Based on the theory of symbolic dynamical systems, we propose a novel computation method to locate and stabilize the unstable periodic points (UPPs) in a two-dimensional dynamical system with a Smale horseshoe. This method directly implies a new framework for controlling chaos. By introducing the subset based correspondence between a planar dynamical system and a symbolic dynamical system, we locate regions sectioned by stable and unstable manifolds comprehensively and identify the specified region containing a UPP with the particular period. Then Newton’s method compensates the accurate location of the UPP with the regional information as an initial estimation. On the other hand, the external force control (EFC) is known as an effective method to stabilize the UPPs. By applying the EFC to the located UPPs, robust controlling chaos is realized. In this framework, we never use ad hoc approaches to find target UPPs in the given chaotic set. Moreover, the method can stabilize UPPs with the specified period regardless of the situation where the targeted chaotic set is attractive. As illustrative numerical experiments, we locate and stabilize UPPs and the corresponding unstable periodic orbits in a horseshoe structure of the Duffing equation. In spite of the strong instability of UPPs, the controlled orbit is robust and the control input retains being tiny in magnitude.

中文翻译：

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定位和稳定嵌入在马蹄形地图中的不稳定周期轨道

基于符号动力系统理论，我们提出了一种新的计算方法来定位和稳定具有 Smale 马蹄铁的二维动力系统中的不稳定周期点 (UPP)。这种方法直接暗示了一种控制混沌的新框架。通过引入平面动力系统和符号动力系统之间基于子集的对应关系，我们全面定位了由稳定和不稳定流形划分的区域，并识别出包含具有特定周期的UPP的指定区域。然后牛顿法以区域信息作为初始估计补偿UPP的准确位置。另一方面，外力控制（EFC）被认为是稳定UPP的有效方法。通过将 EFC 应用于定位的 UPP，实现了鲁棒控制混沌。特设在给定的混沌集中找到目标 UPP 的方法。此外，无论目标混沌集有吸引力的情况如何，该方法都可以在指定周期内稳定 UPP。作为说明性数值实验，我们在 Duffing 方程的马蹄形结构中定位并稳定 UPP 和相应的不稳定周期轨道。尽管 UPP 具有很强的不稳定性，但受控轨道是稳健的，并且控制输入的量级仍然很小。