Beyond Carleman Linearization of Nonlinear Dynamical System: Insights from a Case Study

TL;DR

This paper explores advanced linearization techniques for nonlinear dynamical systems, focusing on Carleman and Carleman-Fourier methods, supported by a detailed case study on finite-section approximation of a simple trigonometric polynomial system.

Contribution

It introduces a comprehensive linearization framework emphasizing Carleman methods and provides a case study on finite-section approximation for specific nonlinear systems.

Findings

Enhanced understanding of Carleman linearization techniques.

Effective finite-section approximation for a trigonometric polynomial system.

Insights into the limitations and potential of linearization methods.

Abstract

Nonlinear dynamical systems are widely encountered in various scientific and engineering fields. Despite significant advances in theoretical understanding, developing complete and integrated frameworks for analyzing and designing these systems remains challenging, which underscores the importance of efficient linearization methods. In this paper, we introduce a general linearization framework with emphasis on Carleman linearization and Carleman-Fourier linearization. A detailed case study on finite-section approximation to the lifted infinite-dimensional dynamical system is provided for the dynamical system with its governing function being a trigonometric polynomial of degree one.