以最小信息传输控制混沌
Control of chaos with minimal information transfer
AI总结：
本文研究速率受限信道下双曲集的不变性与镇定，揭示最小数据速率下界由总不稳定与内在不稳定之差决定，并在极端情况下证明紧性，应用于 Hénon 马蹄示例。
摘要：
本文研究在速率受限信道下双曲集的不变性与镇定问题。我们的主要结果揭示了一种在线性化分析中无法观察到的现象：使双曲集 $Q$ 能够保持不变性的最小数据速率，其下界由两种不稳定度量之差给出：第一种描述 $Q$ 上的总不稳定性，第二种描述不导致从 $Q$ 退出的内在不稳定性。严格而言，这两个量分别是不稳定李雅普诺夫指数之和与相关纤维随机动力系统的度量熵。两者之间的间隙在动力系统中众所周知，且常与逃逸速率相关。当间隙为零时，对应 $Q$ 内部存在支撑 SRB 测度的奇怪吸引子。在这种情况下，无需向控制器传输信息，因为吸引子本身已保证不变性。我们证明我们的下界在两种极端情况下是紧的：即上述情况以及无内在不稳定的情况。此外，我们将我们的技术应用于双曲集的局部一致镇定问题，并讨论了一个基于 Hénon 马蹄的示例。
英文摘要：
This paper studies set-invariance and stabilization of hyperbolic sets over rate-limited channels. Our main results reveal a phenomenon which cannot be seen from a linearized analysis: the smallest data rate above which a hyperbolic set $Q$ can be made invariant is bounded below by the difference between two measures of instability: the first one describing the total instability on $Q$, and the second one describing the intrinsic instability which does not lead to exit from $Q$. In rigorous terms, these two quantities are the sum of unstable Lyapunov exponents and the metric entropy of an associated bundle random dynamical system, respectively. The gap between the two is well-known in dynamical systems and is often related to escape rates. A vanishing gap corresponds to the existence of a strange attractor inside $Q$ supporting an SRB measure. In this case, no information transfer to the controller is necessary, because the attractor already guarantees invariance. We prove that our lower bound is tight in two extreme cases, the one just described and the one without intrinsic instability. Furthermore, we apply our techniques to the problem of local uniform stabilization to a hyperbolic set and discuss an example built on the Hénon horseshoe.
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