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 
where formula_2 is the bifurcation parameter. The transcritical bifurcation 
near formula_4 can be converted to the normal form 
with the transformation formula_6.
See also canonical form for use of the terms canonical form, normal form, or standard form more generally in mathematics.