发表机构
School of Mathematical Sciences, Beijing Normal University; College of Mathematics and Physics, Beijing University of Chemical Technology（北京师范大学数学科学学院; 北京化工大学理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于Laurent多项式的代数符号算法，通过代数运算替代符号积分高效计算平面切换系统的Lyapunov常数，并应用于广义切换四次Liénard系统（证明10个极限环）和Alpazur振荡器电路，验证了方法的有效性与实用性。
AI 中文摘要
Lyapunov常数是分析切换系统分岔行为的关键工具。本文提出了一种高效的代数符号算法，用于计算平面切换系统中的Lyapunov常数。通过将向量场映射到复数域，我们构造了基于Laurent多项式的正规形算法。经典的Poincaré映射方法需要重复的符号积分，这在高阶情况下可能导致大量的计算困难。我们的方法通过用直接的代数运算替代连续积分来提高计算效率。我们将此方法应用于研究两个切换模型。对于广义切换四次Liénard系统，代数方法提取了中心条件并证明了10个小振幅极限环的存在性，为其环性建立了新的下界。此外，我们还研究了对Alpazur振荡器进行物理动机的电路级修改。我们证明，最多有一个小振幅极限环可以从中心分岔产生，并且这个上界是可达到的，这说明了所提算法在实际切换电路中的适用性。这一应用表明，我们的方法为分析和解释实际分段光滑系统中出现的复杂动力学现象提供了一个系统框架。
英文摘要
Lyapunov constants are a crucial tool for analyzing bifurcation behavior in switching systems. This paper proposes an efficient algebraic symbolic algorithm for computing Lyapunov constants in planar switching systems. By mapping the vector fields into the complex domain, we construct a normal form algorithm based on Laurent poly?nomials. The classical Poincaré map method requires repeated symbolic integration, which may lead to substantial computational difficulties at high orders. Our ap?proach improves computational efficiency by replacing continuous integration with direct algebraic operations. We apply this approach to investigate two switching models. For a generalized switching quartic Liénard system, the algebraic method extracts the center conditions and proves the existence of 10 small-amplitude limit cycles, establishing a new lower bound for its cyclicity. Furthermore, a physically motivated circuit-level modification of the Alpazur oscillator is investigated. We show that at most one small-amplitude limit cycle can bifurcate from the center and that this upper bound is attainable, illustrating the applicability of the proposed algorithm to a practical switching circuit. This application shows that our method provides a systematic framework for analyzing and interpreting complex dynamical phenomena arising in practical piecewise smooth systems.
