Bogdanov–Takens bifurcation explained

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

\begin{align} y _{1'} & = y _{2,} \\ y _{2'} & = \beta _{1} + \beta _{2} y _{1} + y ^{2} _{1} \pm y _{1} y _{2. \end{align}}

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

References

• Bogdanov, R. "Bifurcations of a Limit Cycle for a Family of Vector Fields on the Plane." Selecta Math. Soviet 1, 373 - 388, 1981.
• Kuznetsov, Y. A. Elements of Applied Bifurcation Theory. New York: Springer-Verlag, 1995.
• Takens, F. "Forced Oscillations and Bifurcations." Comm. Math. Inst. Rijksuniv. Utrecht 2, 1 - 111, 1974.
• Dumortier F., Roussarie R., Sotomayor J. and Zoladek H., Bifurcations of Planar Vector Fields, Lecture Notes in Math. vol. 1480, 1 - 164, Springer-Verlag (1991).

External links

• Bogdanov–Takens Bifurcation. Guckenheimer. John. Yuri A. Kuznetsov. 2007. Scholarpedia. 2. 1854. 10.4249/scholarpedia.1854. free. John Guckenheimer.