Hopf Meets Hamilton Under Whitney’s Umbrella

Abstract

In Hamiltonian mechanics, the classical Hopf bifurcation theorem is not directly applicable. Instead, there is an analogous “Hamiltonian-Hopf bifurcation theorem” in which two pairs of complex conjugate eigenvalues approach the imaginary axis symmetrically from the left and right, then merge in double purely imaginary eigenvalues and separate along the imaginary axis (or the reverse). This phenomenon has codimension one within the class of Hamiltonian systems. In the general case of non-Hamiltonian vector fields, the occurrence of double imaginary eigenvalues has codimension three. This paper presents a first investigation of the interface between these two cases. They meet in an interesting topological singularity known as Whitney’s umbrella. We show that the Hamiltonian case lies on the “handle” of Whitney’s umbrella. This allows us to investigate near-Hamiltonian or weakly dissipative systems that lie in a tubular neigh-borhood of the handle of Whitney’s umbrella.