Computing Arnol′d tongue scenarios

A famous phenomenon in circle-maps and synchronisation problems leads to a two-parameter bifurcation diagram commonly referred to as the Arnol′d tongue scenario. One considers a perturbation of a rigid rotation of a circle, or a system of coupled oscillators. In both cases we have two natural parameters, the coupling strength and a detuning parameter that controls the rotation number/frequency ratio. The typical parameter plane of such systems has Arnol′d tongues with their tips on the decoupling line, opening up into the region where coupling is enabled, and in between these Arnol′d tongues, quasi-periodic arcs. In this paper, we present unified algorithms for computing both Arnol′d tongues and quasi-periodic arcs for both maps and ODEs. The algorithms generalise and improve on the standard methods for computing these objects. We illustrate our methods by numerically investigating the Arnol′d tongue scenario for representative examples, including the well-known Arnol′d circle map family, a periodically forced oscillator caricature, and a system of coupled Van der Pol oscillators.