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Periodic solutions of nonlinear delay differential equations using spectral element method. (English) Zbl 1246.65102
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.

##### MSC:
 65L03 Numerical methods for functional-differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L12 Finite difference and finite volume methods for ordinary differential equations 65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations 34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)