Piecewise linear perturbations of a linear center
Piecewise linear perturbations of a linear center
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DOI：
10.3934/dcds.2013.33.3915
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发表时间：
2013-03
影响因子：
1.1
通讯作者：
C. Buzzi;C. Pessoa;Joan Torregrosa
中科院分区：
文献类型：
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作者：
C. Buzzi;C. Pessoa;Joan Torregrosa
本文主要研究了在两个区域内利用分段线性扰动可以从一个线性中心分叉的极限环。我们考虑了两个区域被一条直线隔开，且未扰动系统的奇点在π中的情形。证明了在一个七阶扰动下，极限环的最大数目是三个。而且这个上界也达到了。这一结果证实了这些系统具有比预期更多的极限环。最后，中心和等时性问题也研究了系统，其中包括一阶扰动。对于这些最后的系统，它也证明了，当周期函数，定义在中心的周期环，是不单调的，那么它最多有一个临界周期。而且这个上界也达到了。
This paper is mainly devoted to study the limit cycles that can bifurcate from a linear center using a piecewise linear perturbation in two zones. We consider the case when the two zones are separated by a straight line Σ and the singular point of the unperturbed system is in Σ. It is proved that the maximum number of limit cycles that can appear up to a seventh order perturbation is three. Moreover this upper bound is reached. This result confirm that these systems have more limit cycles than it was expected. Finally, center and isochronicity problems are also studied in systems which include a first order perturbation. For these last systems it is also proved that, when the period function, defined in the period annulus of the center, is not monotone, then it has at most one critical period. Moreover this upper bound is also reached.