Bifurcation of critical periods from Pleshkan's isochrones
Bifurcation of critical periods from Pleshkan's isochrones
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DOI：
10.1112/jlms/jdp062
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发表时间：
2008-11
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影响因子：
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通讯作者：
M. Grau;J. Villadelprat
中科院分区：
文献类型：
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作者：
M. Grau;J. Villadelprat
Pleshkan 在 1969 年证明，在线性变换和时间不断缩放的情况下，具有齐次非线性的立方中心族中有四个等时线𝒷3。在本文中，我们证明，如果我们扰动 𝒷3 内的任何等时线，则最多有两个关键周期从其周期环中分叉。此外，我们表明，对于每个 k=0, 1, 2，存在恰好产生 k 个关键周期的扰动。作为副产品，我们获得了二次中心族 𝒷2 中类似问题的部分结果。 Loud 在 1964 年证明，在线性变换和时间不断重新缩放的情况下，𝒷2 中有四个等时线。我们证明，如果我们在 𝒷2 内扰动其中的三个，那么最多有一个关键周期从其周期环中分叉。此外，对于每个 k=0, 1，我们表明存在恰好产生 k 个关键周期的扰动。我们不考虑的二次等时中心显示出一些特性，这些特性将在本文末尾讨论。
Pleshkan proved in 1969 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in the family of cubic centers with homogeneous nonlinearities 𝒷3. In this paper we prove that if we perturb any of these isochrones inside 𝒷3, then at most two critical periods bifurcate from its period annulus. Moreover, we show that, for each k=0, 1, 2, there are perturbations giving rise to exactly k critical periods. As a byproduct, we obtain a partial result for the analogous problem in the family of quadratic centers 𝒷2. Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in 𝒷2. We prove that if we perturb three of them inside 𝒷2, then at most one critical period bifurcates from its period annulus. In addition, for each k=0, 1, we show that there are perturbations giving rise to exactly k critical periods. The quadratic isochronous center that we do not consider displays some peculiarities that are discussed at the end of the paper.