A spectral theorem for reversible second order equations with periodic coefficients
type: Item
instance of: scholarly article
title: A spectral theorem for reversible second order equations with periodic coefficients
zbMATH Open document ID: 0761.58040
author: Daniel C. Offin
published in: Differential and Integral Equations
publication date: 1992-10-04T00:00:00Z
review text: The differential operator \(L=-d^ 2/dt^ 2+Q(t)\) is considered where \(Q\) is a symmetric \(n\) by \(n\) matrix over the Sobolev space of functions \(x:[0,T]\mapsto \mathbb{R}^ n\) satisfying \(x(0)=x(T)\). The Morse index of \(L\) is the number of its negative eigenvalues; the system \(Lx=0\) is said to be reversible if \(Q(-t)=Q(t)\). The main theorem proved in this paper says that if \(Lx=0\) is reversible, the index of \(L\) is zero, and \(Lx=0\) has no symmetric \(T\)-periodic solution then the monodromy matrix \(M(T)\) has no eigenvalues on the unit circle. This means that if, in particular, \(Lx=0\) is the variational system of an autonomous nonlinear system of differential equations with respect to a periodic solution then the latter is hyperbolic, i.e. the system exhibits an exponential dichotomy. The results can be applied to the characterization of periodic solutions of certain Hamiltonian systems with respect to the manifold of solutions with fixed total energy.
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 68018
zbMATH Keywords: second order linear differential operator
zbMATH Keywords: spectrum
zbMATH Keywords: monodromy matrix
zbMATH Keywords: periodic solution
reviewed by: M. Farkas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 68018