# An example on Lyapunov stability and linearization

## TL;DR

This paper constructs an example of an infinite-dimensional discrete dynamical system where the fixed point is exponentially stable but the linearization suggests instability, addressing an open question in Lyapunov stability theory.

## Contribution

It provides the first known example of such a system, demonstrating a discrepancy between nonlinear stability and linearized spectral analysis in infinite dimensions.

## Key findings

- The origin is exponentially asymptotically stable in the nonlinear system.
- The derivative at the origin has spectral radius greater than one.
- This example resolves an open question in the theory of stability in infinite-dimensional systems.

## Abstract

The purpose of this paper is to present an example of a C1 (in the Fr\'echet sense) discrete dynamical system in a infinite-dimensional separable Hilbert space for which the origin is an exponentially asymptotically stable fixed point, but such that its derivative at the origin has spectral radius larger than unity, and this means that the origin is unstable in the sense of Lyapunov for the linearized system. The possible existence or not of an example of this kind has been an open question until now, to our knowledge. The construction is based on a classical example in Operator Theory due to Kakutani.