Phase reduction of weakly perturbed limit cycle oscillations in time-delay systems

The phase reduction method is applied to a general class of weakly perturbed time-delay systems exhibiting periodic oscillations. The adjoint equation with an appropriate initial condition for the infinitesimal phase response curve of a time-delay system is derived. The method is demonstrated numerically for the Mackey–Glass equation as well as for a chaotic Rössler system subject to a delayed feedback control (DFC). We show that the profile of the phase response curve of a periodic orbit stabilized by the DFC algorithm does not depend on the control matrix. This property is universal and holds for any dynamical system subject to the DFC.

► The phase reduction method is applied to weakly perturbed time-delay systems.
► The adjoint equation and initial condition for a phase response curve (PRC) are derived.
► Chaotic systems under delayed feedback control are considered.
► The profile of the PRC of a stabilized orbit does not depend of the control matrix.