= Improper node
{title2=$\dot u=\lambda u,\quad\dot v=\lambda v+cu,\qquad \lambda c\ne0$}
A planar <node> whose <Jacobian matrix> has a repeated nonzero real <eigenvalue> and only one independent <eigenvector> is an improper node. In suitable real coordinates its <linearization of a dynamical system> has the displayed form, with solutions
$$
u(t)=Ae^{\lambda t},\qquad v(t)=(B+cAt)e^{\lambda t}.
$$
It is a <stable node> for $\lambda<0$ and an <unstable node> for $\lambda>0$. The factor $t$ makes generic trajectories tangent to the unique eigendirection as they approach the equilibrium in the attracting time direction. For a nonlinear <autonomous system> with this <hyperbolic equilibrium>, the sign of $\lambda$ still determines local stability; the local <phase portrait> has the same topological type as its linearization.
Back to article page
