Normal form (bifurcation theory)

In mathematics, the normal form of a bifurcation is a simple dynamical system which is equivalent to all systems exhibiting this bifurcation.

Normal forms are defined for local bifurcations. It is assumed that the system is first reduced to the center manifold of the equilibrium point at which the bifurcation takes place. The reduced system is then 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.

References

Guckenheimer, John; Holmes, Philip (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Section 3.3, ISBN 0-387-90819-6 .
Kuznetsov, Yuri A. (1998), Elements of Applied Bifurcation Theory (Second ed.), Springer, Section 2.4, ISBN 0-387-98382-1 .