The hyperbolic umbilic singularity in fast-slow systems
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866914916031528960 |
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| author | Jardón-Kojakhmetov, Hildeberto Kuehn, Christian Steinert, Maximilian |
| author_facet | Jardón-Kojakhmetov, Hildeberto Kuehn, Christian Steinert, Maximilian |
| contents | Fast-slow systems with three slow variables and gradient structure in the fast variables have, generically, hyperbolic umbilic, elliptic umbilic or swallowtail singularities. In this article we provide a detailed local analysis of a fast-slow system near a hyperbolic umbilic singularity. In particular, we show that under some appropriate non-degeneracy conditions on the slow flow, the attracting slow manifolds jump onto the fast regime and fan out as they cross the hyperbolic umbilic singularity. The analysis is based on the blow-up technique, in which the hyperbolic umbilic point is blown up to a 5-dimensional sphere. Moreover, the reduced slow flow is also blown up and embedded into the blown-up fast formulation. Further, we describe how our analysis is related to classical theories such as catastrophe theory and constrained differential equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_01662 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The hyperbolic umbilic singularity in fast-slow systems Jardón-Kojakhmetov, Hildeberto Kuehn, Christian Steinert, Maximilian Dynamical Systems 34E15, 34A26, 58K35 Fast-slow systems with three slow variables and gradient structure in the fast variables have, generically, hyperbolic umbilic, elliptic umbilic or swallowtail singularities. In this article we provide a detailed local analysis of a fast-slow system near a hyperbolic umbilic singularity. In particular, we show that under some appropriate non-degeneracy conditions on the slow flow, the attracting slow manifolds jump onto the fast regime and fan out as they cross the hyperbolic umbilic singularity. The analysis is based on the blow-up technique, in which the hyperbolic umbilic point is blown up to a 5-dimensional sphere. Moreover, the reduced slow flow is also blown up and embedded into the blown-up fast formulation. Further, we describe how our analysis is related to classical theories such as catastrophe theory and constrained differential equations. |
| title | The hyperbolic umbilic singularity in fast-slow systems |
| topic | Dynamical Systems 34E15, 34A26, 58K35 |
| url | https://arxiv.org/abs/2202.01662 |