Sep 17 2012 - 00:00
Sep 21 2012 - 23:59

Centro di Ricerca Matematica "Ennio De Giorgi", Pisa
Conservatorio di Santa Chiara, San Miniato (Pisa), Italy

Short description of the event: 

The “Summer School of Mathematics for Economics and Social Sciences” aims to improve the knowledge of mathematical methods among graduate students in economics and social sciences, with a focus on those techniques which albeit widespread in use are not properly covered in typical graduate programmes. The School is an interdisciplinary venue intended to foster the interaction of people coming from the too often separated communities of mathematical and social scientists. It is organized by the Mathematics Research Center “Ennio De Giorgi” and supported by the PhD programme in Economics of the Scuola Superiore Sant'Anna.

Topics: Structural stability; Codimension one bifurcations of vector fields; Codimension one bifurcations of maps; Introduction to codimension two bifurcations

Lecturer: Florian Wagner, CeNDEF, University of Amsterdam

Participation subject to selection. Only 20-25 positions available. For applications link to or write to

Deadline for application: 5 August, 2012

Topics considered prerequisite for the course: complex numbers, eigenvectors, eigenvalues and diagonalisation, Taylor’s theorem, implicit function theorem, linear ODEs and one-dimensional autonomous nonlinear ODEs.

Scientific Committee: Giulio Bottazzi, Giorgio Fagiolo (Scuola Superiore Sant'Anna), Davide Fiaschi (Università di Pisa), Stefano Marmi (Scuola Normale Superiore)

Syllabus of the course

- Review of linear theory

Structural stability
- Equivalence classes of linear dynamics
- Differentiable and topological equivalence
- Hartman-Grobman theorem
- Structural stability
- General notion of bifurcation

Codimension one bifurcations of vector fields
- Saddle-node bifurcation
- Normal forms
- Hopf bifurcation
- Invariant manifolds
- Homoclinic and heteroclinic bifurcations
- Symmetry
- Pitchfork bifurcation

Codimension one bifurcations of maps
- Poincaré section
- Saddle-node bifurcation
- Neimark-Sacker bifurcation
- Resonances
- Homoclinic tangencies
- Horseshoes

Introduction to Codimension two bifurcations