# Preprint of the project: DFG Priority Research Program: DANSE

## Connecting orbits in highdimensional dynamical systems

50/96 Wolf-Jürgen Beyn, Thorsten Pampel (Göke).
A note on the Hopf stability formula under nonresonance conditions

The Hopf stability formula determines the super- or subcriticality of periodic branches at a Hopf bifurcation point as well as the stability of the emanating periodic orbits. The standard approach to this formula uses the center manifold reduction and hence requires that there are no further eigenvalues on the imaginary axis except for the complex conjugate pair. In this paper we give a complete derivation of this formula under the weaker nonresonance condition which only excludes integer multiples of the purely imaginary pair. We show that the shooting type approach in [Amann 90] can be corrected in such a way that a full algorithmic treatment of the stability formula becomes possible.