Stability of linear periodic systems

Abstract

A method for calculating the solution, in Floquet form, to a system of linear differential equations with periodic parameters is developed. As a result, both the periodic and exponential parts of the solution are developed as power series in a small parameter, ϵ. From this solution, approximate expressions for the characteristic exponents are derived and hence the stability of the system can be determined. The technique is analogous and parallels that developed by Cesari, Hale, Gambill, et al. By returning to the basis of the underlying process of casting out the secular terms, a method is developed which has advantages from the technical and computational point of view. The expressions for the characteristic exponents are explicit, namely, series of powers of ϵ, whose terms can be easily written, and which converge for ϵ sufficiently small. More importantly, from a practical standpoint, an estimate of the error which results when the series is truncated is derived. Using such an estimate, we draw a circle on the complex plane, centered at the approximate characteristic exponents, within which the exact characteristic exponents must lie.