This paper addresses the issues on the dynamics of nonlinear damping gyros subjected to a quintic nonlinear parametric excitation. The fixed points and their stability are analyzed for the autonomous gyros equation. The number of fixed points of the system varies from one to six. The approximate equation of gyros is considered by expanding the nonlinear restoring force and parametric excitation for the study of the dynamics of gyros. Amplitude and frequency of possible resonances are found by using the multiple scales method. Also obtained are the principal parametric resonance and orders 4 and 6 subharmonic resonances. The stability conditions for each of these resonances are also obtained. Chaotic oscillations, multistability, hysteresis, and coexisting attractors are found using the bifurcation diagrams, the Lyapunov exponents, the phase portraits, the Poincaré section and the time histories. The effects of the damping parameter, the angular spin velocity and the parametric nonlinear excitation are analyzed. Results obtained by using the approximate gyros equation are compared to the dynamics obtained with the exact equation of gyros. The analytical investigations are complemented by numerical simulations.

中文翻译：

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非线性阻尼陀螺的非线性振荡：共振、滞后和多稳态

本文解决了非线性阻尼陀螺在五次非线性参数激励下的动力学问题。对自主陀螺方程的不动点及其稳定性进行了分析。系统的固定点数量从 1 个到 6 个不等。通过扩展非线性恢复力和参数激励来考虑陀螺的近似方程，用于陀螺动力学的研究。通过使用多尺度方法找到可能的共振的幅度和频率。还获得了主参量共振和 4 次和 6 次次谐波共振。还获得了这些共振中的每一个的稳定性条件。使用分岔图、李雅普诺夫指数、相图、庞加莱部分和时间历史。分析了阻尼参数、角自旋速度和参数非线性激励的影响。将使用近似陀螺方程获得的结果与使用精确陀螺方程获得的动力学进行比较。分析研究由数值模拟补充。