Anharmonic resonances with recursive delay feedback

TL;DR

This paper explores how recursive delay feedback can control nonlinear oscillators, revealing resonances that significantly affect oscillation coherence, with analytical and numerical validation on van der Pol models.

Contribution

It develops a general theory for noisy limit-cycle oscillators with recursive delay feedback, highlighting the role of anharmonicity and identifying resonances that influence coherence.

Findings

Recursive delay feedback induces anharmonic resonances.

Resonances significantly affect oscillation coherence.

Analytical results agree with numerical simulations.

Abstract

We consider application of the multiple time delayed feedback for control of anharmonic (nonlinear) oscillators subject to noise. In contrast to the case of a single delay feedback, the multiple one exhibits resonances between feedback and nonlinear harmonics, leading to a resonantly strong or weak oscillation coherence even for a small anharmonicity. Analytical results are confirmed numerically for van der Pol and van der Pol-Duffing oscillators. Highlights: > We construct general theory of noisy limit-cycle oscillators with linear feedback. > We focus on coherence and "reliability" of oscillators. > For recursive delay feedback control the theory shows importance of anharmonicity. > Anharmonic resonances are studied both numerically and analytically.