We address the dynamics of near-integrable Hamiltonian systems with 3 degrees of freedom in extended vicinities of unperturbed resonant invariant Liouville tori. The main attention is paid to the case where the unperturbed torus satisfies two independent resonance conditions. In this case the average dynamics is 4-dimensional, reduced to a generalised motion under a conservative force on the 2-torus and is generically non-integrable. Methods of differential topology are applied to full description of equilibrium states and phase foliations of the average system. The results are illustrated by a simple model combining the non-degeneracy and non-integrability of the isoenergetically reduced system.

中文翻译：

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三自由度哈密顿系统的共振

我们在无扰动共振不变Liouville花托的扩展附近处理具有3个自由度的近可积分哈密顿系统的动力学。主要注意的是不扰动的圆环满足两个独立的共振条件的情况。在这种情况下，平均动力学是4维的，在保守的2托力作用下降低为广义运动，并且通常是不可积分的。差分拓扑的方法被用于全面描述平均系统的平衡状态和相叶。通过结合等能量还原系统的非简并性和非可积性的简单模型来说明结果。