We consider two oscillators with delayed direct and velocity coupling. The oscillators have frequencies close or equal to 1:1 resonance. Due to the coupling the oscillations of the subsystems are in or out of phase. For these synchronized and anti-phase solutions, we use averaging for analytical stability results for small parameters. We also determine bifurcation curves of the delay system numerically. We identify regions in the parameter space (two coupling constants and the delay) where both solutions are stable or only one. For small parameters the averaging and numerical results are in good agreement. For larger values of the delay, we find multiple synchronized and anti-phase solutions. For small detuning we show that a minimal coupling value is needed to have almost synchronous or anti-phase behaviour.
|Title of host publication||Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011)|
|Editors||D. Bernardini, G. Rega, F. Romeo|
|Number of pages||2|
|Publication status||Published - Jul 2011|
- Delay equations
- Numerical bifurcation analysis
Meijer, H. G. E., Kalmar-Nagy, T., & van Gils, S. A. (2011). On phase-locking of oscillators with delay coupling. In D. Bernardini, G. Rega, & F. Romeo (Eds.), Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011) (pp. MS-11). EUROMECH.