Optimal waveform for the entrainment of a weakly forced oscillator

TL;DR

This paper develops a theoretical method to determine the optimal waveform for entraining a weakly forced oscillator with minimal power, validated through chemical experiments showing different waveforms near and far from bifurcation.

Contribution

It introduces a combined phase model and calculus of variation approach to derive minimal-power entrainment waveforms based on the phase response curve.

Findings

Optimal waveforms depend on proximity to bifurcation.

Sinusoidal waveforms are optimal near bifurcation.

Higher harmonic waveforms are optimal further from bifurcation.

Abstract

A theory for obtaining waveform for the effective entrainment of a weakly forced oscillator is presented. Phase model analysis is combined with calculus of variation to derive a waveform with which entrainment of an oscillator is achieved with minimum power forcing signal. Optimal waveforms are calculated from the phase response curve and a solution to a balancing condition. The theory is tested in chemical entrainment experiments in which oscillations close to and further away from a Hopf bifurcation exhibited sinusoidal and higher harmonic nontrivial optimal waveforms, respectively.