Nonlinear network systems, November 2021- March 2022

Teachers: Giacomo Como, Fabio Fagnani, Mark Jeeninga

The aim of this course is to introduce the PhD students to the theory of nonlinear dynamical systems, with particular focus on techniques for the analysis of (large-scale) network systems. A number of timely applications in the natural, engineering, and socio-economic sciences will also be presented and studied.

The course will be structured in 15 lectures (30 hours total) to be held weekly at DISMA. The first lecture is scheduled on Wednesday, November 10 10:30-12:30 in Aula Buzano, Politecnico di Torino. 

Tentative Syllabus: 

• Introduction to Linear and Nonlinear Dynamical Systems
• Contraction principle and applications, well posedness results for ODEs. Maximal solutions, a-priori-bounds, Gromwall's lemma
•  Autonomous systems, flows and semi-flows. Equilibria and periodic solutions. First integrals, invariant regions. Poincare Bendixson theorem. Asymptotics of linear systems.
• Lyapunov Stability Theory, Invariance Principle, Linearization
• Topological Equivalence, Hartman-Grobman Theorem, Structural Stability 
• Center Manifold Theory and Applications
• Bifurcations Theory and Classification 
• Invariant Measures of Dynamical Systems, Ergodic Theorem, Poincare Recurrence Theorem 
• Monotone and Positive Systems, Contractive Systems  
• Epidemic Systems on Networks
• Dynamical Flow Networks  
• DC Electrical Networks and the Power Flow Problem 
• Kuramoto Oscillators and AC Power Networks 
• Opinion Dynamics in Social Networks