Pleshkan等时线的临界周期分岔
Bifurcation of critical periods from Pleshkan's isochrones
摘要：
Pleshkan证明，在1969年，一个线性变换和一个恒定的时间重新标度，有四个等时线在家庭的立方中心与齐次非线性$\mathscr C_3.$在本文中，我们证明，如果我们扰动任何这些等时线内$\mathscr C_3,$然后在最多两个临界周期分叉从其周期环。此外，我们表明，对于每个$k=0,1,2,$有扰动引起的正是$k$的关键时期。作为一个副产品，我们得到了一个部分的结果，类似的问题，在家庭的二次中心$\mathscr C_2.$劳德证明在1964年，直到一个线性变换和一个恒定的时间重新标度，有四个等时线在$\mathscr C_2.$我们证明，如果我们扰动其中三个内$\mathscr C_2,$，然后在最多一个临界周期分叉从其周期环。此外，对于每个$k=0,1,$，我们表明，有扰动引起的正是$k$的关键时期。二次等值线中心，我们不考虑显示一些特殊性，讨论在文章的最后。
英文摘要：
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 $\mathscr C_3.$ In this paper we prove that if we perturb any of these isochrones inside $\mathscr C_3,$ 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 $\mathscr C_2.$ Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in $\mathscr C_2.$ We prove that if we perturb three of them inside $\mathscr C_2,$ 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.
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