On rank one perturbations of Hamiltonian system with periodic coefficients

TL;DR

This paper develops a theory for rank one perturbations of Hamiltonian systems with periodic coefficients, showing how such perturbations affect the fundamental solutions and stability, supported by numerical examples.

Contribution

It introduces a new theoretical framework for understanding rank one perturbations in Hamiltonian systems with periodic coefficients, extending previous work by C. Mehl et al.

Findings

Rank one perturbation of the fundamental solution is itself a solution of the perturbed system.

Strong stability of Hamiltonian systems influences their rank one perturbations.

Numerical examples validate the theoretical results.

Abstract

From a theory developed by C. Mehl, et al., a theory of the rank one perturbation of Hamiltonian systems with periodic coefficients is proposed. It is showed that the rank one perturbation of the fundamental solution of Hamiltonian system with periodic coefficients is solution of its rank one perturbation. Some results on the consequences of the strong stability of these types of systems on their rank one perturbation is proposed. Two numerical examples are given to illustrate this theory.