The properties of long, numerically determined periodic orbits of two low-dimensional chaotic systems, the Lorenz equations and the Kuramoto-Sivashinsky system in a minimal-domain configuration, are examined. The primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite-amplitude parameter perturbations. To address this question, an inventory of thousands of orbits at least two orders of magnitude longer than the shortest admissible cycles is constructed. The expectation of period averages, Floquet exponents, and sensitivities over such set is then obtained. It is shown that all these quantities converge to a limiting value as the orbit period is increased. However, while period averages and Floquet exponents appear to converge to analogous quantities computed from chaotic trajectories, the limiting value of the sensitivity is not necessarily consistent with the response of the chaotic state, similar to observations made with other shadowing algorithms.

中文翻译：

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混沌系统长周期轨道的灵敏度

检查了两个低维混沌系统的长数值确定周期轨道的性质，即最小域配置的Lorenz方程和Kuramoto-Sivashinsky系统。首要问题是要确定周期平均量对在长轨道上计算出的参数扰动的敏感度是否可以用作混沌状态对有限振幅参数扰动响应的足够好的代理。为了解决这个问题，构建了数千个轨道的清单，该清单至少比最短允许周期长两个数量级。然后，可以得出周期平均值，Floquet指数和对此类集合的敏感度的期望值。可以看出，随着轨道周期的增加，所有这些量都收敛到极限值。然而，