Zubov's method is a technique for computing the basin of attraction for a set of ordinary differential equations (a dynamical system). The domain of attraction is the set
{
x
:
v
(
x
)
<
1
}
, where
v
(
x
)
is the solution to a partial differential equation known as the Zubov equation. 'Zubov's method' can be used in a number of ways.
Zubov's theorem states that:
If
x
′
=
f
(
x
)
,
t
∈
R
is an ordinary differential equation in
R
n
with
f
(
0
)
=
0
, a set
A
containing 0 in its interior is the domain of attraction of zero if and only if there exist continuous functions
v
,
h
such that:
v
(
0
)
=
h
(
0
)
=
0
,
0
<
v
(
x
)
<
1
for
x
∈
A
∖
{
0
}
,
h
>
0
on
R
n
∖
{
0
}
for every
γ
2
>
0
there exist
γ
1
>
0
,
α
1
>
0
such that
v
(
x
)
>
γ
1
,
h
(
x
)
>
α
1
, if
|
|
x
|
|
>
γ
2
v
(
x
n
)
→
1
for
x
n
→
∂
A
or
|
|
x
n
|
|
→
∞
∇
v
(
x
)
⋅
f
(
x
)
=
−
h
(
x
)
(
1
−
v
(
x
)
)
1
+
|
|
f
(
x
)
|
|
2
If f is continuously differentiable, then the differential equation has at most one continuously differentiable solution satisfying
v
(
0
)
=
0
.
