Other Bifurcation Theorems

Abstract

Several authors have published generalizations of the Hopf Bifurcation. In particular, Chafee [1] has eliminated the condition that the eigenvalue λ(µ) cross the imaginary axis with nonzero speed. In this case, bifurcation to periodic orbits occurs, but it is not possible to predict from eigenvalue conditions exactly how many families of periodic orbits will bifurcate from the fixed point. Chafee’s result gives a good description of the behavior of the flow of the vector field near the bifurcation point. See also Bautin [1] and Section 3C.