AI 中文总结
研究分段光滑单值系统中区分中心与焦点、确定焦点阶数及退化霍普夫分岔的范式方法，证明李雅普诺夫常数与范式系数系列代数等价，据此设计算法计算范式系数系列并构建稳定切换信号。
AI 中文摘要
我们研究分段光滑单值系统中的范式方法，以区分中心和焦点，确定焦点阶数以及退化霍普夫分岔。对于先前出版物中给出的拓扑等价意义下的分段光滑单值系统范式，我们证明了原始分段光滑系统的李雅普诺夫常数与其范式系数系列之间的代数等价性。利用这种代数等价性，我们更倾向于计算范式系数系列以避免在计算李雅普诺夫常数时繁琐的三角函数积分，并找到分段光滑系统及其子系统焦点在阶数和稳定性方面的关系。此外，我们设计了计算范式系数系列的算法，并将其与获得的稳定性关系相结合，为状态依赖切换非线性系统构建稳定切换信号，这是控制理论中的一个基本问题。
英文摘要
We investigate the normal form method in piecewise-smooth monodromic systems to distinguish center from focus and determine the order of focus as well as degenerate Hopf bifurcations. For the normal form of piecewise-smooth monodromic systems given in previous publications in the sense of topologically equivalence, we prove the algebraic equivalence between the Lyapunov constants of the original piecewise-smooth system and the coefficient series of its normal form. With this algebraic equivalence, we prefer to compute the normal form coefficient series to avoid cumbersome integrals of trigonometric functions in the computation of Lyapunov constants, and find the relations of foci of the piecewise-smooth system and its subsystems: among the orders and among the stabilities. Moreover, we design algorithms for computing the normal form coefficient series and combine them with the obtained stability relation to construct stabilizing switching signals for state-dependent switched nonlinear systems, one basic problem in the control theory.
