Nilpotent feed forward network dynamics

In this paper, we explore the normal form of fully inhomogeneous feed forward network dynamical systems, characterized by a nilpotent linear component. We introduce a new normal form method, termed the triangular sl 2 -style, to categorize this normal form. We aim to establish a comprehensive normal form theory, extending to quadratic terms across all dimensions. To accomplish this, we leverage mathematical tools including Hermite reciprocity, transvectants, and insights derived from the well-known 3-dimensional normal form related to Sylvester’s work on generating functions for quadratic covariants. Furthermore, we formulate the orbital normal form in a general Lie algebraic context as the result of an outer transformation and expand it into block-triangular outer normal form. This extended framework is subsequently employed in both 2D and 3D scenarios, leading to noteworthy simplifications that prepare these systems for the study of bifurcations.