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arXiv · 2406.00910

Stability of phase portrait for a gradient ODE with memory

Piotr Kalita·Piotr Zgliczyński
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Abstract

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 $\varepsilon>0$.

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Piotr Kalita, Piotr Zgliczyński. 2024-06-03. Stability of phase portrait for a gradient ODE with memory. https://arxiv.org/abs/2406.00910

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