Reversible linear systems and their versal deformations

Abstract

Let R be a fixed linear involution (R 2=id) of the spaceR n. A linear operator M is said to bereversible with respect to R if RM R=M−1 and infinitesimally reversible with respect to R if M R=−RM. A linear differential equation dx/dt=B(t)x is said to be reversible with respect to R if V(t)R ≡−RV(−t). We construct normal forms and versal deformations for reversible and infinitesimally reversible operators. The results are applied to describe the homotopy classes of strongly stable reversible linear differential equations with periodic coefficients. The analogous theory for linear Hamiltonian systems was developed by J. Williamson, M. G. Krein, I.M. Gel'fand, V. B. Lidskii, D. M. Galin, and H. Koçak.