A class of n-dimensional Poisson systems reducible to an unperturbed harmonic oscillator shall be considered. In such case, perturbations leaving invariant a given symplectic leaf shall be investigated. Our purpose will be to analyze the bifurcation phenomena of periodic orbits as a result of these perturbations in the period annulus associated to the unperturbed harmonic oscillator. This is accomplished via the averaging theory up to an arbitrary order in the perturbation parameter ε. In that theory we shall also use both branching theory and singularity theory of smooth maps to analyze the bifurcation phenomena at points where the implicit function theorem is not applicable. When the perturbation is given by a polynomial family, the associated Melnikov functions are polynomial and tools of computational algebra based on Gröbner basis are employed in order to reduce the generators of some polynomial ideals needed to analyze the bifurcation problem. When the most general perturbation of the harmonic oscillator by a quadratic perturbation field is considered, the complete bifurcation diagram (except at a high codimension subset) in the parameter space is obtained. Examples are given.

中文翻译：

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Casimir 不变流形上的扰动秩 2 泊松系统和周期轨道

应考虑一类可归约为无扰动谐振子的 n 维泊松系统。在这种情况下，应研究使给定辛叶不变的扰动。我们的目的是分析由于与未受扰动的谐振子相关的周期环中的这些扰动而导致的周期轨道的分叉现象。这是通过在扰动参数 ε 中达到任意阶数的平均理论来实现的。在该理论中，我们还将同时使用平滑映射的分支理论和奇点理论来分析隐函数定理不适用的点的分岔现象。当扰动由多项式族给出时，相关的 Melnikov 函数是多项式的，并采用基于 Gröbner 基的计算代数工具，以减少分析分岔问题所需的一些多项式理想的生成器。当考虑二次扰动场对谐振子的最一般扰动时，得到参数空间中的完整分岔图（除了在高余维子集）。给出了例子。