Frequency locking in the injection-locked frequency divider equation

TL;DR

This paper analyzes the injection-locked frequency divider model, deriving explicit formulas for locking plateaux widths and confirming that the largest plateaux occur at even integer ratios, aligning with previous experimental and numerical findings.

Contribution

It provides the first explicit analytical formulas for locking widths in the injection-locked frequency divider model, confirming prior numerical and experimental results.

Findings

Largest locking plateaux occur at even integer ratios

Explicit formulas for plateau widths are derived

Results align with previous experiments and simulations

Abstract

We consider a model for the injection-locked frequency divider, and study analytically the locking onto rational multiples of the driving frequency. We provide explicit formulae for the width of the plateaux appearing in the devil's staircase structure of the lockings, and in particular show that the largest plateaux correspond to even integer values for the ratio of the frequency of the driving signal to the frequency of the output signal. Our results prove the experimental and numerical results available in the literature.