, about 
,
The Taylor expansion (substituting eq. (4.16) into
eq. (4.2)) about gives
It seems reasonable that the motion near the fixed point should be governed by the linear system
since 
.
If 
is an equilibrium point, then
is a matrix with constant entries. We can immediately
write down the solution to this linear system as
where 
is the evolution operator
for a linear system. If we let 
denote the constant 
matrix, then the linear evolution
operator takes the form
where 
denotes the identity matrix.
The asymptotic stability of a fixed point can be determined by the
eigenvalues of the linearized vector field 
at .
In particular, we have the following test for asymptotic stability:
an equilibrium solution of a nonlinear vector field is
asymptotically stable if all the
eigenvalues of the linearized vector field
have negative real parts.
If the real part of at least one eigenvalue exactly equals zero (and all the others are strictly less than zero) then the system is still linearly stable, but the original nonlinear system may or may not be stable.
