Nonlinear Dyn DOI 10.1007/s11071-011-0017-3 ORIGINAL PAPER Periodic solutions of nonlinear delay differential equations using spectral element method Firas A. Khasawneh · David A.W. Barton · Brian P. Mann Received: 21 September 2010 / Accepted: 7 March 2011 © Springer Science+Business Media B.V. 2011 Abstract We extend the temporal spectral element method further to study the periodic orbits of gen- eral autonomous nonlinear delay differential equations (DDEs) with one constant delay. Although we de- scribe the approach for one delay to keep the presenta- tion clear, the extension to multiple delays is straight- forward. We also show the underlying similarities be- tween this method and the method of collocation. The spectral element method that we present here can be used to find both the periodic orbit and its stability. This is demonstrated with a variety of different exam- ples, namely, the delayed versions of Mackey–Glass equation, Van der Pol equation, and Duffing equation. For each example, we show the method’s convergence behavior using both p and h refinement and we pro- vide comparisons between equal size meshes that have different distributions. Keywords Nonlinear equations · Delay differential equations · Spectral element F.A. Khasawneh ( ) · B.P. Mann Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA e-mail: firas.khasawneh@duke.edu D.A.W. Barton Department of Engineering Mathematics, University of Bristol, Bristol, BS8 1TR, UK Delay differential equations (DDEs) have been suc- cessfully used to model many different phenomena occurring in science and engineering contexts. Spe- cific areas in which DDEs have been used extensively include machining dynamics [7, 10, 40], systems bi- ology [2, 8, 34] and laser systems [26, 37]. In some models, for example in systems biology related appli- cations, delays are used to avoid modeling certain pro- cesses that are known to take a predetermined amount of time but otherwise contribute little to the dynamics. In other models, for example in machining dynamics, the delay is intrinsic to the system and cannot be re- moved. DDEs are infinite dimensional dynamic systems whose state-space is typically taken to be the space of continuous functions. Therefore, DDEs require a func- tion segment over a period of time as an initial condi- tion rather than a point value at time zero as with an ordinary differential equation. The infinite dimension- ality of DDEs significantly complicates the resulting analysis from both an analytical and numerical per- spective [20, 39]. Furthermore, complicated behavior can be readily observed in seemingly low-order equa- tions [4, 21]. Due to the difficulties associated with the analyt- ical aspects of DDEs, there has been significant fo- cus on their numerical solution. The majority of this work has focused on initial value solvers, mostly ex- tending Runge–Kutta solvers to DDEs [5, 16, 18]. Al- ternatively, other solvers based on an implicit Radau method, e.g., RADAR5 by Guglielmi and Hairer [17],