Abstract

The article discusses the problems about the main scenarios of dynamical systems’ local bifurcations described by non-autonomous differential equations with periodic right-hand side. New formulae for calculating Lyapunov quantities are proposed. The proposed formulae are obtained on the basis of the general operator method for studying local bifurcations and do not require a transition to normal forms and the usage of the theorems about the central manifold. The indicated method allowed for obtaining new bifurcation formulae for studying the main scenarios of local bifurcations. The paper shows how these bifurcation formulae lead to new formulae for calculating Lyapunov quantities in the problems about bifurcations of forced oscillations and Andronov–Hopf bifurcations.

中文翻译：

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非自治周期动力系统局部分支问题中的李雅普诺夫量

摘要

本文讨论了由具有周期右手边的非自治微分方程描述的动力系统局部分支的主要情形的问题。提出了计算李雅普诺夫数量的新公式。拟议的公式是基于研究局部分支的一般算子方法获得的，不需要转换为标准形式和有关中心流形的定理。所示方法允许获得新的分叉公式，以研究局部分叉的主要情况。本文说明了这些分叉公式如何导致在强迫振动的分叉和Andronov-Hopf分叉的问题中计算Lyapunov量的新公式。