Dynamics of an oscillator with delay parametric excitation

•A new form of parametric excitation for DDEs is presented.•Parametric forcing frequency is within the delay, which varies sinusoidally in time.•Subharmonic resonance occurs when forcing frequency is twice the unforced frequency.•A diverse set of bifurcations occurs in the neighborhood of subharmonic resonance.

This paper involves the dynamics of a delay limit cycle oscillator being driven by a time-varying perturbation in the delay:ẋ=−x(t−T(t))−ϵx3with delay T(t)=π2+ϵk+ϵcosωt. This delay is chosen to periodically cross the stability boundary for the x=0 equilibrium in the constant-delay system.For most of parameter space, the system is non-resonant, leading to quasiperiodic behavior. However, a region of 2:1 resonance is shown to exist where the system׳s response frequency is entrained to half of the forcing frequency ω. By a combination of analytical and numerical methods, we find that the transition between quasiperiodic and entrained behavior consists of a variety of local and global bifurcations, with corresponding regions of multiple stable and unstable steady-states.