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

Edition 1.0, Jun 1, 2022
210 pages and 50 exercises
Kindle Direct Publishing
ISBN 979-8836646806


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 in its initial version 1.0 on June 1, 2022. It is envisioned that future updates will include additional chapters.

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.

Book versions and purchase/download information

The book may be purchased or downloaded in the following versions:

Additionally, the following documents are downloadable:

Citation information
  author =    {F. Bullo},
  title =     {Contraction Theory for Dynamical Systems},
  year =      2022,
  edition =   {{1.0}},
  publisher = {Kindle Direct Publishing},
  ISBN =      {979-8836646806},
  url =       {},

Youtube lectures

(Forthcoming, likely early Fall 2022)

Copyright information

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:

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



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