Optimal waveform for the entrainment of oscillators perturbed by an amplitude-modulated high-frequency force

TL;DR

This paper develops a theoretical framework to determine the optimal envelope waveform for entraining oscillators using amplitude-modulated high-frequency forces, maximizing the frequency range of entrainment.

Contribution

It introduces a novel optimization approach using phase reduction and Pontryagin's principle to find bang-bang waveforms for oscillator entrainment.

Findings

Optimal envelope waveform is bang-bang type.

Inversion symmetry relates signals with different powers.

Numerical validation on FitzHugh-Nagumo oscillators confirms theory.

Abstract

We analyze limit cycle oscillators under perturbation constructed as a product of two signals, namely, an envelope with a period close to natural period of an oscillator and a high-frequency carrier signal. A theory for obtaining an envelope waveform that achieves the maximal frequency interval of entrained oscillators is presented. The optimization problem for fixed power and maximal allowed amplitude is solved by employing the phase reduction method and the Pontryagin's maximum principle. We have shown that the optimal envelope waveform is a bang-bang-type solution. Also, we have found "inversion" symmetry that relates two signals with different powers, but the same interval of entrained frequencies. The theoretical results are confirmed numerically on FitzHugh-Nagumo oscillators.