Abstract Symmetries are ubiquitous in a wide range of nonlinear systems. Particularly in systems whose dynamics is determined by a Lagrangian or Hamiltonian function. For hybrid systems which possess a continuous-time dynamics determined by a Lagrangian function, with a cyclic variable, the degrees of freedom for the corresponding hybrid Lagrangian system can be reduced by means of a method known as hybrid Routhian reduction. In this paper we study sufficient conditions for the existence of periodic orbits in hybrid Routhian systems which also exhibit a time-reversal symmetry. Likewise, we explore some stability aspects of such orbits through the characterization of the eigenvalues for the corresponding linearized Poincare map. Finally, we apply the results to find periodic solutions in underactuated hybrid Routhian control systems.

中文翻译：

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简单混合 Routhian 系统中的对称性和周期轨道

摘要 对称性在广泛的非线性系统中无处不在。特别是在动力学由拉格朗日函数或哈密顿函数确定的系统中。对于具有由拉格朗日函数确定的连续时间动力学和循环变量的混合系统，相应混合拉格朗日系统的自由度可以通过称为混合劳斯约简的方法来降低。在本文中，我们研究了混合 Routhian 系统中存在周期轨道的充分条件，该系统也表现出时间反转对称性。同样，我们通过表征相应线性化庞加莱映射的特征值来探索此类轨道的一些稳定性方面。最后，我们将结果应用到欠驱动混合 Routhian 控制系统中寻找周期解。