 Normal form (dynamical systems)

A simplified local vector field obtained by smooth changes of state variables, and sometimes time or parameter, retaining the terms that cannot be removed at the required order. Near a simple zero eigenvalue, reduction to a centre manifold can give the transcritical form $\dot{v}=ϵv-v^2$ or a reflection-symmetric pitchfork form $\dot{x}=ϵx\pm x^3$. The nonzero leading coefficients and permitted coordinate changes are part of the classification.