The Lyapunov stability theorem gives sufficient conditions for stability of a fixed point of an autonomous system using a Lyapunov function. It applies to nonlinear systems without requiring linearization.
Let 
be an open neighborhood of the origin
in 
,
and let 
be a continuously differentiable
function with 
. Consider the system of ordinary
differential equations
 |
(1)
|
Suppose there is a scalar function 
that is a continuously
differentiable function, with 
and 
for every 
in 
. Its derivative along solutions
is
 |
(2)
|
where 
is the gradient and the multiplication
denotes the dot product. If 
throughout 
, then the zero solution is stable. This means that for every
there is a 
such that 
implies 
for all 
,
where 
denotes the Euclidean vector norm. If 
for every 
in 
, then the zero solution is asymptotically stable. In addition
to stability, solutions starting sufficiently near the origin
converge to it as 
(Iyer 2026).
The conclusions above are local. If the hypotheses hold on all of 
, the derivative is strictly
negative away from the origin, and 
is radially unbounded, meaning 
as 
, then the zero solution is globally asymptotically
stable (Yang 2023).
For example, the ordinary differential equation 
has the Lyapunov function 
, with
 |
(3)
|
The zero solution is globally asymptotically stable, although linearization gives 
and is inconclusive.
