Fast optimal entrainment of limit-cycle oscillators by strong periodic inputs via phase-amplitude reduction and Floquet theory

TL;DR

This paper introduces two novel methods based on phase-amplitude reduction and Floquet theory for achieving fast and robust entrainment of limit-cycle oscillators with strong periodic inputs, surpassing traditional phase-only approaches.

Contribution

The paper presents two new techniques for optimal entrainment using strong inputs, incorporating amplitude feedback and penalty methods within Floquet theory, enhancing speed and range of entrainment.

Findings

Methods achieve faster entrainment than traditional phase-only approaches.

Proposed techniques extend the entrainment range significantly.

Validated on van der Pol and Rössler oscillators with various Floquet exponents.

Abstract

Optimal entrainment of limit-cycle oscillators by strong periodic inputs is studied on the basis of the phase-amplitude reduction and Floquet theory. Two methods for deriving the input waveforms that keep the system state close to the original limit cycle are proposed, which enable the use of strong inputs for entrainment. The first amplitude-feedback method uses feedback control to suppress deviations of the system state from the limit cycle, while the second amplitude-penalty method seeks an input waveform that does not excite large deviations from the limit cycle in the feedforward framework. Optimal entrainment of the van der Pol and Willamowski-R\"ossler oscillators with real or complex Floquet exponents are analyzed as examples. It is demonstrated that the proposed methods can achieve considerably faster entrainment and provide wider entrainment ranges than the conventional method that relies only on phase reduction.