4First-order differential equations
IA Differential Equations
4.7 Fixed (equilibrium) points and stability
Definition (Equilibrium/fixed point). An equilibrium point or a fixed point of
a differential equation is a constant solution
y
=
c
. This corresponds to
dy
dt
= 0
for all t.
These are usually the easiest solutions to find, since we are usually given an
expression for
dy
dt
and we can simply equate it to zero.
Definition (Stability of fixed point). An equilibrium is stable if whenever
y
is deviated slightly from the constant solution
y
=
c
,
y → c
as
t → ∞
. An
equilibrium is unstable if the deviation grows as t → ∞.
The objective of this section is to study how to find equilibrium points and
their stability.
Example. Referring to the differential equation above (
˙y
=
t
(1
− y
2
)), we see
that the solutions converge towards
y
= 1, and this is a stable fixed point. They
diverge from y = −1, and this is an unstable fixed point.
4.7.1 Perturbation analysis
Perturbation analysis is used to determine stability. Suppose
y
=
a
is a fixed
point of
dy
dt
=
f
(
y, t
), so
f
(
a, t
) = 0. Write
y
=
a
+
ε
(
t
), where
ε
(
t
) is a small
perturbation from
y
=
a
. We will later assume that
ε
is arbitrarily small. Putting
this into the differential equation, we have
dε
dt
=
dy
dt
= f(a + ε, t)
= f(a, t) + ε
∂f
∂y
(a, t) + O(ε
2
)
= ε
∂f
∂y
(a, t) + O(ε
2
)
Note that this is a Taylor expansion valid when
ε
1. Thus
O
(
ε
2
) can be
neglected and
dε
dt
∼
=
ε
∂f
∂y
.
We can use this to study how the perturbation grows with time.
This approximation is called a linearization of the differential equation.
Example. Using the example ˙y = t(1 − y
2
) above, we have
∂f
∂y
= −2yt =
(
−2t at y = 1
2t at y = −1
.
At
y
= 1,
˙ε
=
−
2
tε
and
ε
=
ε
0
e
−t
2
. Since
ε →
0 as
t → ∞
,
y →
1. So
y
= 1 is a
stable fixed point.
On the other hand, if we consider
y
=
−
1, then
˙ε
= 2
tε
and
ε
=
ε
0
e
t
2
. Since
ε → ∞ as t → ∞, y = −1 is unstable.
Technically
ε → ∞
is not a correct statement, since the approximation used
is only valid for small
ε
. But we can be sure that the perturbation grows (even
if not → ∞) as t increases.
4.7.2 Autonomous systems
Often, the mechanics of a system does not change with time. So
˙y
is only a
function of y itself. We call this an autonomous system.
Definition (Autonomous system). An autonomous system is a system in the
form ˙y = f(y), where the derivative is only (explicitly) dependent on y.
Near a fixed point
y
=
a
, where
f
(
a
) = 0, write
y
=
a
+
ε
(
t
). Then
˙ε
=
ε
df
dy
(
a
) =
kε
for some constant
k
. Then
ε
=
ε
0
e
kt
. The stability of the
system then depends solely on sign of k.
Example. Consider a chemical reaction NaOH + HCl
→
H
2
O + NaCl. We
have
NaOH + HCl → H
2
O + NaCl
Number of molecules a b c c
Initial number of molecules a
0
b
0
0 0
If the reaction is in dilute solution, then the reaction rate is proportional to
ab
.
Thus
dc
dt
= λab
= λ(a
0
− c)(b
0
− c)
= f(c)
We can plot
dc
dt
as a function of c, and wlog a
0
< b
0
.
c
˙c
a
0
b
0
We can also plot a phase portrait, which is a plot of the dependent variable only,
where arrows show the evolution with time,
c
a
0
b
0
We can see that the fixed point c = a
0
is stable while c = b
0
is unstable.
We can also solve the equation explicitly to obtain
c =
a
0
b
0
[1 − e
−(b
0
−a
0
)λt
]
b
0
− a
0
e
−λ(b
0
−a
0
)t
.
Of course, this example makes no sense physically whatsoever because it assumes
that we can have negative values of
a
and
b
. Clearly in reality we cannot have,
say,
−
1 mol of NaOH, and only solutions for
c ≤ a
0
are physically attainable,
and in this case, any solution will tend towards c = a
0
.
4.7.3 Logistic Equation
The logistic equation is a simple model of population dynamics. Suppose we
have a population of size
y
. It has a birth rate
αy
and a death rate
βy
. With
this model, we obtain
dy
dt
= (α − β)y
y = y
0
e
(α−β)t
Our population increases or decreases exponentially depending on whether the
birth rate exceeds death rate or vice versa.
However, in reality, there is fighting for limited resources. The probability of
some piece of food (resource) being found is
∝ y
. The probability of the same
piece of food being found by two individuals is
∝ y
2
. If food is scarce, they fight
(to the death), so death rate due to fighting (competing) is γy
2
for some γ. So
dy
dt
= (α − β)y − γy
2
dy
dt
= ry
1 −
y
Y
,
where
r
=
α − β
and
Y
=
r/γ
. This is the differential logistic equation. Note
that it is separable and can be solved explicitly.
However, we find the phase portrait instead. If
r
=
α − β >
0, then the
graph is a quadratic parabola, and we see that Y is a stable fixed point.
y
f
O Y
y
O Y
Now when the population is small, we have
˙y ' ry
So the population grows exponentially. Eventually, the stable equilibrium
Y
is
reached.