 Bogdanov–Takens bifurcation

A Bogdanov–Takens bifurcation has a double zero eigenvalue with a nontrivial Jordan block and generic two-parameter unfolding. Its local diagram includes saddle-node bifurcations, Hopf bifurcations, and a curve of small homoclinic orbits. A weakly perturbed Hamiltonian scaling can locate the homoclinic curve through energy balance.

The Bogdanov–Takens bifurcation is a significant phenomenon in the study of dynamical systems, particularly in the context of the behavior of nonlinear systems. It describes a scenario in which a system undergoes a bifurcation, leading to the simultaneous occurrence of a transcritical bifurcation (where the stability of fixed points is exchanged) and a Hopf bifurcation (where a fixed point becomes unstable and bifurcates into a periodic orbit).