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


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

with the transformation $$ u = x -1, \mu = r + 1 $$.

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