Possible participants: Young researchers, PhD students.

Prerequisites. Basic linear algebra (vector spaces, matrices and linear operators, eigenvalues and eigenvectors, complex numbers). Basic notions of calculus with one and several variables.

Suggested readings before attending the Tutorial Workshop. An undergraduate textbook of general maths (e.g., Simon & Blume) and an introductory textbook on Dynamical Systems (e.g., Devaney).

Place: Department of Economics, Palazzo Battiferri, via Saffi n. 42, Urbino (Italy)

Room: Aula Blu

**9:00 -10:45 Gian Italo Bischi**

Introduction to one-dimensional smooth discrete dynamical systems: fixed points, cycles, stability, local bifurcations, basins and their bifurcations**Break**

**11:00 -12:45.Laura Gardini.**

Global properties of one-dimensional smooth discrete dynamical systems. Noninvertible maps, homoclinic bifurcations and chaotic sets, trapping intervals and critical points.

**13: Lunch**

**15:00 -16:45. Iryna Sushko.**

One-dimensional continuous piecewise-smooth dynamical systems. Border-collision bifurcations.

**Break**

**17:00 -18:00. Fabio Lamantia.**

Examples and applications of one-dimensional discrete dynamical systems in economic, social and ecological systems.

**18:00 -19:00. Discussion with the Participants.**

**9:00-10:45. Gian Italo Bischi.**

Two-dimensional discrete dynamical systems, local bifurcations, critical curves, absorbing areas, chaotic areas, basins of attraction and related bifurcations due to contacts with critical curves.

**Break**

**11:00-12:45. Laura Gardini.**

Homoclinic bifurcations and global bifurcations of attractors and basins in two-dimensional noninvertible maps.

**13: Lunch**

**15:00-16:45. Anna Agliari.**

Closed invariant curves and their bifurcations in two-dimensional maps.**Break**

**17:00-18:00. Iryna Sushko.**

Two-dimensional piecewise-smooth maps.

**18:00-19:00. Discussion with the Participants.**

**9:00-10:45. Laura Gardini and Fabio Tramontana.**

One-dimensional discontinuous dynamical systems.

**Break**

**11-12:45 Roberto Dieci.**

Applications to Economics: modeling interacting markets with higher-dimensional discrete dynamical systems.

**13: Lunch**

**15:00-16:00. Gian Italo Bischi and Anna Agliari.**

Global bifurcations in applied models.

**16-16.45 Roberto Dieci.**

Applications to financial markets: higher-dimensional dynamical systems in heterogeneous-agent asset pricing models.

**Break**

**17:00-18:00. Fabio Tramontana.**

Examples and applications of discrete dynamical systems in economic, social and ecological systems.

**18:00-19:00. Discussion with the Participants.**

**9:00-10:00. Fabio Tramontana and Fabio Lamantia.**

Numerical tools.

**10:00-12:45. Participants' space: discussion, open problems, examples.**

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