Invariant surfaces play a crucial role in the dynamics of mechanical systems separating regions filled with chaotic behavior. Cases where such surfaces can be found are rare enough. Perhaps the most famous of these is the so-called Hess case in the mechanics of a heavy rigid body with a fixed point.We consider here the motion of a non-autonomous mechanical pendulum-like system with one degree of freedom. The conditions of existence for invariant surfaces of such a system corresponding to non-split separatrices are investigated. In the case where an invariant surface exists, combination of regular and chaotic behavior is studied analytically via the Poincaré-Mel’nikov separatrix splitting method, and numerically using the Poincaré maps.

中文翻译：

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非自治摆型系统的渐近不变曲面

不变的表面在机械系统的动力学中起着至关重要的作用，该系统将充满混沌行为的区域分开。可以发现这种表面的情况很少见。其中最著名的也许是具有固定点的重型刚体的力学中的所谓的赫斯（Hess）情况。在这里，我们考虑具有一个自由度的非自治机械摆式系统的运动。研究了与非分裂分离相对应的这种系统的不变表面的存在条件。在存在不变曲面的情况下，通过Poincaré-Mel'nikov分离线分割方法并使用Poincaré映射进行数值研究，分析规则行为和混沌行为的组合。