# Contraction Theory for Dynamical Systems

### Francesco Bullo

Department of Mechanical Engineering
Center for Control, Dynamical-Systems, and Computation
University of California, Santa Barbara
bullo at ucsb.edu

Edition 1.1, Mar 1, 2023
229 pages and 75 exercises
Kindle Direct Publishing
ISBN 979-8836646806

## Abstract

These lecture notes provide a mathematical introduction to contraction theory for dynamical systems. Special emphasis is given to continuous-time differential equations arising in the study of network multi-agent systems, monotone dynamics, and semi-contracting systems. This document is version 1.1 on March 1, 2023.

These notes cover (i) the manifold properties of the induced norms and logarithmic norms of matrices, (ii) contracting dynamics over finite-dimensional vector spaces endowed with Euclidean and non-Euclidean norms, (iii) weakly-contracting dynamics and monotone dynamics, and (iv) semicontracting, perpendicularly and partially contracting systems. Numerous examples are presented in some detail, including Hopfield neural networks, systems in Lure’ form, interconnected contracting systems, gradient and primal dual flows of convex functions, Lotka-Volterra population dynamics, Daganzo traffic models, averaging flows, and diffusively-coupled synchronizing systems.

• printed paperback version [Amazon link], publishedby Kindle Direct Publishing (7inx10in, gray-scale, x + 210 pages), ISBN 979-8836646806

• tablet version [PDF file], optimized for viewing on tablets (3x4 aspect ratio, color), freely downloadable from the book website

• slide version [PDF file], that is, reformatted version suited for displaying on a classroom projector, freely downloadable from the book website

Citation information
@Book{FB-CTDS,
author =    {F. Bullo},
title =     {Contraction Theory for Dynamical Systems},
year =      2023,
edition =   {{1.1}},
publisher = {Kindle Direct Publishing},
ISBN =      {979-8836646806},
url =       {https://fbullo.github.io/ctds},
}


Minicourse on Contraction Theory: introductory slides (PDF) and youtube lectures (10h in 4 lectures, with chapters)

This book is intended for personal non-commercial use only: you may not use this material for commercial purposes and you may not copy and redistribute this material in any medium or format.

## Why I decided to self-publish a print-on-demand book

There are several reasons why I decided to self-publish this book via the print-on-demand service by Kindle Direct Publishing (former Amazon CreateSpace). I appreciate the ability to:

• retain the full copyrights of all the material,

• make the document freely available online in multiple versions (see above),

• retain direct anytime control over the paperback publication costs, retain a majority of the royalties after low fix costs, and set a low price,

• update the book revision at any time (simply re-upload the PDF file, minimal turn-around time, no need for any permission, and never out-of-print), and

• make available a high-quality paperback book through a broad distribution network.

As a combination of this flexibility, I typically polish and enrich the book every time I teach the course.

Similar arguments are presented in the write-up Why I Self-Publish My Mathematics Texts With Amazon by Robert Ghrist

## History

• Winter 2018: first time I taught the course

• Fall 2020: second time I taught the course and first book draft

• Spring 2022: third time and second book draft

• Jul 1, 2022: first edition online, version 1.0

• (Jul-Aug, 2022: document updated with corrected typos and minor additions)

• March 2023: version 1.1

### Feedback

I am thankful for any feedback information, including suggestions, evaluations, error descriptions, or comments about teaching or research uses. Please email bullo at ucsb.edu