Analytic weak-stable manifolds in unfoldings of saddle-nodes
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915034731380736 |
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| author | Kristiansen, Kristian Uldall Szmolyan, Peter |
| author_facet | Kristiansen, Kristian Uldall Szmolyan, Peter |
| contents | Any attracting, hyperbolic and proper node of a two-dimensional analytic vector-field has a unique strong-stable manifold. This manifold is analytic. The corresponding weak-stable manifolds are, on the other hand, not unique, but in the nonresonant case there is a unique weak-stable manifold that is analytic. As the system approaches a saddle-node (under parameter variation), a sequence of resonances (of increasing order) occur. In this paper, we give a detailed description of the analytic weak-stable manifolds during this process. In particular, we relate a ``flapping-mechanism'', corresponding to a dramatic change of the position of the analytic weak-stable manifold as the parameter passes through the infinitely many resonances, to the lack of analyticity of the center manifold at the saddle-node. Our work is motivated and inspired by the work of Merle, Raphaël, Rodnianski, and
Szeftel, where this flapping mechanism is the crucial ingredient in the construction of $C^\infty$-smooth self-similar solutions of the compressible Euler equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01488 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analytic weak-stable manifolds in unfoldings of saddle-nodes Kristiansen, Kristian Uldall Szmolyan, Peter Dynamical Systems 34C23, 34C45, 37G05, 37G10 Any attracting, hyperbolic and proper node of a two-dimensional analytic vector-field has a unique strong-stable manifold. This manifold is analytic. The corresponding weak-stable manifolds are, on the other hand, not unique, but in the nonresonant case there is a unique weak-stable manifold that is analytic. As the system approaches a saddle-node (under parameter variation), a sequence of resonances (of increasing order) occur. In this paper, we give a detailed description of the analytic weak-stable manifolds during this process. In particular, we relate a ``flapping-mechanism'', corresponding to a dramatic change of the position of the analytic weak-stable manifold as the parameter passes through the infinitely many resonances, to the lack of analyticity of the center manifold at the saddle-node. Our work is motivated and inspired by the work of Merle, Raphaël, Rodnianski, and Szeftel, where this flapping mechanism is the crucial ingredient in the construction of $C^\infty$-smooth self-similar solutions of the compressible Euler equations. |
| title | Analytic weak-stable manifolds in unfoldings of saddle-nodes |
| topic | Dynamical Systems 34C23, 34C45, 37G05, 37G10 |
| url | https://arxiv.org/abs/2403.01488 |