Stability of phase portrait for a gradient ODE with memory

Abstract

We consider the problem governed by the gradient ODE $x^{\prime }=\nabla F(x)$ 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 $$x^{\prime }(t)=\nabla F(x(t))+\epsilon {\int }_{-\infty }^{t}M(t-s)x(s)ds$$ where $\epsilon >0$ 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 $\epsilon >0$.

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