论文标题
具有代数不变曲线的Kukles系统中的小和大振幅限制周期
Small- and large-amplitude limit cycles in Kukles systems with algebraic invariant curves
论文作者
论文摘要
几十年来，平面多项式载体场的极限周期一直是一个积极的研究领域。对周期性相关动态的兴趣来自希尔伯特的第16个问题，以及在应用中经常发现振荡状态的事实。我们通过lyapunov的数量和第一阶和二阶的梅尔尼科夫功能研究了两个库克斯系统家族中极限周期的存在及其与代数曲线的共存。我们展示了中心条件，以及在其中一个家庭中产生的小幅度和大振幅限制周期之间的联系，其中墨尼科夫函数的第一个系数与第一个Lyapunov的数量相对应。我们还提供了一个平面多项式系统的示例，其中循环性不受第一个非零Melnikov函数的完全控制。
Limit cycles of planar polynomial vector fields have been an active area of research for decades; the interest in periodic-orbit related dynamics comes from Hilbert's 16th problem and the fact that oscillatory states are often found in applications. We study the existence of limit cycles and their coexistence with invariant algebraic curves in two families of Kukles systems, via Lyapunov quantities and Melnikov functions of first and second order. We show center conditions, as well as a connection between small- and large-amplitude limit cycles arising in one of the families, in which the first coefficients of the Melnikov function correspond to the first Lyapunov quantities. We also provide an example of a planar polynomial system in which the cyclicity is not fully controlled by the first nonzero Melnikov function.
