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

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

where $\mu$ is the bifurcation parameter. The transcritical bifurcation

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

near $x=1$ can be converted to the normal form

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

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

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