The $k$-Compound of a Difference-Algebraic System

TL;DR

This paper introduces the concept of the $k$-compound system for differential-algebraic systems and explores its applications in analyzing discrete-time dynamical systems governed by difference-algebraic equations.

Contribution

It extends the theory of $k$-compounds to differential-algebraic systems and demonstrates their usefulness in analyzing discrete-time difference-algebraic systems.

Findings

Introduced the $k$-compound system for differential-algebraic systems.

Applied $k$-compounds to analyze discrete-time difference-algebraic systems.

Provided new tools for studying the evolution of $k$-dimensional structures in complex systems.

Abstract

The multiplicative and additive compounds of a matrix have important applications in geometry, linear algebra, and the analysis of dynamical systems. In particular, the $k$-compounds allow to build a $k$-compound dynamical system that tracks the evolution of $k$-dimensional parallelotopes along the original dynamics. This has recently found many applications in the analysis of non-linear systems described by ODEs and difference equations. Here, we introduce the $k$-compound system corresponding to a differential-algebraic system, and describe several applications to the analysis of discrete-time dynamical systems described by difference-algebraic equations.