Differential analysis of nonlinear systems: revisiting the pendulum example

TL;DR

This paper explores differential analysis techniques applied to the nonlinear pendulum, highlighting recent advances like differential Lyapunov functions and differential positivity to infer global behaviors from local linearizations.

Contribution

It revisits the pendulum example to demonstrate the application of differential analysis methods, including new frameworks for contraction and positivity analysis.

Findings

Differential Lyapunov functions help assess stability of nonlinear systems.

Differential positivity provides insights into system behavior and convergence.

The approach offers a systematic way to infer global properties from local analysis.

Abstract

Differential analysis aims at inferring global properties of nonlinear behaviors from the local analysis of the linearized dynamics. The paper motivates and illustrates the use of differential analysis on the nonlinear pendulum model, an archetype example of nonlinear behavior. Special emphasis is put on recent work by the authors in this area, which includes a differential Lyapunov framework for contraction analysis and the concept of differential positivity.