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.