Optimal entrainment with smooth, pulse, and square signals in weakly forced nonlinear oscillators

Abstract

A physical limit of entrainability of nonlinear oscillators is considered for an external weak signal (forcing). This limit of entrainability is characterized by the optimization problem maximizing the width of the Arnold tongue (the frequency-locking range versus forcing magnitude) under certain practical constraints. Here we show a solution to this optimization problem, thanks to a direct link to Hölder’s inequality. This solution defines an ideal forcing realizing the entrainment limit, and as the result, a fundamental limit of entrainment is clarified as follows. For 1:1 entrainment, we obtain (i) a construction of the global optimal forcing and a condition for its uniqueness in Lp-space with p>1, and (ii) a construction of the global optimal pulse-like forcings in L1-space, and for m:n entrainment (m≠n), some informations about the non-existence of the ideal forcing. (iii) In addition, we establish definite algorithms for obtaining the global optimal forcings for 1<p≤∞ and these pulse-like forcings for p=1. These theoretical findings are verified by systematic, extensive numerical calculations and simulations.