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arXiv · 2408.07447

Nilpotent Feed Forward Network Dynamics

Fahimeh Mokhtari
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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.

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Fahimeh Mokhtari. 2024-08-14. Nilpotent Feed Forward Network Dynamics. https://arxiv.org/abs/2408.07447

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