Bogdanov–Takens bifurcation

In bifurcation theory, a field within mathematics, a Bogdanov–Takens bifurcation is a well-studied example of a bifurcation with co-dimension two, meaning that two parameters must be varied for the bifurcation to occur. It is named after Rifkat Bogdanov and Floris Takens, who independently and simultaneously described this bifurcation.

A system y' = f(y) undergoes a Bogdanov–Takens bifurcation if it has a fixed point and the linearization of f around that point has a double eigenvalue at zero (assuming that some technical nondegeneracy conditions are satisfied).

Three codimension-one bifurcations occur nearby: a saddle-node bifurcation, an Andronov–Hopf bifurcation and a homoclinic bifurcation. All associated bifurcation curves meet at the Bogdanov–Takens bifurcation.

The normal form of the Bogdanov–Takens bifurcation is

${\displaystyle {\begin{aligned}y_{1}'&=y_{2},\\y_{2}'&=\beta _{1}+\beta _{2}y_{1}+y_{1}^{2}\pm y_{1}y_{2}.\end{aligned}}}$

There exist two codimension-three degenerate Takens–Bogdanov bifurcations, also known as Dumortier–Roussarie–Sotomayor bifurcations.