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

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.

A Taylor series-based continuation method for solutions of dynamical systems

“…The first step is to introduce two auxiliary vectors y τ a (t) = y(t − τ ) and y α a = D α (y) in order to isolate the time delays and the fractional order derivatives. These two terms are rewritten quadratically following the frequency-domain implementation developed in [42] and [23] respectively. The system (10) can now be written…”

“…The two last equations are treated separately using appropriate methods (see [42] and [23]). The first equations are a set of Implicit Differential Algebraic Equations (IDAE).…”

“…The IDE formulation (25) has directly the energy as its continuation parameter. The DAE formulation (23) gives the tension of the string without post-processing as it is an unknown of the equations. The ODE formulation (20) can be used to compute the stability of the solutions obtained.…”

“…Two other types of systems have been studied using ANM and HBM. The delay differential systems [23] and the systems with fractional order derivatives [42]. To summarize these type of systems, a Van der Pol oscillator with feedback is exploredẍ…”

“…The bottom figure shows the relative error between the theoretical value (see text) and the computed value of the period of the oscillations in logarithmic scales. In blue the ODE system (21) is solved, in dashed red the DAE system(23) and in dotted yellow the IDE system(26).…”