Phase-Locked Solutions of a Coupled Pair of Nonidentical Oscillators

Kuan Wei Chen, Chih Wen Shih*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We start with a simple example of two coupled phase oscillators. In this example, stable phase-locking occurs if and only if coupling is sufficiently strong. We then add amplitude (radial) variables to the phase oscillators in the most straightforward possible way. For symmetric coupling, stable phase-locking still requires sufficiently strong coupling. For asymmetric coupling, however, stable phase-locking now becomes possible for arbitrarily weak coupling. We also give an exact formula for the common frequency of the two oscillators in the phase-locked state. By examining the degenerate Routh-Hurwitz criterion for Hopf bifurcation, we confirm the presence of periodic solutions. By solving the coupled system in polar coordinates, the conditions for the existence of phase-locked solutions are derived. This leads us to demonstrate various oscillatory scenarios in several parameter ranges. Stability or instability of the phased-locked solutions is justified in some cases. We further develop a new approach which combines the monotone dynamics theory and sequential contracting technique to conclude the attraction of the stable phase-locked solution and locate its basin of attraction. The associations between the findings from the Hopf bifurcation theory and those from the polar-coordinate setting are also addressed.

Original languageEnglish
Article number14
JournalJournal of Nonlinear Science
Issue number1
StatePublished - Feb 2024


  • Amplitude-phase equation
  • Collective oscillation
  • Coupled system
  • Hopf bifurcation
  • Phase-locked solution


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