Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems.
Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems.
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DOI：
10.1063/1.4977195
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发表时间：
2017-01
期刊：
影响因子：
2.9
通讯作者：
Sho Shirasaka;W. Kurebayashi;H. Nakao
中科院分区：
文献类型：
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作者：
Sho Shirasaka;W. Kurebayashi;H. Nakao
基于等时线的极限循环系统的相减框架已成为分析节律现象的有力工具。最近，等稳的概念被引入，它通过表征系统状态的幅度，即与极限环吸引子的偏差来补充等时线，以描述极限环周围的瞬时动力学[Wilson和Moehlis，Phys。修订本E 94,052213(2016年)]。在这项研究中，我们介绍了一种简化的相幅描述稳定极限环系统的暂态动力学的框架。与前面的研究不同，等稳态的处理方式与Koopman算符分析完全一致，这使得我们能够避免等稳态的不连续性，并将该框架应用于远离极限环的系统状态。我们还提出了一种新的、方便的双正交化方法来获得幅值的响应函数，它可以解释为伴随协变Lyapunov向量在极限环系统的瞬时动力学中的推广。在一个生化振荡器模型中，我们通过估计外部输入的最优注入时机，有效地抑制了系统状态与极限环的偏差，说明了所提出的简化框架的实用性。
Phase reduction framework for limit-cycling systems based on isochrons has been used as a powerful tool for analyzing the rhythmic phenomena. Recently, the notion of isostables, which complements the isochrons by characterizing amplitudes of the system state, i.e., deviations from the limit-cycle attractor, has been introduced to describe the transient dynamics around the limit cycle [Wilson and Moehlis, Phys. Rev. E 94, 052213 (2016)]. In this study, we introduce a framework for a reduced phase-amplitude description of transient dynamics of stable limit-cycling systems. In contrast to the preceding study, the isostables are treated in a fully consistent way with the Koopman operator analysis, which enables us to avoid discontinuities of the isostables and to apply the framework to system states far from the limit cycle. We also propose a new, convenient bi-orthogonalization method to obtain the response functions of the amplitudes, which can be interpreted as an extension of the adjoint covariant Lyapunov vector to transient dynamics in limit-cycling systems. We illustrate the utility of the proposed reduction framework by estimating the optimal injection timing of external input that efficiently suppresses deviations of the system state from the limit cycle in a model of a biochemical oscillator.