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Consider an autonomous differential equation
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(1) |
An equilibrium point  of (1) is such that  . Conversely a regular point of (1) is such that
 .
If the linearization  has no eigenvalue with zero real part,  is said to be a hyperbolic equilibrium, whereas if there exists an eigenvalue with zero real part, the equilibrium point is nonhyperbolic.
An equilibrium point  is said to be stable if for every neighborhood  , there exists a neighborhood of  ,
 such that every solution of (1) with initial condition in  (i.e.
 ), satisfies
for all  .
Consequently an equilibrium point  is said to be unstable if it is not stable.
Moreover an equilibrium point  is said to be asymptotically stable if it is stable and there exists  such that every solution of (1) with initial condition in  (i.e.
 ) satisfies
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