Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs

TL;DR

This paper develops Hill-type and Krein-type trace formulas for S-periodic solutions in first-order ODEs, extending classical formulas and exploring their relationship, especially for symmetric and quasi-periodic solutions.

Contribution

It derives the Hill-type formula for S-periodic orbits in first-order ODEs and constructs a Krein-type trace formula as a non-self-adjoint extension, revealing their connection.

Findings

Established Hill-type formula for S-periodic solutions in first-order ODEs.

Constructed Krein-type trace formula based on Hill-type formula.

Linked the formulas to symmetric and quasi-periodic solutions.

Abstract

The present paper is devoted to studying the Hill-type formula and Krein-type trace formula for ODE, which is a continuous work of our previous work for Hamiltonian systems \cite{HOW}. Hill-type formula and Krein-type trace formula are given by Hill at 1877 and Krein in 1950's separately. Recently, we find that there is a closed relationship between them \cite{HOW}. In this paper, we will obtain the Hill-type formula for the $S$-periodic orbits of the first order ODEs. Such a kind of orbits is considered naturally to study the symmetric periodic and quasi-periodic solutions. By some similar idea in \cite{HOW}, based on the Hill-type formula, we will build up the Krein-type trace formula for the first order ODEs, which can be seen as a non-self-adjoint version of the case of Hamiltonian system.