周期扰动平面自治系统的周期解：拓扑方法
Periodic solutions of periodically perturbed planar autonomous systems: A topological approach
摘要：
本文研究了具有极限环x_0的非线性平面自治系统在小参数T_1-周期扰动下周期解的存在性.当T_0/T_1为有理数l/k，且为l，k个素数时，当参数摄动e>0足够小时，给出了系统存在klT_0-周期解x_e的条件，当e->0时，该解x_e收敛于极限环的轨迹x_1.此外，还给出了T=klT_0是周期解x_e的最小周期的条件.我们还提出了一个简单的标准，确保这些条件得到验证。最后，当T_0/T_1为无理数时，证明了当T>0和e>0时，不存在收敛于x_1的扰动系统的T-周期解x_e。所采用的方法是基于拓扑度理论。
英文摘要：
Aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle x_0 of least period T_0>0 when it is perturbed by a small parameter, T_1-periodic, perturbation. In the case when T_0/T_1 is a rational number l/k, with l, k prime numbers, we provide conditions to guarantee, for the parameter perturbation e>0 sufficiently small, the existence of klT_0-periodic solutions x_e of the perturbed system which converge to the trajectory x_1 of the limit cycle as e->0. Moreover, we state conditions under which T=klT_0 is the least period of the periodic solutions x_e. We also suggest a simple criterion which ensures that these conditions are verified. Finally, in the case when T_0/T_1 is an irrational number we show the nonexistence, whenever T>0 and e>0, of T-periodic solutions x_e of the perturbed system converging to x_1. The employed methods are based on the topological degree theory.
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