Nilpotent Feed Forward Network Dynamics

TL;DR

This paper develops a new normal form theory for nilpotent feedforward network dynamics, introducing a triangular $rak{sl}_2$-style method and extending it to quadratic terms across dimensions, aiding bifurcation analysis.

Contribution

It introduces a novel triangular $rak{sl}_2$-style normal form method for nilpotent feedforward networks and extends the theory to quadratic terms in all dimensions.

Findings

Developed a comprehensive normal form theory for nilpotent feedforward systems.

Extended the normal form to include quadratic terms across all dimensions.

Simplified the systems for bifurcation analysis in 2D and 3D cases.

Abstract

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 $\mathfrak{sl}_2$-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 $2$D and $3$D scenarios, leading to noteworthy simplifications that prepare these systems for the study of bifurcations.