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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2408.06548 |
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| _version_ | 1866916665390792704 |
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| author | Ivanov, Anatoli F. Lani-Wayda, Bernhard |
| author_facet | Ivanov, Anatoli F. Lani-Wayda, Bernhard |
| contents | For a class of $(N+1)$-dimensional systems of differential delay equations with a cyclic and monotone negative feedback structure, we construct a two-dimensional invariant manifold, on which phase curves spiral outward towards a bounding periodic orbit. For this to happen we assume essentially only instability of the zero equilibrium. Methods of the Poincaré-Bendixson theory due to Mallet-Paret and Sell are combined with techniques used by Walther for the scalar case $(N = 0)$. Statements on the attractor location and on parameter borders concerning stability and oscillation are included. The results apply to models for gene regulatory systems, e.g. the `repressilator' system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06548 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The strong unstable manifold and periodic solutions in differential delay equations with cyclic monotone negative feedbck Ivanov, Anatoli F. Lani-Wayda, Bernhard Dynamical Systems 34K13 For a class of $(N+1)$-dimensional systems of differential delay equations with a cyclic and monotone negative feedback structure, we construct a two-dimensional invariant manifold, on which phase curves spiral outward towards a bounding periodic orbit. For this to happen we assume essentially only instability of the zero equilibrium. Methods of the Poincaré-Bendixson theory due to Mallet-Paret and Sell are combined with techniques used by Walther for the scalar case $(N = 0)$. Statements on the attractor location and on parameter borders concerning stability and oscillation are included. The results apply to models for gene regulatory systems, e.g. the `repressilator' system. |
| title | The strong unstable manifold and periodic solutions in differential delay equations with cyclic monotone negative feedbck |
| topic | Dynamical Systems 34K13 |
| url | https://arxiv.org/abs/2408.06548 |