Given a system of ODEs,

du

dt

= F(u,a),     u Î Rn,a Î R    F: Rn+1® Rn .

an equilibrium is a point such that F(u,a)=0.

Eigenvalues of an equilibrium are roots of the characteristic equation

where

is the Jacobian matrix andis the n x n unit matrix. The equilibrium is hyperbolic if there is no eigenvalue with zero real part. A hyperbolic
equilibrium has invariant stable and unstable manifolds of dimension equal to the number of eigenvalues with negative and positive real part, respectively. If there is an eigenvalue with zero real part, the equilibrium is called nonhyperbolic. A nonhyperbolic equilibrium has a center manifold of dimension equal to the number of eigenvalues of the imaginary axis. The invariant manifolds are tangent to the corresponding generalized eigenspaces at.