Contraction Theory for Dynamical Systems

These lecture notes provide a mathematical introduction to contractiontheory for dynamical systems. Special emphasis is given to continuous-timedifferential equations arising in the study of network multi-agent systems,monotone dynamics, and semi-contracting systems. This document is version 1.3on January 1, 2026. These notes cover (i) the manifold properties of the induced norms andlogarithmic norms of matrices, (ii) contracting dynamics overfinite-dimensional vector spaces endowed with Euclidean and non-Euclideannorms, (iii) weakly-contracting dynamics and monotone dynamics, and (iv)semicontracting, perpendicularly and partially contractingsystems. Numerous examples are presented in some detail, including Hopfieldneural networks, systems in Lure' form, interconnected contracting systems,gradient and primal dual flows of convex functions, Lotka-Volterrapopulation dynamics, Daganzo traffic models, averaging flows, anddiffusively-coupled synchronizing systems. Read more