具有非光滑周期扰动的平面自治系统的周期解
Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations
摘要：
本文考虑一类平面自治系统，其最小周期T>0的孤立极限环x_0使得其周围的线性化系统只有一个绝对值为1的特征乘子.我们考虑两个功能，定义的线性化系统的伴随的本征函数的装置，我们制定的条件，以便有两个几何不同的家庭的自治系统的T-周期解的存在时，它是由非光滑的T-周期非线性项的小振幅扰动。我们还表明，这些周期解的收敛到x_0的扰动消失，我们提供了一个估计的收敛速率。所采用的方法主要是基于理论的拓扑度和它的属性，允许较少的规律性的数据比所需的方法，通常采用在现有的文献中关于这个问题，基于各种版本的隐函数定理。
英文摘要：
In this paper we consider a class of planar autonomous systems having an isolated limit cycle x_0 of smallest period T>0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T-periodic solutions of the autonomous system when it is perturbed by nonsmooth T-periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to x_0 as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem.
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