A geometric approach to exponentially small splitting: The generic zero-Hopf bifurcation of co-dimension two

TL;DR

This paper introduces a geometric, dynamical-systems approach to analyze exponentially small splitting in zero-Hopf bifurcations, emphasizing the role of analyticity and the blowup method without explicit time-parametrization.

Contribution

It provides a new geometric proof for exponentially small splitting, linking it to analyticity issues of center manifolds and utilizing the blowup method in complex phase space.

Findings

Established a geometric proof for exponentially small splitting.

Linked splitting to non-analyticity of center-like manifolds.

Utilized the blowup method to relate dynamics across scales.

Abstract

In this paper, we consider the unfolding of the real-analytic and generic zero-Hopf bifurcation of co-dimension two. It is well-known that in an open set of parameter space the splitting of one-dimensional stable and unstable manifolds is beyond all orders. This paper provides a new geometric dynamical-systems-oriented proof for the exponentially small splitting. As a novel aspect, we relate the exponentially small splitting to the lack of analyticity of center-like manifolds of generalized saddle-nodes. Moreover, the blowup method plays an important technical role as a systematic way to relate dynamics on different orders of magnitude. Finally, as our approach takes place in the (complexified) phase space, we do not rely on an explicit time-parametrization of the unperturbed heteroclinic connection. We therefore believe that our approach has general interest.