Unique resonant normal forms for area-preserving maps at an elliptic fixed point
Unique resonant normal forms for area-preserving maps at an elliptic fixed point
复制标题
椭圆定点处面积保留贴图的独特共振范式
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
N. Gelfreikh
中科院分区：
文献类型：
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作者：
V. Gelfreich;N. Gelfreikh
我们研究了在共振椭圆不动点附近保面积映射的规范形的可能的简化。在一般情况下，我们证明了在所有阶塔肯标准形矢量场都可以变换为形式插值哈密顿量所描述的特别简单的形式，形式级数A和B的形式取决于共振n的阶数。对于每个n⩾3，我们证明了级数中的哪些项可以通过正则代换来消除，并导出了唯一的标准形，它提供了关于坐标正则变化的全部形式不变量。我们将这些结果推广到保面积映射族。然后，形式内插的哈密顿量的形式类似于单个映射的情况，但涉及作用i中的形式幂级数和参数。
We study possible simplifications of normal forms for area-preserving maps near a resonant elliptic fixed point. In the generic case we prove that at all orders the Takens normal form vector field can be transformed to the particularly simple form described by the formal interpolating Hamiltonian The form of formal series A and B depends on the order of the resonance n. For each n ⩾ 3 we establish which terms of the series can be eliminated by a canonical substitution and derive a unique normal form which provides a full set of formal invariants with respect to canonical changes in coordinates. We extend these results to families of area-preserving maps. Then the formal interpolating Hamiltonian takes a form similar to the case of an individual map but involves formal power series in action I and the parameter.