This work aims to study the influence of the rotation of the galaxy which it is modelled as a bi-symmetrical potential consists of a two-dimensional harmonic oscillator with quartic perturbing terms on some dynamics aspects for the problem of the motion of stars. We prove analytically the non-integrability of the motion (i.e., the motion is chaotic) when the parameters meet certain conditions. Poincaré surface of section is introduced as a numerical method that is employed to confirm the obtained analytical results. We present the equilibrium points and examine their stability. We also clarify the force resulting from the rotating frame serves as a stabilizer for the maximum equilibrium points. We illustrate graphically the size of stability zones depends on the value of the angular velocity for the frame. Based on the Lyapunov theorem, the periodic solutions are constructed near the equilibrium point. Additionally, we prove the existence of one or two families of periodic solutions according to the equilibrium point is either saddle or stable, respectively. The permitted zones of possible motion are delimited and they are graphically explained for different values of the parameters.

中文翻译：

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旋转参考系中四次星系势的不可积性、稳定性和周期解

这项工作旨在研究星系自转的影响，它被建模为双对称势，由二维谐振子组成，具有四次扰动项，对恒星运动问题的某些动力学方面的影响。我们分析证明了当参数满足一定条件时运动的不可积性（即运动是混沌的）。截面的庞加莱面作为一种数值方法被引入，用于确认所获得的分析结果。我们提出平衡点并检查它们的稳定性。我们还阐明了旋转框架产生的力作为最大平衡点的稳定器。我们以图形方式说明稳定区的大小取决于框架的角速度值。基于李雅普诺夫定理，周期解是在平衡点附近构建的。此外，我们证明了根据平衡点的一族或两族周期解的存在分别是鞍形或稳定的。可能运动的允许区域被划定，并针对不同的参数值以图形方式解释了这些区域。