ABC流的动力学、周期轨道与$C^1$不可积性
Dynamics, periodic orbits and $C^1$ non-integrability of the ABC flow
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- University of Zielona Góra（热舒夫绿色山大学）
- Universitat Autònoma de Barcelona（巴塞罗那自治大学）
- Reial Acadèmia de Ciències i Arts de Barcelona（巴塞罗那皇家科学与艺术学院）
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中文总结 AI 辅助
研究ABC流在三个可积轴附近的微扰动力学，用一阶平均化证明两个孤立周期轨道存在，并借助特征乘子与Poincaré-Llibre-Valls判据证明其邻域内无$C^1$非平凡首次积分，数值验证了平均化的广泛有效性。
中文摘要 AI 辅助
我们研究Arnold-Beltrami-Childress (ABC)流在其三个基本可积坐标轴附近的微扰区域。从极限可积系统的退化周期流线族出发，我们使用一阶平均化方法证明了扰动流存在两个孤立周期解。这三个微扰区域通过ABC向量场的循环对称性相互关联。在每种情况下，相应的二维平均系统具有两个简单平衡点，一个椭圆型和一个双曲型，具体取决于相关参数比值的符号。庞加莱截面展示了可积极限的退化周期结构在扰动下如何破裂，以及平均化预测的孤立周期轨道如何在周围动力学中涌现。这些周期轨道为微扰分析与可积性问题之间提供了自然联系。线性化平均系统的特征值决定了分岔周期解的非平凡特征乘子的主导阶行为。将此关系与Poincaré-Llibre-Valls判据相结合，我们证明了在每个这样的周期轨道邻域内，不存在沿轨道正则的非恒定首次积分$H\in C^1$。通过直接打靶法、单值矩阵计算以及椭圆和双曲分支的延拓，进一步说明并证实了分析结果，表明一阶平均化近似在比严格渐近区域宽得多的参数范围内保持定量准确。
英文摘要
We study the Arnold-Beltrami-Childress (ABC) flow in perturbative regimes near its three elementary integrable coordinate axes. Starting from a degenerate family of periodic streamlines of the limiting integrable system, we use first-order averaging to prove the existence of two isolated periodic solutions of the perturbed flow. The three perturbative regimes are related by the cyclic symmetry of the ABC vector field. In each case, the corresponding two-dimensional averaged system possesses two simple equilibria, one elliptic and the other hyperbolic, depending on the sign of the relevant parameter ratio. Poincaré sections illustrate how the degenerate periodic structure of the integrable limit breaks under perturbation and how the isolated periodic orbits predicted by averaging emerge within the surrounding dynamics. These periodic orbits provide the natural link between the perturbative analysis and the integrability problem. The eigenvalues of the linearized averaged system determine the leading-order behavior of the nontrivial characteristic multipliers of the bifurcating periodic solutions. Combining this relation with the Poincaré-Llibre-Valls criterion, we prove that, in a neighborhood of each of these periodic orbits, there exists no nonconstant first integral $H\in C^1$ that is regular along the orbit. The analytical results are further illustrated and confirmed by direct shooting, monodromy-matrix computations, and continuation of the elliptic and hyperbolic branches, demonstrating that the first-order averaging approximation remains quantitatively accurate over a substantially wider parameter range than the strict asymptotic regime.
