Maximisation of coherence under weakly noisy synchronisation

Abstract The synchronisability of noisy limit-cycle oscillators under external forcing has been measured in terms of the coherence of oscillations. In this letter, we clarify a fundamental limit on maximising this noisy synchronisability for a given power consumption of the external forcing, which underlies an important and ubiquitous problem in nonlinear science related to the efficient synchronisation of weakly forced nonlinear oscillators in noisy environments. For this purpose, we consider general limit-cycle oscillators subjected to external forcing with white Gaussian noise and systematically analyse their coherence maximisation. Our theory clarifies the maximisation of the coherence of oscillations and the existence of a fundamental limit on maximising noisy synchronisability. An infinite-dimensional optimisation establishes the existence of a fundamental limit on noisy synchronisability, validated through systematic numerical simulations and analysis. The results also offer insights into noisy synchronisation for a weak external forcing with frequency mismatch, which can be experimentally verified through the measured probability density function of the oscillator phase.