A Commentary on the Linearity and Time-Invariance of ODE-Based Systems

TL;DR

This paper clarifies the conditions under which ordinary differential equations (ODEs) represent linear time-invariant (LTI) systems, addressing ambiguities in definitions and providing educational insights for understanding LTI properties.

Contribution

It offers a detailed discussion on the subtle distinctions and criteria for identifying LTI systems from ODE descriptions, which is rarely explored in depth.

Findings

Clarifies the definitions of linearity and time-invariance in the context of ODEs

Provides criteria to determine if an ODE describes an LTI system

Highlights common misconceptions and ambiguities in the literature

Abstract

Linear time-invariant (LTI) systems appear frequently in natural sciences and engineering contexts. Many LTI systems are described by ordinary differential equations (ODEs). For example, biological gene regulation, analog filter circuits, and simple mechanical, electrical, and hydraulic systems can all be described with varying approximations as LTI systems using ODEs. While linearity and time-invariance are straightforward to demonstrate for closed-form system definitions, determining whether an ODE describes a system with LTI properties is less obvious and rarely discussed in depth in the literature. Complications arise due to slightly different definitions of linearity in different contexts. This commentary is intended to provide clarity on this subtle point, and act as an instructional aid or educational supplement.