The Lyapunov–Malkin theorem (named for Aleksandr Lyapunov and Ioel Gilevich Malkin) is a mathematical theorem detailing nonlinear stability of systems.
In the system of differential equations,
x
˙
=
A
x
+
X
(
x
,
y
)
,
y
˙
=
Y
(
x
,
y
)
where,
x
∈
R
m
,
y
∈
R
n
,
A
in an m × m matrix, and X(x, y), Y(x, y) represent higher order nonlinear terms. If all eigenvalues of the matrix
A
have negative real parts, and X(x, y), Y(x, y) vanish when x = 0, then the solution x = 0, y = 0 of this system is stable with respect to (x, y) and asymptotically stable with respect to x. If a solution (x(t), y(t)) is close enough to the solution x = 0, y = 0, then
lim
t
→
∞
x
(
t
)
=
0
,
lim
t
→
∞
y
(
t
)
=
c
.
