92
CHAPTER 3. INTRODUCTION TO FIRST-ORDER SYSTEMS
3.4
Equilibrium points and linearization
Suppose we have a system of equations
dP
= f (P, R)
dt
dR
= g(P, R)
dt
(3.4.1)
linearize:generic
and suppose that (P⇤ , R⇤ ) is an equilibrium point of the system. Our goal is
to find another system of equations, related to the system (3.4.1), such that
1. the new system closely approximates (3.4.1) when (P, R) is close to
(P⇤ , R⇤ ) and
2. the new system is simple enough that we can explicitly solve it.
If we can find such an approximating system, then we can use it to determine
how solutions to (3.4.1) behave when they are close by to the equilibrium
point. In particular, such an approximating system can be used to understand the stability of the equilibrium point.
The main tool we use is that of linear approximation. Recall that for a
smooth function f (x) we have
f (x) ⇡ f (x⇤ ) +
df
(x⇤ ) (x
dx
x⇤ )
when x ⇡ x⇤ .
(3.4.2)
Taylor-1D
p
Example 3.4.1. Suppose that f (x) = x. If x is close to x⇤ = 4 then we
have
p
p
1
1
x ⇡ 4 + p (x 4) = 2 + (x 4).
4
2 4
In particular,
p
1
4.1 ⇡ 2 + (0.1) = 2.025.
4
p
Compare this to the Sage output for 4.1, which is 2.02484567313166. The
approximation is pretty good!
The linear approximation (3.4.2) is, of course, just the first-order Taylor
approximation of the function f centered at x = x⇤ . For systems, we need
the corresponding Taylor approximation for functions of two variables, which
is
f (x, y) ⇡ f (x⇤ , y⇤ ) +
@f
(x⇤ , y⇤ ) (x
@x
which holds when (x, y) ⇡ (x⇤ , y⇤ ).
x⇤ ) +
@f
(x⇤ , y⇤ ) (y
@y
y⇤ ),
(3.4.3)
Taylor-2D
3.4. EQUILIBRIUM POINTS AND LINEARIZATION
93
We now apply (3.4.3) to the functions f and g appearing in (3.4.1). Notice that f (P⇤ , R⇤ ) = 0 and g(P⇤ , R⇤ ) = 0 because (P⇤ , R⇤ ) is an equilibrium
point. Thus we have
@f
(P⇤ , R⇤ ) (P
@P
@g
g(P, R) ⇡
(P⇤ , R⇤ ) (P
@P
@f
(P⇤ , R⇤ ) (R
@R
@g
P⇤ ) +
(P⇤ , R⇤ ) (R
@R
f (P, R) ⇡
P⇤ ) +
R⇤ )
R⇤ ),
when (P, R) ⇡ (P⇤ , R⇤ ). We also have
dP
d
=
[P
dt
dt
P⇤ ]
dR
d
=
[R
dt
dt
and
R⇤ ] ,
simply because P⇤ and R⇤ are constants. Thus when (P, R) are close to the
equilibrium point (P⇤ , R⇤ ) we have
d
[P
dt
d
[R
dt
@f
(P⇤ , R⇤ ) (P
@P
@g
R⇤ ] ⇡
(P⇤ , R⇤ ) (P
@P
P⇤ ] ⇡
@f
(P⇤ , R⇤ ) (R
@R
@g
P⇤ ) +
(P⇤ , R⇤ ) (R
@R
P⇤ ) +
R⇤ )
R⇤ ).
We now make a change of variables, setting
v=P
P⇤
and
w=R
R⇤ .
The functions v and w describe the displacement of (P, R) away from the
equilibrium (P⇤ , R⇤ ).
[picture needed]
With this change of variables, we obtain the linearization of (3.4.1) at
(P⇤ , R⇤ ):
dv
@f
@f
=
(P⇤ , R⇤ ) v +
(P⇤ , R⇤ ) w
dt
@P
@R
(3.4.4)
dw
@g
@g
=
(P⇤ , R⇤ ) v +
(P⇤ , R⇤ ) w.
dt
@P
@R
The system (3.4.4) is also called the “linearized equation” at (P⇤ , R⇤ ).
linearize:linearize
94
CHAPTER 3. INTRODUCTION TO FIRST-ORDER SYSTEMS
Example 3.4.2. Consider the predator-prey system
dP
= P (1
dt
P)
PR
dR
=
dt
R + 2P R.
We linearize this system about the equilibrium point (P⇤ , R⇤ ) = (.5, .5).
The system takes the form (3.4.1) with
f (P, R) = P (1
P)
g(P, R) =
P2
PR = P
P R,
R + 2P R.
We can easily compute
@f
@f
= 1 2P R
= P
@P
@R
@f
@f
(.5, .5) = .5
(.5, .5) = .5
@P
@R
and
@g
@g
= 2R
= 1 + 2P
@P
@R
@g
@g
(.5, .5) = 1
(.5, .5) = 0
@P
@R
Thus, setting v = P .5 and w = R .5, we see that the linearized system
is
dv
dw
= .5v .5w
=v
dt
dt
FindThemAll
Exercise 3.4.1. Find all equilibrium solutions of the following system of
di↵erential equations:
dx
= 2 + 14x + 12y,
dt
dy
=
dt
9 + 12x + 21y.
Then linearize the system about each equilibrium.
EqSol-Last
Exercise 3.4.2. Find all equilibrium solutions of the predator-prey model:
⇣
dx
= 2x 1
dt
x⌘
2
xy,
⇣
dy
= 5y 1
dt
Then linearize the system about each equilibrium.
y⌘
5
4xy.