Abstract
This article presents an extension of the asymptotic numerical method combined with the harmonic balance method to the continuation of periodic orbits of delay differential equations. The equations can be forced or autonomous and possibly of neutral type. The approach developed in this paper requires the system of equations to be written in a quadratic formalism which is detailed. The method is applied to various systems, from Van der Pol and Duffing oscillators to toy models of clarinet and saxophone. The harmonic balance method is ascertained from a comparison to standards time-integration solvers. Bifurcation diagrams are drawn which are sometimes intricate, showing the robustness of this method.
Highlights
The aim of this article is to extend the framework of the numerical continuation of periodic solutions of differential systems using the Harmonic Balance Method (HBM) coupled with the Asymptotic Numerical Method (ANM) to time-delayed systems
As opposed to already existing methods of continuation of delay differential equations (DDE), see [12] and its adaptation for neutral systems [2], the method developed in this paper is based on the frequency representation of the periodic solutions
For the signals represented in figure 6, the acoustic pressure obtained by Harmonic Balance has a mean value around the numerical precision 10−16 while the time integration solver from [18] gives a mean around 10−4 and ddensd gives 10−3
Summary
To cite this version: Louis Guillot, Christophe Vergez, Bruno Cochelin. Continuation of periodic solutions of various types of Delay Differential Equations using Asymptotic Numerical Method and Harmonic Balance Method. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés
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