A powerful technique for the analysis of nonlinear oscillators is the rigorous reduction to phase models, with a single variable describing the phase of the oscillation with respect to some reference state. An analog to phase reduction has recently been proposed for systems with a stable fixed point, and phase reduction for periodic orbits has recently been extended to take into account transverse directions and higher-order terms. This tutorial gives a unified treatment of such phase reduction techniques and illustrates their use through mathematical and biological examples. It also covers the use of phase reduction for designing control algorithms which optimally change properties of the system, such as the phase of the oscillation. The control techniques are illustrated for example neural and cardiac systems.

中文翻译：

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生物系统的相变和基于相的最优控制：教程。

分析非线性振荡器的一项强大技术是严格简化相位模型，其中一个变量描述了相对于某些参考状态的振荡相位。近来已经提出了对于具有稳定的固定点的系统的相位减小的模拟，并且最近已将周期性轨道的相位减小扩展为考虑横向和高阶项。本教程对此类相还原技术进行了统一处理，并通过数学和生物学示例说明了它们的使用。它还涵盖了使用相减来设计控制算法，这些算法可以最佳地改变系统的属性，例如振荡的相位。举例说明了控制技术，例如神经系统和心脏系统。