Significance of floquet eigenvalues and eigenvectors for the dynamics of time-varying systems

The history of the analysis of periodic-coefficient systems is surveyed along with a discussion of the long-standing issue of how to interpret the imaginary part of the Floquet system exponent (which arises from the complex logarithm of the eigenvalues of the Transition Matrix). In particular, it is well-known that one may add or subtract any integer multiple of the fundamental frequency to the imaginary part of the system exponent; and this arbitrariness has resulted both in confusion and in the mistaken idea that there must be one "correct" integer to be added. A similar conundrum has arisen as to how to interpret the case in which the eigenvalues of the Floquet Transition Matrix split on the negative real axis. In that case, the system exponents no longer come in complex-conjugate pairs; and there has been confusion as to how to interpret the non-conjugate roots and whether or not to use different integers for the different roots.This paper will demonstrate that the choice of integer multiple to be added (in either the general case or the case of a negative-real split) is an entirely arbitrary bookkeeping decision. This is due to the fact that the Floquet eigenvector, which multiplies the exponential term, is periodic in time and contains in principle all of the integer-multiple frequencies in varying degrees. Thus, it is the product of the eigenvector and exponential that must be used to determine frequency content; and that product is unique. The uniqueness is established because once an integer is chosen that choice in turn affects the computation of the periodic eigenvector. It follows that the resulting product of the exponential and the eigenvector remains uniqueindependent of the choice of integer. This product contains the true frequency content of the system dynamics including the relative strengths of each integer-multiple harmonic. Examples are given in the paper.