bifurcation theory, a field within mathematics, a Bogdanov-Takens bifurcation is a well-studied example of a bifurcation with co-dimensiontwo, meaning that two parameters must be varied for the bifurcation to occur. It is named after R. I. Bogdanovand 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
eigenvalueat zero (assuming that some technical nondegeneracy conditions are satisfied).
Three codimension-one bifurcations occur nearby: a
saddle-node bifurcation, an Andronov-Hopf bifurcationand a homoclinic bifurcation. All associated bifurcation curves meet at the Bogdanov-Takens bifurcation.
normal formof the Bogdanov-Takens bifurcation is:
It has also been found the existence of a codimension-three degenerate Takens-Bogdanov bifurcation, also known as
*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).
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