A new example on Lyapunov stability

TL;DR

This paper presents a novel example of an infinite-dimensional ODE where the equilibrium is asymptotically stable despite the linearized operator having spectrum in the right half-plane, addressing an open theoretical question.

Contribution

It provides the first known example of such an ODE in Hilbert space, using techniques based on Kakutani's classical operator theory example.

Findings

Demonstrates existence of an asymptotically stable equilibrium with spectrum in the right half-plane

Addresses an open question in the spectral theory of infinite-dimensional dynamical systems

Uses novel construction techniques inspired by classical operator theory

Abstract

The purpose of this paper is to present an example of an Ordinary Differential Equation $x'=F(x)$ in the infinite-dimensional Hilbert space $\ell^2$ with $F$ being of class $\mathcal{C}^1$ in the Fr\'{e}chet sense, such that the origin is an asymptotically stable equilibrium point but the spectrum of the linearized operator $DF(0)$ intersects the half-plane $\Re(z)>0$. The possible existence or not of an example of this kind has been an open question until now, to our knowledge. An analogous example, but of a non-invertible map instead of a flow defined by an ODE was recently constructed by the authors in a recent paper. The two examples use different techniques, but both are based on a classical example in Operator Theory due to S. Kakutani.