THE PROBLEM OF HILL FOR SYSTEMS

Abstract

This chapter discusses Hill's problem for systems. In the special case of the Mathieu equation, the infinite determinant is the determinant of a continuant matrix, and continued fraction methods allow the systematic computation of the characteristic exponents. The appropriate generalization of the infinite determinant for higher-order scalar systems with periodic coefficients is reported in this system. For general linear periodic Hamiltonian systems, a new method can be developed that not only gives the characteristic exponents but also allows for the numerical determination of the associated constant Hamiltonian system given by the Floquet theory. The results are based on finding stable periodic solutions of an associated matrix Riccati equation with periodic coefficient matrices. The determination of these periodic equilibrium solutions actually allows the determination of the constant coefficient matrices for the Riccati equation, which is associated with the Floquet equivalent system.