equations and Their Applications: An Introduction to Applied Mathematics. Springer, 1992.
[2] Jiri Lebl. Notes on Diffy Qs: Differential Equations for Engineers. Createspace, 2014.
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