Stability of phase portrait for a gradient ODE with memory

TL;DR

This paper proves that the qualitative structure of the phase portrait for a gradient ODE with hyperbolic equilibria remains stable under small perturbations involving a memory term.

Contribution

It establishes the preservation of equilibrium connections in a gradient ODE when perturbed by a small memory term, extending classical stability results.

Findings

The structure of connections between equilibria is preserved for small memory perturbations.

The stability of hyperbolic equilibria is maintained under the memory perturbation.

The phase portrait's qualitative features are robust to the introduced memory term.

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 $\varepsilon>0$.