Unstable eigenvectors and reduced amplitude spaces specifying limit cycles of coupled oscillators with simultaneously diagonalizable matrices: with applications from electric circuits to gene regulation,The European Physical Journal B

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Unstable eigenvectors and reduced amplitude spaces specifying limit cycles of coupled oscillators with simultaneously diagonalizable matrices: with applications from electric circuits to gene regulation
The European Physical Journal B ( IF 1.6 ) Pub Date : 2022-09-19 , DOI: 10.1140/epjb/s10051-022-00412-y
S Mongkolsakulvong ₁ , T D Frank _{2,

3}

Affiliation

Abstract

A fascinating phenomenon is the self-organization of coupled systems to a whole. This phenomenon is studied for a particular class of coupled oscillatory systems exhibiting so-called simultaneously diagonalizable matrices. For three exemplary systems, namely, an electric circuit, a coupled system of oscillatory neurons, and a system of coupled oscillatory gene regulatory pathways, eigenvectors and amplitude equations are derived. It is shown that for all three systems, only the unstable eigenvectors and their amplitudes matter for the dynamics of the systems on their respective limit cycle attractors. A general class of coupled second-order dynamical oscillators is presented in which stable limit cycles emerging via Hopf bifurcations are solely specified by appropriately defined unstable eigenvectors and their amplitudes. While the eigenvectors determine the orientation of limit cycles in state spaces, the amplitudes determine the evolution of states along those limit cycles. In doing so, it is shown that the unstable eigenvectors define reduced amplitude spaces in which the relevant long-term dynamics of the systems under consideration takes place. Several generalizations are discussed. First, if stable and unstable system parts exhibit a slow-fast dynamics, the fast variables may be eliminated and approximative descriptions of the emerging limit cycle dynamics in reduced amplitude spaces may be again obtained. Second, the principle of reduced amplitude spaces holds not only for coupled second-order oscillators, but can be applied to coupled third-order and higher order oscillators. Third, the possibility to apply the approach to multifrequency limit cycle attractors and other types of attractors is discussed.

Graphic abstract

中文翻译：

不稳定的特征向量和减少幅度空间指定耦合振荡器的极限循环与同时对角化矩阵：从电路到基因调控的应用

摘要

一个引人入胜的现象是耦合系统自组织成一个整体。这种现象是针对特定类别的耦合振荡系统进行研究的，该系统表现出所谓的同时对角化矩阵。对于三个示例性系统，即电路、振荡神经元耦合系统和耦合振荡基因调节通路系统，推导出特征向量和幅度方程。结果表明，对于所有三个系统，只有不稳定的特征向量及其幅度对系统在各自极限环吸引子上的动力学有影响。提出了一类通用的耦合二阶动力振荡器，其中通过 Hopf 分岔出现的稳定极限环仅由适当定义的不稳定特征向量及其幅度指定。虽然特征向量决定了状态空间中极限环的方向，但幅度决定了状态沿这些极限环的演变。这样做表明，不稳定的特征向量定义了减幅空间，在这些空间中，所考虑的系统的相关长期动力学发生了。讨论了几个概括。首先，如果稳定和不稳定的系统部分表现出慢-快动力学，则可以消除快速变量，并且可以再次获得在减小幅度空间中出现的极限环动力学的近似描述。其次，减小幅度空间的原理不仅适用于耦合的二阶振荡器，而且可以应用于耦合的三阶和更高阶振荡器。第三，

图形摘要

更新日期：2022-09-20

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