Linear Hamilton Systems without Regular Properties. Solving a Problem Stated by M.G. Krein

TL;DR

This paper constructs linear Hamilton systems lacking the usual dichotomy property, revealing complex Lyapunov spectra and trajectory behaviors, impacting the stability analysis in indefinite inner product spaces.

Contribution

It introduces Hamilton systems without regular properties, challenging existing stability methods and expanding understanding of indefinite inner product dynamics.

Findings

Unusual Lyapunov spectra observed

Complex trajectory behaviors demonstrated

Implications for stability theory in indefinite spaces

Abstract

We construct linear Hamilton systems without usual dichotomy property. The Ljapunov spectra of these systems are unfamiliar and conflicting, the behaviour of trajectories is very complicated. The paper's subject refers to some problems of indefinite inner product methods in the stability theory of abstract dynamical equation solutions.