Quasiperiodic Birhythmicity in a Multicycle van der Pol Oscillator

In this work, the dynamics of a time-delayed multicycle van der Pol oscillator is investigated in the case where the delay amplitude is harmonically modulated around a mean value with a certain amplitude and frequency. Using perturbation methods, approximate analytical solutions of quasiperiodic (QP) oscillations and their bifurcation equations are obtained and regions of existence of different solutions are determined. It is revealed that, while monorhythmic and birhythmic periodic oscillations are commonly present in the undelayed or delay-unmodulated multicycle van der Pol oscillator, the delay-modulated oscillator may exhibit QP-monorhythmicity or QP-birhythmicity. Based on the bifurcation diagrams, transitions between different rhythmicities are derived. The results show that for appropriate values of delay parameters, large-amplitude stable (QP-monorhythmic) oscillations can take place over a broadband of the modulation frequency near the delay-induced parametric resonance. Bistability generated by QP-birhythmicity can also occur for different values of delay parameters.