2015/04/08 by Hu, Xijun, Wang, Penghui · 1 citation
#34B09 #34C27 #34L05 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1504.01815
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 \citeHOW. 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 \citeHOW. 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 \citeHOW, 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.