We consider the linear and quadratic higher-order terms associated with the response of the statistical properties of a dynamical system to suitable small perturbations. These terms are related to the first and second derivative of the stationary measure with respect to the changes in the system itself, expressing how the statistical properties of the system vary under the perturbation. We show a general framework in which one can obtain rigorous convergence and formulas for these two terms. The framework is flexible enough to be applied both to deterministic and random systems. We give examples of such an application computing linear and quadratic response for Arnold maps with additive noise and deterministic expanding maps.

中文翻译：

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随机和确定性动力系统的二次响应。

我们考虑与动力学系统的统计特性对适当的小扰动的响应相关的线性和二次高阶项。这些术语与固定度量相对于系统本身的变化的一阶和二阶导数有关，表示系统的统计特性在扰动下如何变化。我们展示了一个通用的框架，在该框架中人们可以获得这两个术语的严格收敛和公式。该框架足够灵活，可以应用于确定性系统和随机系统。我们给出了这样的应用示例，该应用可计算具有加性噪声和确定性扩展图的Arnold映射的线性和二次响应。