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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.00910 |
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| _version_ | 1866910251064754176 |
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| author | Kalita, Piotr Zgliczyński, Piotr |
| author_facet | Kalita, Piotr Zgliczyński, Piotr |
| contents | We consider the problem governed by the gradient ODE $x'=\nabla F(x)$ in $\mathbb{R}^d$ on which we assume that it has a finite number of hyperbolic equilibria whose stable and unstable manifolds intersect transversally. This problem is perturbed by the memory term $$x'(t)=\nabla F(x(t))+\varepsilon\int_{-\infty}^t M(t-s)x(s)\, ds$$ where $\varepsilon>0$ is a small constant. The key result is that the structure of connections between the equilibria of the unperturbed problem is exactly preserved for a small $\varepsilon>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00910 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability of phase portrait for a gradient ODE with memory Kalita, Piotr Zgliczyński, Piotr Dynamical Systems Classical Analysis and ODEs We consider the problem governed by the gradient ODE $x'=\nabla F(x)$ in $\mathbb{R}^d$ on which we assume that it has a finite number of hyperbolic equilibria whose stable and unstable manifolds intersect transversally. This problem is perturbed by the memory term $$x'(t)=\nabla F(x(t))+\varepsilon\int_{-\infty}^t M(t-s)x(s)\, ds$$ where $\varepsilon>0$ is a small constant. The key result is that the structure of connections between the equilibria of the unperturbed problem is exactly preserved for a small $\varepsilon>0$. |
| title | Stability of phase portrait for a gradient ODE with memory |
| topic | Dynamical Systems Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2406.00910 |