Node (dynamical systems)

A node is a planar hyperbolic equilibrium whose two linearized eigenvalues are real and have the same sign. Negative eigenvalues give a stable node; positive eigenvalues give an unstable node. Nearby motion approaches or recedes without the repeated rotation characteristic of a focus.

For $\dot{x}=-ax$, $\dot{y}=-by$ with $a,b>0$, solutions are $x=x_0{e}^{-at}$ and $y=y_0{e}^{-bt}$. Each coordinate decays, while unequal decay rates usually make a trajectory align with the slower direction as it nears the origin. Reversing both signs gives outward motion. Repeated eigenvalues require attention to the number of independent eigenvectors and can add polynomial factors to the exponential response. The node classification describes local behavior near an equilibrium; it does not determine the geometry of remote orbits.

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 solutionsIt 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.