To calculate the stability of a fixed point consider a small perturbation,
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.