On a perturbation theory of Hamiltonian systems with periodic coefficients

TL;DR

This paper develops a perturbation theory for Hamiltonian systems with periodic coefficients, focusing on rank k perturbations of symplectic matrices, and analyzes their effects on system stability and canonical forms.

Contribution

It introduces a new framework for analyzing rank k perturbations of Hamiltonian systems with periodic coefficients using isotropic subspaces.

Findings

Fundamental matrices of perturbed and unperturbed systems are identical.

The Jordan canonical form of the perturbed fundamental matrix is characterized.

Numerical examples demonstrate the impact of perturbations on system stability.

Abstract

A theory of rank $k\ge 2$ perturbation of symplectic matrices and Hamiltonian systems with periodic coefficients using a base of isotropic subspaces, is presented. After showing that the fundamental matrix ${\displaystyle \left(\widetilde{X}(t)\right)_{t\ge 0}}$ of the rank $k$ perturbation of Hamiltonian system with periodic coefficients and the rank $k$ perturbation of the fundamental matrix ${\displaystyle \left(X(t)\right)_{t\ge 0}}$ of the unperturbed system are the same, the Jordan canonical form of ${\displaystyle \left(\widetilde{X}(t)\right)_{t\ge 0}}$ is given. Two numerical examples illustrating this theory and the consequences of rank $k$ perturbations on the strong stability of Hamiltonian systems were also given.