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Periodic solutions occur commonly in linear or nonlinear dynamical systems. In some cases, the stability of a periodic solution holds if the rightmost characteristic root has negative real part. Based on the Lambert W function, this paper presents a simple algorithm for locating the rightmost characteristic root of periodic solutions of some nonlinear oscillators with large time delay. As application, the proposed algorithm is used to study the primary resonance and 1/3 subharmonic resonance of the Duffing oscillator under harmonic excitation and delayed feedback, as well as the control problem of the van der Pol oscillator under harmonic excitation by using delayed feedback, with a number of case studies. The main advantage of this algorithm is that though very simple in implementation, it works effectively with high accuracy even if the delay is large.
Periodic solutions occur generally in linear or nonlinear dynamical systems. In some cases, the stability of a periodic solution holds if the rightmost characteristic root has negative real part. Based on the Lambert W function, this paper presents a simple algorithm for locating the rightmost characteristic root of periodic solutions of some nonlinear oscillators with large time delay. As application, the proposed algorithm is used to study the primary resonance and 1/3 subharmonic resonance of the Duffing oscillator under harmonic excitation and delayed feedback, as well as the control problem of the van der Pol oscillator under harmonic excitation by using delayed feedback, with a number of case studies. The main advantage of this algorithm is that though very simple in implementation, it works effectively with high accuracy even if the delay is large.