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 d x d t = μ + x 2 where μ is the bifurcation parameter. The transcritical bifurcation d x d t = r ln ⁡ x + x − 1 near x = 1 can be converted to the normal form d u d t = μ u − u 2 + O ( u 3 ) with the transformation u = x − 1 , μ = r + 1 .