@TechReport{ it:2003-025,
author = {Torbj{\"o}rn Wigren and Torsten S{\"o}derstr{\"o}m},
title = {Second Order {ODE}s are Sufficient for Modeling of Many
Periodic Signals},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Systems and Control},
year = {2003},
number = {2003-025},
month = apr,
abstract = {Which is the minimum order an autonomous nonlinear
ordinary differential equation (ODE) needs to have to be
able to model a periodic signal? This question is motivated
by recent research on periodic signal analysis, where
nonlinear ODEs are used as models. The results presented
here show that an order of two of the ODE is sufficient for
a large class of periodic signals. More precisely,
conditions on a periodic signal are established that imply
the existence of an ODE that has the periodic signal as a
solution. A criterion that characterizes the above class of
periodic signals by means of the overtone contents of the
signals is also presented. The reason why higher order ODEs
are sometimes needed is illustrated with geometric
arguments. Extensions of the theoretical analysis to cases
with orders higher than two are developed using this
insight.}
}