论文标题
具有不变代数曲线的平面系统的极限周期数量
Number of limit cycles for planar systems with invariant algebraic curves
论文作者
论文摘要
对于平面多项式系统，存在不变代数曲线的存在限制了此曲线中未包含的极限循环的数量。我们提出了一种普遍的方法来证明该代数曲线中未包含的周期性轨道的存在。当该方法应用于具有限制参数值的多项式系统的参数族时，我们的结果会导致有效的代数条件，这些参数迫使不存在周期性轨道的参数。作为应用程序，我们考虑了几个二次系统的家族：具有一些二次不变的代数曲线，已知曲线具有代数极限周期，一个具有立方不变的代数曲线和其他的曲线。对于任何具有两个不变代数曲线的二次系统，我们证明其仅取决于这些曲线程度的极限循环的数量是有限的。我们还考虑一些具有二次或立方体不变的代数曲线和Lienard系统家族的立方系统家族。我们还提供了一个新的简单证明，证明具有不变抛物线的二次系统最多具有一个极限周期。实际上，我们表明的是，这个结果是具有不变直线的二次系统的相似结果的结果。
For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non existence of periodic orbits not contained in this given algebraic curve. When the method is applied to parametric families of polynomial systems that have limit cycles for some values of the parameters, our result leads to effective algebraic conditions on the parameters that force non existence of the periodic orbits. As applications we consider several families of quadratic systems: the ones having some quadratic invariant algebraic curve, the known ones having an algebraic limit cycle, a family having a cubic invariant algebraic curve and other ones. For any quadratic system with two invariant algebraic curves we prove a finiteness result for its number of limit cycles that only depends on the degrees of these curves. We also consider some families of cubic systems having either a quadratic or a cubic invariant algebraic curve and a family of Lienard systems. We also give a new and simple proof of the known fact that quadratic systems with an invariant parabola have at most one limit cycle. In fact, what we show is that this result is a consequence of the similar result for quadratic systems with an invariant straight line.
