@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.}
}