A geometric approach to exponentially small splitting: The generic zero-Hopf bifurcation of co-dimension two
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914481485905920 |
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| author | Kristiansen, Kristian Uldall |
| author_facet | Kristiansen, Kristian Uldall |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_12103 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A geometric approach to exponentially small splitting: The generic zero-Hopf bifurcation of co-dimension two Kristiansen, Kristian Uldall Dynamical Systems 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. |
| title | A geometric approach to exponentially small splitting: The generic zero-Hopf bifurcation of co-dimension two |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2603.12103 |