Bifurcation from multiple complex eigenvalues

Abstract

We shall study bifurcation and stability for nonlinear ordinary differential systems of arbitrary dimension when an equilibrium solution loses its stability by virtue of two pairs (α(λ) ± iβ(λ) ± iδ(λ)) complex conjugate eigenvalues of the linearized system simultaneously crossing the imaginary axis. Such a situation is not at all uncommon, and we shall give applications where this situation is indeed the usual phenomenon encountered. In these situations considerably different and more diverse behavior can occur than in the simpler bifurcation at a simple complex eigenvalue (Hopf bifurcation). As we shall see, the complexity of bifurcating solutions will result principally from the simple fact that superpositions such as βt + sin δt are not periodic if β and δ are incommensurate. The extension of our theory to account for arbitrary numbers of pairs of complex conjugate eigenvalues will be clear.