The numerical method of Kahan applied to quadratic differential equations is known to often generate integrable maps in low dimensions and can in more general situations exhibit preserved measures and integrals. Computerized methods based on discrete Darboux polynomials have recently been used for finding these measures and integrals. However, if the differential system contains many parameters, this approach can lead to highly complex results that can be difficult to interpret and analyze. But this complexity can in some cases be substantially reduced by using aromatic series. These are a mathematical tool introduced independently by Chartier and Murua and by Iserles, Quispel and Tse. We develop an algorithm for this purpose and derive some necessary conditions for the Kahan map to have preserved measures and integrals expressible in terms of aromatic functions. An important reason for the success of this method lies in the equivariance of the map from vector fields to their aromatic funtions. We demonstrate the algorithm on a number of examples showing a great reduction in complexity compared to what had been obtained by a fixed basis such as monomials.

翻译：Kahan数值方法应用于二次微分方程时，已知在低维情形下常能生成可积映射，且在更一般情形中可能呈现守恒测度与积分。基于离散达布多项式的计算机方法近期被用于寻找这些测度与积分。然而，当微分系统包含众多参数时，该方法可能导致高度复杂的、难以解释与分析的结果。但在某些情形下，通过使用芳香级数可显著降低这种复杂性。这类数学工具由Chartier与Murua及Iserles、Quispel与Tse分别独立提出。我们为此开发了一种算法，推导了Kahan映射具有可用芳香函数表达的守恒测度与积分所满足的若干必要条件。该方法成功的重要基础在于向量场到其芳香函数的映射具有等变性。我们通过多个算例展示了该算法，相较于基于单项式等固定基函数的方法，其复杂度显著降低。