Coupling of two Schmitt oscillators: non-periodic solutions are rare

The coupling of two Schmitt oscillators has been studied, but no definitive answer is known on the conditions for which they synchronise giving a periodic solution. A complete answer has been given for the particular case of a periodically forced Schmitt oscillator. In this particular case a complete description of the synchronisation domains (sometime called tongues) was given from a type of continued fraction expansion of the winding number. This continued fraction expansion clearly shows the self-similarity of the organisation of the tongues and shows that the measure of the case where the synchronisation does not take place is zero as soon as the coupling exists. It is shown that the more general case can be deduced from the forcing case. In particular it is possible to formally compute the bounds of the domains where the solutions are periodic. When the integration speeds of the oscillators are varied, it is shown that as long as the coupling is dissymmetric the measure of the case where the solutions are not periodic is zero. Such a result is rather counter-intuitive.