© 2008
Zachary S Tseng
D
-2 - 23
x
′ =
F
(
x
,
y
) ≈
F
(
α
,
β
) +
F
x
(
α
,
β
)(
x
−
α
) +
F
y
(
α
,
β
)(
y
−
β
)
y
′ =
G
(
x
,
y
) ≈
G
(
α
,
β
) +
G
x
(
α
,
β
)(
x
−
α
) +
G
y
(
α
,
β
)(
y
−
β
)
But since (
α
,
β
) is a critical point, so
F
(
α
,
β
) = 0 =
G
(
α
,
β
), the above
linearizations become
x
′ ≈
F
x
(
α
,
β
)(
x
−
α
) +
F
y
(
α
,
β
)(
y
−
β
)
y
′ ≈
G
x
(
α
,
β
)(
x
−
α
) +
G
y
(
α
,
β
)(
y
−
β
)
As before, the critical point could be translated to (0, 0) and still retains its
type and stability, using the substitutions
χ
=
x
−
α
and
γ
=
y
−
β
.
After the
translation, the approximated system becomes
x
′ =
F
x
(
α
,
β
)
x
+
F
y
(
α
,
β
)
y
y
′ =
G
x
(
α
,
β
)
x
+
G
y
(
α
,
β
)
y
It is now a homogeneous linear system with a coefficient matrix
A
=
)
,
(
)
,
(
)
,
(
)
,
(
β
α
β
α
β
α
β
α
y
x
y
x
G
G
F
F
.
That is, it is a matrix calculate by plugging in
x
=
α
and
y
=
β
into the matrix
of first partial derivatives
J
=
y
x
y
x
G
G
F
F
.
This matrix of first partial derivatives,
J
, is often called the
Jacobian
matrix
.
It just needs to be calculated once for each nonlinear system.
For each
critical point of the system, all we need to do is to compute the coefficient
matrix of the linearized system about the given critical point (
x
,
y
) = (
α
,
β
),
and then use its eigenvalues to determine the (approximated) type and
stability.