Using both analytical and numerical methods on the global dynamics, including the existence and uniqueness of solutions, subharmonic bifurcations and dynamic responses, of an elliptically excited pendulum model are investigated in this paper. The heteroclinic orbits, as well as periodic orbits with [Formula: see text] and [Formula: see text] types of unperturbed systems are obtained analytically. Chaotic vibrations arising from heteroclinic intersections are studied by means of the Melnikov method. The critical curves separating the chaotic and nonchaotic regions are plotted for different system parameters. The chaotic feature on the system parameter [Formula: see text], named the ratio between the horizontal and the vertical diameter of the upright ellipse traced out by the pivot during each period, is discussed in detail. The conditions for subharmonic bifurcations with the [Formula: see text] type or the [Formula: see text] type are also presented with the subharmonic Melnikov method. It is proved rigorously that the system can undergo chaotic motions through finite subharmonic bifurcations with the [Formula: see text] type. In addition, chaotic motions can occur through infinite subharmonic bifurcations with the [Formula: see text] type. An interesting dynamical phenomenon, i.e. “controllable frequency”, which decreases monotonically with the system parameter [Formula: see text], is presented. A number of related numerical simulations are given to confirm the analytical results.

中文翻译：

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椭圆激发摆模型的全局动力学

本文利用解析和数值方法对椭圆激发摆模型的全局动力学，包括解的存在性和唯一性、次谐波分岔和动力响应进行了研究。异宿轨道，以及具有[公式：见正文]和[公式：见正文]类型的未扰动系统的周期轨道是解析获得的。由异宿交叉引起的混沌振动通过梅尔尼科夫方法进行了研究。针对不同的系统参数绘制了区分混沌和非混沌区域的临界曲线。详细讨论了系统参数[公式：见正文]的混沌特征，即枢轴在每个周期中绘制的直立椭圆的水平与垂直直径之比。[公式：见正文]类型或[公式：见正文]类型的次谐波分岔条件也用次谐波Melnikov方法给出。严格证明，系统可以通过[公式：见正文]类型的有限次谐波分岔进行混沌运动。此外，混沌运动可以通过[公式：见文本]类型的无限次谐波分岔发生。提出了一种有趣的动态现象，即“可控频率”，它随系统参数[公式：见正文]单调递减。给出了一些相关的数值模拟来证实分析结果。严格证明，系统可以通过[公式：见正文]类型的有限次谐波分岔进行混沌运动。此外，混沌运动可以通过[公式：见文本]类型的无限次谐波分岔发生。提出了一种有趣的动态现象，即“可控频率”，它随系统参数[公式：见正文]单调递减。给出了一些相关的数值模拟来证实分析结果。严格证明，系统可以通过[公式：见正文]类型的有限次谐波分岔进行混沌运动。此外，混沌运动可以通过[公式：见文本]类型的无限次谐波分岔发生。提出了一种有趣的动态现象，即“可控频率”，它随系统参数[公式：见正文]单调递减。给出了一些相关的数值模拟来证实分析结果。见正文]，提出。给出了一些相关的数值模拟来证实分析结果。见正文]，提出。给出了一些相关的数值模拟来证实分析结果。