等时线的高阶近似
Higher order approximation of isochrons
摘要：
相位归约是分析稳定振荡器的一种常用技术，特别是在振荡器网络的同步和锁相研究中。在用于获得单个振荡器的相位缩减的广泛使用的数值方法中，需要获得相位函数的梯度，其本质上提供等时线的线性近似。在本文中，我们扩展的方法获得偏导数的相位函数到任意阶，提供更高的阶近似的等时线。特别是，我们的方法在2阶可以应用到一个稳定的振荡器进行随机扰动的动力学研究，一个主题，将在未来的论文中讨论。我们使用的Stuart-Landau振荡器来说明该方法在二阶。
英文摘要：
Phase reduction is a commonly used techinque for analyzing stable oscillators, particularly in studies concerning synchronization and phase lock of a network of oscillators. In a widely used numerical approach for obtaining phase reduction of a single oscillator, one needs to obtain the gradient of the phase function, which essentially provides a linear approximation of isochrons. In this paper, we extend the method for obtaining partial derivatives of the phase function to arbitrary order, providing higher order approximations of isochrons. In particular, our method in order 2 can be applied to the study of dynamics of a stable oscillator subjected to stochastic perturbations, a topic that will be discussed in a future paper. We use the Stuart-Landau oscillator to illustrate the method in order 2.
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