program

Monday, June 22 (CEST time)

09:00–13:00 - Gian Italo Bischi: Continuous-Time Dynamical Systems
14:00–18:00 - Marwil Davila-Fernandez: Goodwin's Growth-Cycle Model: Theory and Applications

Tuesday, June 23 (CEST time)

09:00–13:00 - Fabio Tramontana: Discrete-Time Dynamical Systems: Theory and Applications
14:00–17:00 - Guided visit to the Leaning Tower of Pisa

Wednesday, June 24 (CEST time)

09:00–13:00 - Davide Radi: Evolutionary Game Theory
14:00–15:00 - Final test

Abstracts of the contributions

Gian Italo Bischi Università di Urbino, Italy

Continuous-time dynamical systems

The lecture introduces the qualitative theory of continuous-time dynamical systems, focusing on ordinary differential equations analyzed through phase portraits and stability properties of invariant sets, including equilibrium points, limit cycles, and chaotic sets. Key concepts such as structural stability, local bifurcations, and basins of attraction in one- and two-dimensional systems are presented, together with selected examples in higher dimensions and applications to physical, ecological, and social systems.

Marwil Davila-Fernandez Università di Siena, Italy

Goodwin's Growth-Cycle Model: Theory and Applications

This module introduces Goodwin's (1967) growth-cycle model, interpreting economic growth and fluctuations as the outcome of a single endogenous dynamic process. Starting from the classical predator–prey framework linking income distribution and capital accumulation, the lecture explores how real–financial interactions affect stability properties and cyclical dynamics. Emphasis is placed on hands-on analysis, combining analytical conditions, such as existence, stability, and bifurcations, with numerical simulations in MATLAB to study parameter sensitivity, providing both theoretical insight and practical tools for macro-dynamic analysis.

Fabio Tramontana Università di Urbino, Italy

Discrete-time dynamical systems

The lecture presents the qualitative theory of discrete-time dynamical systems, illustrating how to identify and analyze the stability of equilibria and other invariant sets in one- and two-dimensional systems. Analytical, geometrical, and simulation-based methods are combined, with applications to economic, social, and financial models.

Davide Radi Università Cattolica del Sacro Cuore, Italy

Evolutionary Game Theory

This lecture focuses on nonlinear dynamics in discrete-time economic models and evolutionary games. Applications of discrete-time dynamical systems in economics are presented, including examples from environmental economics and finance. The module also introduces evolutionary game theory and discusses non-smooth models, with particular attention to non-smooth bifurcations relevant for economic and financial applications.