In bifurcation theory

Bifurcation theory

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...

, a field within mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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

Floris Takens

Floris Takens was a Dutch mathematician known for contributions to the theory of chaotic dynamical systems.Together with David Ruelle he predicted that fluid turbulence could develop through a strange attractor, a term they coined, as opposed to the then-prevailing theory of accretion of modes....

, 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

Saddle-node bifurcation

In the mathematical area of bifurcation theory a saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to...

, an Andronov-Hopf bifurcation and a homoclinic bifurcation

Homoclinic bifurcation

In mathematics, a homoclinic bifurcation is a global bifurcation which often occurs when a periodic orbit collides with a saddle point.The image below shows a phase portrait before, at, and after a homoclinic bifurcation in 2D. The periodic orbit grows until it collides with the saddle point...

. All associated bifurcation curves meet at the Bogdanov-Takens bifurcation.

The normal form

Normal form

Normal form may refer to:* Normal form * Normal form * Normal form * Normal form In formal language theory:* Beta normal form* Chomsky normal form* Greibach normal form* Kuroda normal form...

 of the Bogdanov-Takens bifurcation is

It has also been found the existence of a codimension-three degenerate Takens-Bogdanov bifurcation, also known as Dumortier-Roussarie-Sotomayor bifurcation.