Continuation of periodic solutions of various types of delay differential equations using asymptotic numerical method and harmonic balance method

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.