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.

A reflection acting as $(A,B)\mapsto (A,-B)$ forces an equivariant vector field to have $\dot{A}$ even and $\dot{B}$ odd in $B$. With the zero solution preserved, its quadratic normal form is $\dot{A}={\lambda }_{1}A+a_1{A}^2+a_2{B}^2$, $\dot{B}={\lambda }_{2}B+a_{3}AB$. For positive coefficients, every mixed equilibrium with $B\ne 0$ has Jacobian determinant $-2a_2{a}_3{B}^2<0$, hence is a saddle equilibrium. The two invariant-axis equilibria meet in a transcritical bifurcation, and the mixed branches meet them in pitchfork bifurcations.

A near-identity transformation is a local change of variables whose derivative at the reference point is the identity. In a normal form of a dynamical system, substituting $x=y+h(y)$ and solving $(I+Dh(y))\dot{y}=f(y+h(y))$ determines which nonlinear coefficients can be removed. The transformation is locally invertible by the inverse function theorem. Terms whose removal equations are singular are resonant and must be retained; a computed zero resonant coefficient cannot be assigned an arbitrary nonzero value by normalization.