Title: Hamiltonian structure and Darboux theorem for families of generalized Lotka-Volterra systems Show Chinese title

Title: 哈密顿结构和广义洛特卡-沃尔泰拉系统的达布定理

Abstract: This work is devoted to the establishment of a Poisson structure for a format of equations known as Generalized Lotka-Volterra systems. These equations, which include the classical Lotka-Volterra systems as a particular case, have been deeply studied in the literature. They have been shown to constitute a whole hierarchy of systems, the characterization of which is made in the context of simple algebra. Our main result is to show that this algebraic structure is completely translatable into the Poisson domain. Important Poisson structures features, such as the symplectic foliation and the Darboux' canonical representation, arise as result of rather simple matrix manipulations. Show Chinese abstract

Abstract: 这项工作致力于建立一种称为广义Lotka-Volterra系统的方程格式的Poisson结构。 这些方程包括经典的Lotka-Volterra系统作为特例，在文献中已被深入研究。 它们已被证明构成一个完整的系统层次结构，其特征在简单代数的背景下进行描述。 我们的主要结果是表明这种代数结构可以完全转化为Poisson域。 重要的Poisson结构特性，如辛叶层和Darboux的标准表示，作为相当简单的矩阵运算的结果出现。