一类 van der Pol-Duffing 振荡器的动态性质
Dynamic properties of a class of van der Pol-Duffing oscillators
AI总结：
本文利用定性理论和分岔理论，研究了具有五次项的 van der Pol-Duffing 振荡器的分岔存在性及其准周期解，分析了自治系统的局部与全局分岔及极限环的存在性与稳定性，并通过 KAM 定理证明了周期受迫系统周期解和准周期解的存在性。
摘要：
本文利用定性理论和分岔理论，研究了具有五次项的 van der Pol-Duffing 振荡器的分岔存在性及其准周期解。首先，我们分析了自治系统，发现它具有两种局部分岔和一种全局分岔：叉形分岔、Hopf 分岔和同宿分岔。值得注意的是，同宿轨道的消失与一个大极限环的出现是同步的。然后，通过讨论无穷远处平衡点的稳定性和轨迹的方向，结合 Poincaré-Bendixson 定理和指标理论，分析了自治系统极限环的存在性和稳定性。给出了自治系统在不同参数值下的全局相图和数值模拟。最后，通过 KAM 定理证明了周期受迫系统周期解和准周期解的存在性。
英文摘要：
In this paper, we study the existence of bifurcation of a van der Pol-Duffing oscillator with quintic terms and its quasi-periodic solutions by means of qualitative and bifurcation theories. Firstly, we analyze the autonomous system and find that it has two kinds of local bifurcations and a global bifurcation: pitchfork bifurcation, Hopf bifurcation, homoclinic bifurcation. It is worth noting that the disappearance of the homoclinic orbit is synchronized with the emergence of a large limit cycle. Then, by discussing the stability of equilibria at infinity and the orientation of the trajectory, the existence and stability of limit circles of the autonomous system are analyzed by combining the Poincaré-Bendixson theorem and the index theory. The global phase portrait and the numerical simulation of the autonomous system in different parameter values are given. Finally, the existence of periodic and quasi-periodic solutions to periodic forced system is proved by a KAM theorem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
