Fascinating paper by Lord Rayleigh from 1883 whereby he clearly describes the phenomenon of period doubling. He also describes the Mathieu equation, which was not widely recognized at the time, having been described just a decade earlier.
> Rayleigh, Lord. "XXXIII. on maintained vibrations." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 15, no. 94 (1883): 229-235.
Review by Dino Boccaletti, Giuseppe Pucacco in Theory of Orbits: Perturbative and Geometrical Methods (2013)
Rayleigh used the Mathieu equation in a much more general context than Mathieu, who only applied it to a specific elliptical-shaped membrane in 1868.