非线性阻尼下光滑不连续振子极限环的全局演化

针对光滑不连续振子，提出了一种优化的广义谐波函数摄动法，得到其极限环的振幅与系统参数之间的解析关系式以及极限环的解析近似解。同时，基于微分方程定性理论，建立了该振子极限环特征量的解析计算公式。利用上述结果，可围绕极限环何时产生、如何分岔、在何处消失以及稳定性如何等问题，对具有复杂非线性阻尼项的光滑不连续振子极限环的全局演化过程展开定量分析。通过将本文所得之结果与龙格-库塔法之结果进行对比，验证了所提优化方法的可行性和可靠性，为研究强非线性振动系统解的全局演化问题，提供了新的参考方法。

Global Evolution of Limit Cycle of Smooth and Discontinuous Oscillator with Nonlinear Damping

An optimized generalized harmonic function perturbation method was proposed for the smooth discontinuity (SD for short) oscillator. Via the method, an analytical relationship between amplitude of limit cycle and the system parameters can be derived. Meanwhile, based on the qualitative theorem of ordinary differential equations, an analytical calculation formula of the limit cycle characteristic quantity for this oscillator was established. With the above results, the global evolution of limit cycle for SD oscillator with complex nonlinear damping can be quantitatively analyzed, such as when the limit cycle emerges, how it bifurcates, whether it is stable and where it disappears. In addition, the analytical approximate solution of the limit cycle was also obtained. By comparing the results obtained in this paper with those of the Runge-Kutta method, the feasibility and reliability of the proposed optimization method were verified, and a new reference method was provided for the study of the global evolution of the solutions of strongly nonlinear vibration systems.