# Linear stability

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 r is the perturbation to the steady state, *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**.[1][2] 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".[3]