Fast optimal entrainment of limit-cycle oscillators by strong periodic inputs via phase-amplitude reduction*

Entrainment of self-sustained oscillators by periodic inputs is widely observed in the real world, including the entrainment of circadian rhythms to sunlight and injection locking of electrical oscillators to clock signals. When the perturbation given to the oscillator is sufficiently weak, the phase reduction theory can be employed to approximate the multidimensional nonlinear dynamics of the oscillator by a one-dimensional phase equation. Recently, optimization of periodic inputs has been studied by using the phase equation to realize fast entrainment and high energy efficiency. However, this method does not work well for strong inputs because the amplitude deviations of the oscillator state from the original orbit become large, and the phase-only approximation of the system breaks down. In this study, using phase-amplitude reduction, we propose two methods to find the input waveforms that can suppress the amplitude deviations even for strong inputs. We demonstrate that the proposed method allows us to use stronger inputs, thereby achieving faster convergence to the target phase-locking point than the conventional method based only on the phase equation.