1:1 and 2:1 phase entrainment in a system of two coupled limit cycle oscillators
The authors propose a model of two coupled limit cycle oscillators which takes into account both 1:1 and 2:1 phase entrainment: \[ (1)\quad {\dot\theta }_ 1=\omega_ 1-\alpha \sin (\theta_ 1-\theta_ 2)-(1- \alpha)\sin (\theta_ 1-2\theta_ 2), \] \[ (2)\quad {\dot\theta }_ 2=\omega_ 2+\alpha \sin (\theta_ 1-\theta_ 2)+(1-\alpha)\sin (\theta_ 1-2\theta_ 2), \] where \(\theta_ 1\) and \(\theta_ 2\) represent the respective phases of the two oscillators and \(\alpha\) is a coupling coefficient. They show that the two oscillator model given by equations (1), (2) exhibits 1:1 and 2:1 phase locking when \(\alpha =1\) and 0, respectively, and that this model also exhibits a 3:2 phase locked motion when \(\alpha =1/2\), and that no other phase locked motions are possible. In the neighbourhood of \(\alpha =1\), where the 1:1 coupling term predominates, the 1:1 phase-entrainment region is centered around a natural frequency ratio of \(\omega_ 1/\omega_ 2=1\). In the neighborhood of \(\alpha =0\), the 2:1 phase-entrainment region is centered around a frequency ratio of \(\omega_ 1/\omega_ 2=2\). For a given value of \(\omega_ 1\), the two regions are separated by a narrow band which becomes vanishingly small as \(\omega_ 1\to 0.\) A change in magnitude of the coupling \(\alpha\), or of the natural frequency \(\omega_ 1\) (or more generally of the ratio \(\omega_ 1/\omega_ 2)\) can lead to a transition from 1:1 to 2:1 phase- entrainment. This is of significance in modeling a pathological condition occurring in cardiac pacemakers known as 2:1 AV block.
scientific article; zbMATH DE number 3873121