Normal form (dynamical systems)

In mathematics, the normal form of a dynamical system is a simplified form that can be useful in determining the system's behavior.

Normal forms are often used for determining local bifurcations in a system. All systems exhibiting a certain type of bifurcation are locally (around the equilibrium) topologically equivalent to the normal form of the bifurcation. For example, the normal form of a saddle-node bifurcation is

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=\mu +x^{2}}$

where ${\displaystyle \mu }$ is the bifurcation parameter. The transcritical bifurcation

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=r\ln x+x-1}$

near ${\displaystyle x=1}$ can be converted to the normal form

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=Ru-u^{2}+O(u^{3})}$

with the transformation ${\displaystyle u={\frac {r}{2}}(x-1),R=r+1}$.^[1]

See also canonical form for use of the terms canonical form, normal form, or standard form more generally in mathematics.