= Node
{disambiguate=dynamical systems}
= Node
{synonym}
= Node equilibrium
{synonym}
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 $x=x_0e^{-at}$ and $y=y_0e^{-bt}$. 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>.
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