Uppsala University Department of Information Technology

Technical Report 2003-025

Second Order ODEs are Sufficient for Modeling of Many Periodic Signals

Torbjörn Wigren and Torsten Söderström

April 2003

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.

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