 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.