用安全函数控制二维混沌瞬态
Controlling two-dimensional chaotic transients with the safety function
摘要：
在这项工作中，我们处理的Hénon和Lozi映射的参数选择，他们表现出瞬态混沌。接近混沌鞍的轨道会混沌一段时间，最终逃逸到外部吸引子。传统上，为了防止这种逃逸，已经应用了部分控制技术。这种方法突出考虑干扰（噪声）影响的地图，并找到一个特殊的区域的相空间，称为安全集，其中所需的控制，以维持轨道是小的。然而，在这项工作中，我们将采用一种新的方法的部分控制方法，最近开发的。这种新方法是基于找到一个称为安全功能的特殊函数，它允许自动找到避免轨道逃逸所需的最小控制。此外，我们将展示安全函数和经典安全集之间的强联系。为了说明这一点，我们将首次计算二维Hénon和Lozi映射的安全函数，其中我们还显示了该函数与影响映射的扰动大小的强烈依赖性，以及这种变化如何极大地影响受控轨道。
英文摘要：
In this work we deal with the Hénon and the Lozi map for a choice of parameters where they show transient chaos. Orbits close to the chaotic saddle behave chaotically for a while to eventually escape to an external attractor. Traditionally, to prevent such an escape, the partial control technique has been applied. This method stands out for considering disturbances (noise) affecting the map and for finding a special region of the phase space, called the safe set, where the control required to sustain the orbits is small. However, in this work we will apply a new approach of the partial control method that has been recently developed. This new approach is based on finding a special function called the safety function, which allows to automatically find the minimum control necessary to avoid the escape of the orbits. Furthermore, we will show the strong connection between the safety function and the classical safe set. To illustrate that, we will compute for the first time, safety functions for the two-dimensional Hénon and Lozi maps, where we also show the strong dependence of this function with the magnitude of disturbances affecting the map, and how this change drastically impacts the controlled orbits.
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