Graduate School of Information Science and Engineering, Tokyo Institute of Technology, O-okayama 2-12-1, Meguro, Tokyo 152-8552, Japan.
Faculty of Software and Information Technology, Aomori University, Kobata 2-3-1, Aomori, Aomori 030-0943, Japan.
Phys Rev E. 2017 Jan;95(1-1):012212. doi: 10.1103/PhysRevE.95.012212. Epub 2017 Jan 23.
Hybrid dynamical systems characterized by discrete switching of smooth dynamics have been used to model various rhythmic phenomena. However, the phase reduction theory, a fundamental framework for analyzing the synchronization of limit-cycle oscillations in rhythmic systems, has mostly been restricted to smooth dynamical systems. Here we develop a general phase reduction theory for weakly perturbed limit cycles in hybrid dynamical systems that facilitates analysis, control, and optimization of nonlinear oscillators whose smooth models are unavailable or intractable. On the basis of the generalized theory, we analyze injection locking of hybrid limit-cycle oscillators by periodic forcing and reveal their characteristic synchronization properties, such as ultrafast and robust entrainment to the periodic forcing and logarithmic scaling at the synchronization transition. We also illustrate the theory by analyzing the synchronization dynamics of a simple physical model of biped locomotion.
具有离散切换平滑动力学特性的混合动态系统已被用于模拟各种节律现象。然而，相位降阶理论作为分析节律系统中极限环振荡同步的基本框架，主要局限于平滑动力学系统。在这里，我们为混合动力学系统中弱受扰极限环开发了一个通用的相位降阶理论，该理论便于分析、控制和优化那些平滑模型不可用或难以处理的非线性振荡器。基于这个广义理论，我们分析了周期性激励对混合极限环振荡器的注入锁定，并揭示了它们的特征同步特性，例如对周期性激励的超快和稳健的同步以及在同步转变处的对数标度。我们还通过分析双足运动的简单物理模型的同步动力学来说明该理论。
