Computing Arnol′d tongue scenarios

Abstract

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, quasiperiodic 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.

On the Numerical Continuation of Isolas of Equilibria

“…For each node, we can replace the equilibrium condition with a boundary-value problem of the form The corresponding condition for the node distribution can be written as d(y i , y i+1 ) = y i − y i+1 , where the Euclidean norm in (5) is replaced by a suitable norm on C 1 ([0, 1], R n ) × R [28]. In this case, an integral phase condition for the isola can also be defined in order to close the problem.…”

“…We point out that more sophisticated adaptations are also possible. In [28], the authors propose a hybrid between uniform node distribution and uniform errors distribution. We illustrate the effect of node adaptivity with the toy model…”

“…More recently, a polygonal representation of closed curves was used to trace Arnol'd tongues [28]. As we shall see, isolas can be computed as zeroes of a suitable functional equation that can be discretized using Schilder & Peckham's approach.…”

Mode locking in a periodically forced resonate-and-fire neuron model

“…6 for I 0 = 2.27 and = 2.65 where one expects to see three spikes, namely φ 0 , φ 1 , and φ 2 within every two periods of I app (t). Moreover, it is clear that the graze as a local maximum arrives between φ 0 and φ 1 For a complete discussion on the computing of Arnol'd tongues see [40] and for discussion on the smooth circle map, its bifurcations, and the corresponding Arnol'd tongues see [5,31]. Grazing of Type two also occurs whenever varying a parameter causes the voltage to reach the threshold but with dv dt = 0 at that time, rather than dv dt > 0, see Fig.…”

A new method for the determination of the locking range of oscillators

“…Various analytical and numerical approaches have been suggested for predicting the locking range, e.g. the classic Adler approach in [14] and more recently [17]- [19].…”