Synchronization and locking in oscillators with flexible periods

• Published in: Chaos (Woodbury, N.Y.) N.Y.), 2021-03, Vol.31 (3), p.033143-033143
• Main Authors: Savinov, Mariya, Swigon, David, Ermentrout, Bard
• Format: Article
• Language: English
• Subjects: Dynamical systems; Entrainment; Locking; Nonlinear dynamics; Oscillators; Stability; Synchronism
• ISSN: 1054-1500; 1089-7682
• DOI: 10.1063/5.0021836

Entrainment of a nonlinear oscillator by a periodic external force is a much studied problem in nonlinear dynamics and characterized by the well-known Arnold tongues. The circle map is the simplest such system allowing for stable N: M entrainment where the oscillator produces N cycles for every M stimulus cycles. There are a number of experiments that suggest that entrainment to external stimuli can involve both a shift in the phase and an adjustment of the intrinsic period of the oscillator. Motivated by a recent model of Loehr et al. [J. Exp. Psychol.: Hum. Percept. Perform. 37, 1292 (2011)], we explore a two-dimensional map in which the phase and the period are allowed to update as a function of the phase of the stimulus. We characterize the number and stability of fixed points for different N: M-locking regions, specifically, 1:1, 1:2, 2:3, and their reciprocals, as a function of the sensitivities of the phase and period to the stimulus as well as the degree that the oscillator has a preferred period. We find that even in the limited number of locking regimes explored, there is a great deal of multi-stability of locking modes, and the basins of attraction can be complex and riddled. We also show that when the forcing period changes between a starting and final period, the rate of this change determines, in a complex way, the final locking pattern.