an:06057853
Zbl 1246.65102
Khasawneh, Firas A.; Barton, David A. W.; Mann, Brian P.
Periodic solutions of nonlinear delay differential equations using spectral element method
EN
Nonlinear Dyn. 67, No. 1, 641-658 (2012).
00293582
2012
j
65L03 65L60 65L20 65L12 65L50 34K28
mesh refinement; numerical examples; spectral element method; periodic orbits; autonomous nonlinear delay differential equations; stability; Mackey-Glass equation; Van der Pol equation; Duffing equation; convergence
Summary: We extend the temporal spectral element method further to study the periodic orbits of general autonomous nonlinear delay differential equations (DDEs) with one constant delay. Although we describe the approach for one delay to keep the presentation clear, the extension to multiple delays is straightforward. We also show the underlying similarities between 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 examples, 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 provide comparisons between equal size meshes that have different distributions.