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.
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The authors want to thank Tom Colinot for the help with the saxophone model and its associated time-integration method. This work has been carried out in the framework of the Labex MEC (ANR-10-LABX-0092) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the Investissements d’Avenir French Government program managed by the French National Research Agency (ANR). Conflict of Interest: The authors declare that they have no conflict of interest.
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Guillot, L., Vergez, C. & Cochelin, B. Continuation of periodic solutions of various types of delay differential equations using asymptotic numerical method and harmonic balance method. Nonlinear Dyn 97, 123–134 (2019). https://doi.org/10.1007/s11071-019-04958-y
- Nonlinear dynamics
- Delay equations
- Harmonic balance method
- Numerical continuation
- Periodic solutions
- Quadratic recast
- Asymptotic numerical method