We consider a nonholonomic system that describes the rolling without slipping of a spherical shell inside which a frame rotates with constant angular velocity (this system is one of the possible generalizations of the problem of the rolling of a Chaplygin sphere). After a suitable scale transformation of the radius of the shell or the mass of the system the equations of motion can be represented as a perturbation of the integrable Euler case in rigid body dynamics. Using this representation, we explicitly calculate a Melnikov integral, which contains an isolated zero under some restrictions on the system parameters. Thereby we prove the absence of an additional integral in this system and the existence of chaotic trajectories. We conclude by presenting numerical experiments that illustrate the system dynamics depending on the behavior of the Melnikov function.

我们考虑一种非完整系统，该系统描述了球壳不打滑且框架以恒定角速度旋转的情况下的滚动（该系统是Chaplygin球滚动问题的可能概括之一）。在对壳的半径或系统的质量进行适当的比例转换之后，运动方程可以表示为刚体动力学中可积分Euler情况的扰动。使用此表示，我们显式计算梅尔尼科夫积分，其中包含一些受系统参数限制的孤立零。因此，我们证明了该系统中不存在附加积分，并且存在混沌轨迹。