Normal form (bifurcation theory)

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 ${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=\mu +x^{2}}$ where ${\displaystyle \mu }$ is the bifurcation parameter. The transcritical bifurcation ${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=r\ln x+x-1}$ near ${\displaystyle x=1}$ can be converted to the normal form ${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=\mu u-u^{2}+O(u^{3})}$ with the transformation ${\displaystyle u=x-1,\mu =r+1}$.

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