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.