Linear stability

In mathematics, in the theory of <a href="/facts/Differential_equations/9MGbE3Ca">differential equations</a> and <a href="/facts/Dynamical_systems/782gREl5">dynamical systems</a>, a particular <a href="/facts/Stationary_state/HBQvbb6s">stationary or quasistationary solution</a> to a nonlinear system is called linearly unstable if the <a href="/facts/Linearization/ohYSeQR3">linearization</a> of the equation at this solution has the form 
 
 
 
 d
 r
 
 /
 
 d
 t
 =
 A
 r
 
 
 {\displaystyle dr/dt=Ar}
 
, where r is the perturbation to the steady state, A is a linear <a href="/facts/Operator_(mathematics)/1VWgEdjS">operator</a> whose <a href="/facts/Spectrum_(functional_analysis)/cA1evTo6">spectrum</a> 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 exists 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".

Linear stability open-in-new

Linear stability