In mathematics, in the theory of
differential equations and
dynamical systems, a particular
stationary or quasistationary solution to a nonlinear system is called
linearly unstable if the
linearization of the equation at this solution has the form , where
A is a linear
operator whose
spectrum contains eigenvalues with
positive real part. If all the eigenvalues have
negative real part, then the solution is called
linearly stable. Other names for linear stability include
exponential stability or
stability in terms of first approximation. If there exist an eigenvalue with
zero real part then the question about stability cannot be solved on the basis of the first approximation and we approach the so-called "centre and focus problem".