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

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

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

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

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

d u d t = μ u − u 2 + O ( u 3 ) {\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=\mu u-u^{2}+O(u^{3})}

with the transformation u = x − 1 , μ = r + 1 {\displaystyle u=x-1,\mu =r+1} .

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