Dynamics of an oscillator with delay parametric excitation

This paper involves the dynamics of a delay limit cycle oscillator being driven by a time-varying perturbation in the delay: _ x 1⁄4 x t TðtÞ ð Þ εx3 with delay TðtÞ 1⁄4 2þεkþε cos ωt. This delay is chosen to periodically cross the stability boundary for the x1⁄40 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. & 2015 Elsevier Ltd. All rights reserved.