Schmitt trigger oscillator and its synchronisation by an external square oscillator

The properties of synchronisation of the Schmitt oscillator forced by an external square wave input are presented. The synchronisation occurs for all rational period ratio p/q, p and q odd, in a finite interval of forcing period and for a large domain of forcing amplitude. The bounds of these domains of synchronisation are given explicitly from a kind of continuous fraction. The set of synchronisation domains has a self-similarity property. The measure of the set of non-periodic solutions is zero and the evolution of the period ratio versus the forced period is a devil's staircase. The hierarchy of the width of the synchronisation domain depends on the amplitude of the forcing input. Then, independently of the practical aspects of the synchronisation, this Schmitt oscillator gives an example of a nonlinear differential equation in which some theoretical results are easily shown and computed without any approximations in a large domain of the parameters, a rather rare case.