quality:correction
Dynamical systems

Consider a dynamical system of the form x ˙ = F (x) {\displaystyle {\dot {\mathbf {x} }}=F(\mathbf {x})}, where x ˙ {\displaystyle {\dot {\mathbf {x} }}} is the (component-wise) derivative of x {\displaystyle \mathbf {x} } with respect to the evolution parameter t {\displaystyle t} (time), and F: R n → R n {\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}} is differentiable. If F (x 0) = 0 {\displaystyle F(\mathbf {x} _{0})=0}, then x 0 {\displaystyle \mathbf {x} _{0}} is a stationary point (also called a steady state). By the Hartman–Grobman theorem, the behavior of the system near a stationary point is related to the eigenvalues of J F (x 0) {\displaystyle \mathbf {J} _{F}\left(\mathbf {x} _{0}\right)}, the Jacobian of F {\displaystyle F} at the stationary point. Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point. If any eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the eigenvalues is zero, the Jacobian matrix does not allow for an evaluation of the stability.